The identification of unknown parameters from noisy observations arises in various areas of application, e.g., engineering systems, biological models, environmental systems. In recent years, Bayesian inference has become a popular approach to model inverse problems [39], i.e., noisy observations are used to update the knowledge of unknown parameters from a prior distribution to the posterior distribution. The latter is then the solution of the Bayesian inverse problem and obtained by conditioning the prior distribution on the data. This approach is very appealing in various fields of applications, since uncertainty quantification can be performed, once the prior distribution is updated—barring the fact that Bayesian credible sets are not in a one-to-one correspondence to classical confidence sets, see [7, 40].

To ensure the applicability of the Bayesian approach to computationally demanding models, there has been a lot of research effort towards improved algorithms allowing for effective sampling or integration w.r.t. the resulting posterior measure. For example, the computational burden of expensive forward or likelihood models can be reduced by surrogates or multilevel strategies [14, 20, 27, 34] and for many classical sampling or integration methods such as Quasi-Monte Carlo [12], Markov chain Monte Carlo [6, 32, 42], and numerical quadrature [5, 35] we now know modifications and conditions which ensure a dimension-independent efficiency.

However, a completely different, but very common challenge for many numerical methods has drawn surprisingly less attention so far: the challenge of concentrated posterior measures such as

$$\begin{aligned} \mu _n(\mathrm {d}x) = \frac{1}{Z_n} \exp \left( -n \varPhi _n(x)\right) \mu _0(\mathrm {d}x), \quad Z_n :=\int _{\mathbb {R}^d} \exp \left( -n \varPhi _n(x)\right) \mu _0(\mathrm {d}x), \end{aligned}$$

Here, \(n\gg 1\) and \(\mu _0\) denotes a reference or prior probability measure on \(\mathbb {R}^d\) and \(\varPhi _n:\mathbb {R}^d\rightarrow [0,\infty )\) are negative log-likelihood functions resulting, e.g., from n observations.

From a modeling point of view the concentration effect of the posterior is a highly desirable situation due to large data sets and less remaining uncertainty about the parameter to be inferred. From a numerical point of view, on the other hand, this can pose a delicate situation, since standard integration methods may perform worse and worse if the concentration increases due to \(n\rightarrow \infty \). Hence, understanding how sampling or quadrature methods for \(\mu _n\) behave as \(n\rightarrow \infty \) is a crucial task with immediate benefits for purposes of uncertainty quantification. Since small noise yields “small” uncertainty, one might be tempted to consider only optimization-based approaches in order to compute a point estimator (i.e., the maximum a-posteriori estimator) for the unknown parameter which is usually computationally much cheaper than a complete Bayesian inference. However, for quantifying the remaining risk, e.g., computing the posterior failure probability for some quantity of interest, we still require efficient integration methods for concentrated posteriors as \(\mu _n\). Nonetheless, we will use well-known preconditioning techniques from numerical optimization in order to derive such robust integration methods for the small noise setting.

Numerical methods are often based on the prior \(\mu _0\), since \(\mu _0\) is usually a simple measure allowing for direct sampling or explicit quadrature formulas. However, for large n most of the corresponding sample points or quadrature nodes will be placed in regions of low posterior importance missing the needle in the haystack—the minimizers of \(\varPhi _n\). An obvious way to circumvent this is to use a numerical integration w.r.t. another reference measure which can be straightforwardly computed or sampled from and concentrates around those minimizers and shrinks like the posterior measures \(\mu _n\) as \(n\rightarrow \infty \). In this paper we consider numerical methods based on a Gaussian approximation of \(\mu _n\)—the Laplace approximation.

When it comes to integration w.r.t. an increasingly concentrated function, the well-known and widely used Laplace’s method provides explicit asymptotics for such integrals, i.e., under certain regularity conditions [44] we have for \(n\rightarrow \infty \) that

$$\begin{aligned} \int _{\mathbb {R}^d} f(x) \ \exp (-n \varPhi (x)) \mathrm {d}x \ = \ f(x_\star ) \frac{(2\pi )^{d/2}\, \exp (-n \varPhi (x_\star ))}{n^{d/2} \sqrt{\det \left( \nabla ^2 \varPhi (x_\star ) \right) }}\ \left( 1+\mathcal O(n^{-1})\right) \end{aligned}$$

where \(x_\star \in \mathbb {R}\) denotes the assumed unique minimizer of \(\varPhi :\mathbb {R}^d\rightarrow \mathbb {R}\). This formula is derived by approximating \(\varPhi \) by its second-order Taylor polynomial at \(x_\star \). We could now use (2) and its application to \(Z_n\) in order to derive that \(\int _{\mathbb {R}^d} f(x) \ \mu _n(\mathrm {d}x) \rightarrow f(x_\star ) \) as \(n\rightarrow \infty \). However, for finite n this is only of limited use, e.g., consider the computation of posterior probabilities where f is an indicator function. Thus, in practice we still rely on numerical integration methods in order to obtain a reasonable approximation of the posterior integrals \(\int _{\mathbb {R}^d} f(x) \ \mu _n(\mathrm {d}x)\). Nonetheless, the second-order Taylor approximation employed in Laplace’s method provides us with (a guideline to derive) a Gaussian measure approximating \(\mu _n\).

This measure itself is often called the Laplace approximation of \(\mu _n\) and will be denoted by \(\mathcal L_{\mu _n}\). Its mean is given by the maximum a-posteriori estimate (MAP) of the posterior \(\mu _n\) and its covariance is the inverse Hessian of the negative log posterior density. Both quantities can be computed efficiently by numerical optimization and since it is a Gaussian measure it allows for direct samplings and easy quadrature formulas. The Laplace approximation is widely used in optimal (Bayesian) experimental design to approximate the posterior distribution (see, for example, [1]) and has been demonstrated to be particularly useful in the large data setting, see [25, 33] and the references therein for more details. Moreover, in several recent publications the Laplace approximation was already proposed as a suitable reference measure for numerical quadrature [5, 38] or importance sampling [2]. Note that preconditioning strategies based on Laplace approximation are also referred to as Hessian-based strategies due to the equivalence of the inverse covariance and the Hessian of the corresponding optimization problem, cp. [5]. In [38], the authors showed that a Laplace approximation-based adaptive Smolyak quadrature for Bayesian inference with affine parametric operator equations exhibits a convergence rate independent of the size of the noise, i.e., independent of n.

This paper extends the analysis in [38] for quadrature to the widely applied Laplace-based importance sampling and Laplace-based quasi-Monte Carlo (QMC) integration.

Before we investigate the scale invariance or robustness of these methods we examine the behaviour of the Laplace approximation and in particular, the density \(\frac{\mathrm {d}\mu _n}{\mathrm {d}\mathcal L_{\mu _n}}\). The reason behind is that, for importance sampling as well as QMC integration, this density naturally appears in the methods, hence, if it deteriorates as \(n\rightarrow \infty \), this will be reflected in a deteriorating efficiency of the method. For example, for \(\varPhi _n \equiv \varPhi \) the density w.r.t. the prior measure \(\frac{\mathrm {d}\mu _n}{\mathrm {d}\mu _0} = \exp (-n\varPhi )/Z_n\) deteriorates to a Dirac function at the minimizer \(x_\star \) of \(\varPhi \) as \(n\rightarrow \infty \) which causes the shortcomings of Monte Carlo or QMC integration w.r.t. \(\mu _0\) as \(n\rightarrow \infty \). However, for the Laplace approximation we show that the density \(\frac{\mathrm {d}\mu _n}{\mathrm {d}\mathcal L_{\mu _n}}\) converges \(\mathcal L_{\mu _n}\)-almost everywhere to 1 which in turn results in a robust—and actually improving—performance w.r.t. n of related numerical methods. In summary, the main results of this paper are the following:

  1. 1.

    Laplace Approximation: Given mild conditions the Laplace approximation \(\mathcal L_{\mu _n}\) converges in Hellinger distance to \(\mu _n\):

    $$\begin{aligned} d_\mathrm {H}(\mu _n, \mathcal L_{\mu _n}) \in \mathcal O(n^{-1/2}). \end{aligned}$$

    This result is closely related to the well-known Bernstein–von Mises theorem for the posterior consistency in Bayesian inference [41]. The significant difference here is that the covariance in the Laplace approximation depends on the data and the convergence holds for the particularly observed data whereas in the classical Bernstein–von Mises theorem the covariance is the inverse of the expected Fisher information matrix and the convergence is usually stated in probability.

  2. 2.

    Importance Sampling: We consider integration w.r.t. measures \(\mu _n\) as in (1) where \(\varPhi _n(x) = \varPhi (x) - \iota _n\) for a \(\varPhi :\mathbb {R}^d \rightarrow [0,\infty )\) and \(\iota _n \in \mathbb {R}\).

    • Prior-based Importance Sampling: We consider the case of prior-based importance sampling, i.e., the prior \(\mu _0\) is used as the importance distribution for computing the expectation of smooth integrands \(f\in L^2_{\mu _0}(\mathbb {R})\). Here, the asymptotic variance w.r.t. such measures \(\mu _n\) deteriorates like \(n^{d/2-1}\).

    • Laplace-based Importance Sampling. The (random) error \(e_{n,N}(f)\) of Laplace-based importance sampling for computing expectations of smooth integrands \(f\in L^2_{\mu _0}(\mathbb {R})\) w.r.t. such measures \(\mu _n\) using a fixed number of samples \(N\in \mathbb {N}\) decays in probability almost like \(n^{-1/2}\), i.e.,

      $$\begin{aligned} n^\delta e_{n,N}(f) \xrightarrow [n\rightarrow \infty ]{\mathbb {P}} 0, \qquad \delta < 1/2. \end{aligned}$$
  3. 3.

    Quasi-Monte Carlo: We focus for the analysis of the quasi-Monte Carlo methods on the bounded case of \(\mu _0 = \mathcal U([\frac{1}{2},\frac{1}{2}]^d)\).

    • Prior-based Quasi-Monte Carlo: The root mean squared error estimate for computing integrals of the form (2) by QMC using randomly shifted Lattice rules deteriorates like \(n^{d/4}\) as \(n\rightarrow \infty \).

    • Laplace-based Quasi-Monte Carlo: If the lattice rule is transformed by an affine mapping based on the mean and the covariance of the Laplace approximation, then the resulting root mean squared error decays like \(n^{-d/2}\) for integrals of the form (2).

The outline of the paper is as follows: in Sect. 2 we introduce the Laplace approximation for measures of the form (1) and the notation of the paper. In Sect. 2.2 we study the convergence of the Laplace approximation. We also consider the case of singular Hessians or perturbed Hessians and provide some illustrative numerical examples. At the end of the section, we shortly discuss the relation to the classical Bernstein–von Mises theorem. The main results about importance sampling and QMC using the prior measure and the Laplace approximation, respectively, are then discussed in Sect. 3. We also briefly comment on existing results for numerical quadrature and provide several numerical examples illustrating our theoretical findings. The appendix collects the rather lengthy and technical proofs of the main results.

Convergence of the Laplace approximation

We start with recalling the classical Laplace method for the asymptotics of integrals.

Theorem 1

(variant of [44, Section IX.5]) Set

$$\begin{aligned} J(n) :=\int _D f(x) \exp (-n \varPhi (x)) \mathrm {d}x, \qquad n\in \mathbb {N}, \end{aligned}$$

where \(D\subseteq \mathbb {R}^d\) is a possibly unbounded domain and let the following assumptions hold:

  1. 1.

    The integral J(n) converges absolutely for each \(n\in \mathbb {N}\).

  2. 2.

    There exists an \(x_\star \) in the interior of D such that for every \(r > 0\) there holds

    $$\begin{aligned} \delta _r :=\inf _{x \in B^c_r(x_\star )} \varPhi (x) - \varPhi (x_\star ) > 0, \end{aligned}$$

    where \(B_r(x_\star ) :=\{x \in \mathbb {R}^d:\Vert x-x_\star \Vert \le r \}\) and \(B^c_r(x_\star ) :=\mathbb {R}^d {\setminus } B_r(x_\star )\).

  3. 3.

    In a neighborhood of \(x_\star \) the function \(f:D\rightarrow \mathbb R\) is \((2p+2)\) times continuously differentiable and \(\varPhi :\mathbb {R}^d \rightarrow \mathbb {R}\) is \((2p+3)\) times continuously differentiable for a \(p\ge 0\), i.e., and the Hessian \(H_\star :=\nabla ^2 \varPhi (x_\star )\) is positive definite.

Then, as \(n\rightarrow \infty \), we have

$$\begin{aligned} J(n) = \mathrm e^{-n \varPhi (x_\star )}\, n^{-d/2}\, \left( \sum _{k=0}^p c_k(f) n^{- k} + \mathcal O\left( n^{-p-1}\right) \right) \end{aligned}$$

where \(c_k(f) \in \mathbb {R}\) and, particularly, \(c_0(f) = \sqrt{\det (2\pi H^{-1}_\star )}\, f(x_\star )\).

Remark 1

As stated in [44, Section IX.5] the asymptotic

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{J(n)}{c_0(f)\, \exp (-n \varPhi (x_\star ))\, n^{-d/2} } = 1 \end{aligned}$$

with \(c_0(f)\) is as above, already holds for \(f:\mathbb {R}^d\rightarrow \mathbb {R}\) being continuous and \(\varPhi :\mathbb {R}^d\rightarrow \mathbb {R}\) being twice continuously differentiable in a neighborhood of \(x_\star \) with positive definite \(\nabla ^2 \varPhi (x_\star )\)—given that the first two assumptions of Theorem 1 are also satisfied.

Assume that \(\varPhi (x_\star ) = 0\), then the above theorem and remark imply

$$\begin{aligned} \left| \int _{\mathbb {R}^d} f(x)\ \exp (-n\varPhi (x)) \ \mathrm {d}x - \int _{\mathbb {R}^d} f(x)\ \exp (- \frac{n}{2} \Vert x-x_\star \Vert ^2_{H_\star }) \ \mathrm {d}x\right| \in o(n^{-d/2}) \end{aligned}$$

for continuous and integrable \(f:\mathbb {R}^d\rightarrow \mathbb {R}\), where \(\Vert \cdot \Vert _A=\Vert A^{1/2}\cdot \Vert \) for a symmetric positive definite matrix \(A\in \mathbb {R}^{d\times d}\). This is similar to the notion weak convergence (albeit with two different non-static measures). If we additionally claim that \(f(x_\star ) > 0,\) then also

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{\int _{\mathbb {R}^d} f(x)\ \exp (-n\varPhi (x)) \ \mathrm {d}x}{ \int _{\mathbb {R}^d} f(x)\ \exp (- \frac{n}{2} \Vert x-x_\star \Vert ^2_{H_\star }) \ \mathrm {d}x} = 1 \qquad \text { as } n\rightarrow \infty , \end{aligned}$$

which is sort of a relative weak convergence. In other words, the asymptotic behaviour of \(\int f\ \mathrm e^{-n\varPhi }\ \mathrm {d}x\), in particular, its convergence to zero, is the same as of the integral of f w.r.t. an unnormalized Gaussian density with mean in \(x_\star \) and covariance \( (nH_\star )^{-1}\).

If we consider now probability measures \(\mu _n\) as in (1) but with \(\varPhi _n \equiv \varPhi \) where \(\varPhi \) satisfies the assumptions of Theorem 1, and if we suppose that \(\mu _0\) possesses a continuous Lebesgue density \(\pi _0:\mathbb {R}\rightarrow [0,\infty )\) with \(\pi _0(x_\star ) > 0\), then Theorem 1 and Remark 1 will imply for continuous and integrable \(f:\mathbb {R}^d\rightarrow \mathbb {R}\) that

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _{\mathbb {R}^d} f(x) \mu _n(\mathrm {d}x)&= \lim _{n\rightarrow \infty } \frac{\int _{\mathbb {R}^d} f(x) \pi _0(x) \exp (-n \varPhi (x))\ \mathrm {d}x}{\int _{\mathbb {R}^d} \pi _0(x) \exp (-n \varPhi (x))\ \mathrm {d}x}\\&= \lim _{n\rightarrow \infty } \frac{c_0(f\pi _0)\, n^{-d/2}}{c_0(\pi _0)\, n^{-d/2}} = \frac{c_0(f\pi _0)}{c_0(\pi _0)} = f(x_\star ). \end{aligned}$$

The same reasoning applies to the expectation of f w.r.t. a Gaussian measure \(\mathcal N(x_\star , (nH_\star )^{-1})\) with unnormalized density \(\exp (- \frac{n}{2} \Vert x-x_\star \Vert ^2_{H_\star })\). Thus, we obtain the weak convergence of \(\mu _n\) to \(\mathcal N(x_\star , (nH_\star )^{-1})\), i.e., for any continuous and bounded \(f:\mathbb {R}^d\rightarrow \mathbb {R}\) we have

$$\begin{aligned} \lim _{n\rightarrow \infty } \left| \int _{\mathbb {R}^d} f(x) \ \mu _n(\mathrm {d}x) - \int _{\mathbb {R}^d} f(x) \ \mathcal N_{x_\star , (nH_\star )^{-1}}(\mathrm {d}x)\right| = 0, \end{aligned}$$

where \(\mathcal N_{x,C}\) is short for \(\mathcal N(x,C)\). In fact, for twice continuously differentiable \(f:\mathbb {R}^d \rightarrow \mathbb {R}\) we get by means of Theorem 1 the rate

$$\begin{aligned} \left| \int _{\mathbb {R}^d} f(x) \ \mu _n(\mathrm {d}x) - \int _{\mathbb {R}^d} f(x) \ \mathcal N_{x_\star , (nH_\star )^{-1}}(\mathrm {d}x)\right| \in \mathcal O(n^{-1}). \end{aligned}$$

Note that due to normalization we do not need to assume \(\varPhi (x_\star ) = 0\) here. Hence, this weak convergence suggests to use \(\mathcal N_{x_\star , (nH_\star )^{-1}}\) as a Gaussian approximation to \(\mu _n\). In the next subsection we derive similar Gaussian approximation for the general case \(\varPhi _n \not \equiv \varPhi \), whereas Sect. 2.2 includes convergence results of the Laplace approximation in terms of the Hellinger distance.

Bayesian inference We present some context for the form of equation (1) in the following. Integrals of the form (1) arise naturally in the Bayesian setting for inverse problems with large amount of observational data or informative data. Note that the mathematical results for the Laplace approximation given in Sect. 2 are derived in a much more general setting and are not restricted to integrals w.r.t. the posterior in the Bayesian inverse framework. We refer to [8, 21] and the references therein for a detailed introduction to Bayesian inverse problems.

Consider a continuous forward response operator \({\mathcal {G}}:\mathbb {R}^d \rightarrow \mathbb R^K\) mapping the unknown parameters \(x\in \mathbb {R}^d\) to the data space \(\mathbb R^K\), where \(K\in \mathbb N\) denotes the number of observations. We investigate the inverse problem of recovering unknown parameters \(x\in \mathbb {R}^d\) from noisy observations \(y\in \mathbb {R}^K\) given by

$$\begin{aligned} y = {\mathcal {G}}(x)+\eta , \end{aligned}$$

where \(\eta \sim {\mathcal {N}}(0,\varGamma )\) is a Gaussian random variable with mean zero and covariance matrix \(\varGamma \), which models the noise in the observations and in the model.

The Bayesian approach for this inverse problem of inferring x from y (which is ill-posed without further assumptions) works as follows: For fixed \(y\in \mathbb {R}^K\) we introduce the least-squares functional (or negative loglikelihood in the language of statistics) \(\varPhi (\cdot ;y):\mathbb R^d\rightarrow \mathbb R\) by

$$\begin{aligned} \varPhi (x ; y)=\frac{1}{2}\Vert y-\mathcal G(x) \Vert _{\varGamma _{-1}}^2\,. \end{aligned}$$

with \(\Vert \cdot \Vert _{\varGamma ^{-1}}:=\Vert \varGamma ^{-\frac{1}{2}}\cdot \Vert \) denoting the weighted Euclidean norm in \(\mathbb R^K\). The unknown parameter x is modeled as a \(\mathbb {R}^d\)-valued random variable with prior distribution \(\mu _0\) (independent of the observational noise \(\eta \)), which regularizes the problem and makes it well-posed by application of Bayes’ theorem: The pair (xy) is a jointly varying random variable on \(\mathbb R^d \times \mathbb R^K\) and hence the solution to the Bayesian inverse problem is the conditional or posterior distribution \(\mu \) of x given the data y where the law \(\mu \) is given by

$$\begin{aligned} \mu (\mathrm {d}x)=\frac{1}{Z}\exp (-\varPhi (x ; y))\mu _0(\mathrm {d}x) \end{aligned}$$

with the normalization constant \(Z :=\int _{\mathbb R^d}\exp (-\varPhi (x;y))\mu _0(\mathrm {d}x)\). If we assume a decaying noise-level by introducing a scaled noise covariance \(\varGamma _n = \frac{1}{n} \varGamma \), the resulting noise model \(\eta _n \sim N(0,\varGamma _n)\) yields an n-dependent log-likelihood term which results in posterior measures \(\mu _n\) of the form (1) with \(\varPhi _n(x) = \varPhi (x ; y)\). Similarly, an increasing number \(n\in \mathbb {N}\) of data \(y_1, \ldots , y_n \in \mathbb {R}^k\) resulting from n observations of \({\mathcal {G}}(x)\) with independent noises \(\eta _1,\ldots ,\eta _n\sim N(0,\varGamma )\) yields posterior measures \(\mu _n\) as in (1) with \(\varPhi _n(x) = \frac{1}{n} \sum _{j=1}^n \varPhi (x ; y_j)\).

The Laplace approximation

Throughout the paper, we assume that the prior measure \(\mu _0\) is absolutely continuous w.r.t. Lebesgue measure with density \(\pi _0:\mathbb {R}^d\rightarrow [0,\infty )\), i.e.,

$$\begin{aligned} \mu _0(\mathrm {d}x)= \pi _0(x) \mathrm {d}x, \quad \text { and we set } \quad \mathrm {S}_0 :={\{x\in \mathbb {R}^d:\pi _0(x) > 0\} = \mathrm {supp}\,\mu _0}. \end{aligned}$$

Hence, also the measures \(\mu _n\) in (1) are absolutely continuous w.r.t. Lebesgue measure, i.e.,

$$\begin{aligned} \mu _n(\mathrm {d}x) = \frac{1}{Z_n}\ {\mathbf {1}}_{\mathrm {S}_0}(x)\ \exp \left( -n I_n(x)\right) \mathrm {d}x \end{aligned}$$

where \(I_n:\mathrm {S}_0 \rightarrow \mathbb {R}\) is given by

$$\begin{aligned} I_n(x) :=\varPhi _n(x) - \frac{1}{n}\log \pi _0(x). \end{aligned}$$

In order to define the Laplace approximation of \(\mu _n\) we need the following basic assumption.

Assumption 1

There holds \(\varPhi _n, \pi _0 \in C^2(\mathrm {S}_0,\mathbb {R})\), i.e., the mappings \(\pi _0,\varPhi _n:\mathrm {S}_0\rightarrow \mathbb {R}\) are twice continuously differentiable. Furthermore, \(I_n\) has a unique minimizer \(x_n \in \mathrm {S}_0\) satisfying

$$\begin{aligned} I_n(x_n) = 0, \quad \nabla I_n(x_n) = 0, \quad \nabla ^2 I_n(x_n) > 0, \end{aligned}$$

where the latter denotes positive definiteness.

Remark 2

Assuming that \(\min _{x\in \mathrm {S}_0} I_n(x) = 0\) is just a particular (but helpful) normalization and in general not restrictive: If \(\min _{x\in \mathrm {S}_0} I_n(x) = c >-\infty \), then we can simply set

$$\begin{aligned} {\hat{\varPhi }}_n(x) :=\varPhi _n(x) - c, \qquad {\hat{I}}_n(x) :={\hat{\varPhi }}_n(x) - \log \pi _0(x) \end{aligned}$$

for which we obtain

$$\begin{aligned} \mu _n(\mathrm {d}x) = \frac{1}{{\hat{Z}}_n} \exp \left( -n {\hat{\varPhi }}_n(x)\right) \mu _0(\mathrm {d}x), \quad {\hat{Z}}_n = \int _{\mathbb {R}^d} \exp \left( -n {\hat{\varPhi }}_n(x)\right) \mu _0(\mathrm {d}x), \end{aligned}$$

and \(\min _{x\in \mathrm {S}_0} {\hat{I}}_n(x) = \min _x {\hat{\varPhi }}_n(x) - \frac{1}{n} \log \pi _0(x) = 0\).

Given Assumption 1 we define the Laplace approximation of \(\mu _n\) as the following Gaussian measure

$$\begin{aligned} \mathcal L_{\mu _n}:=\mathcal N (x_n, n^{-1} C_n), \qquad C^{-1}_n :=\nabla ^2 I_n(x_n). \end{aligned}$$

Thus, we have

$$\begin{aligned} \mathcal L_{\mu _n}(\mathrm {d}x) = \frac{1}{{\widetilde{Z}}_n}\exp \left( - \frac{n}{2} \Vert x-x_n\Vert ^2_{C_n^{-1}}\right) \ \mathrm {d}x, \qquad {\widetilde{Z}}_n :=n^{-d/2} \sqrt{\det (2\pi C_n)}, \end{aligned}$$

where we can view

$$\begin{aligned} {\widetilde{I}}_n(x) :=\frac{1}{2} \Vert x-x_n\Vert ^2_{C_n^{-1}} = \underbrace{I_n(x_n)}_{=0} + \underbrace{\nabla I_n(x_n)^\top }_{=0} (x-x_n) + \frac{1}{2} \Vert x-x_n\Vert ^2_{\nabla ^2 I_n(x_n)} \end{aligned}$$

as the second-order Taylor approximation \({\widetilde{I}}_n = T_2 I_n(x_n)\) of \(I_n\) at \(x_n\). This point of view is crucial for analyzing the approximation

$$\begin{aligned} \mu _n \approx \mathcal L_{\mu _n}, \qquad \frac{1}{Z_n} \exp (-n I_n(x)) \approx \frac{1}{{\widetilde{Z}}_n} \exp \left( - n {\widetilde{I}}_n(x)\right) . \end{aligned}$$

Notation and recurring equations Before we continue, we collect recurring important definitions and where they can be found in Table 1 and provide the following important equations cheat sheet

$$\begin{aligned} \mu _0(\mathrm d x)&= \pi _0(x) \mathrm d x&\text { (relative to Lebesgue measure)} \\ \mu _n(\mathrm d x)&= Z_n^{-1}\exp (-n\varPhi _n(x))\mu _0(\mathrm d x)&\text { (relative to }\mu _0\text {)}\\&= Z_n^{-1}\exp (-nI_n(x)){\mathbf {1}}_{S_0}(x)\mathrm d x&\text { (relative to Lebesgue measure)} \\ {\mathcal {L}}_{\mu _n}(\mathrm d x)&= {\widetilde{Z}}_n^{-1} \exp (-n T_2\varPhi _n(x; x_n))\mu _0(\mathrm d x)&\text { (relative to }\mu _0\text {)}\\&= {\widetilde{Z}}_n^{-1} \exp (-\frac{n}{2}\Vert x-x_n\Vert _{\nabla ^2 I_n(x_n)}^2)\mathrm d x&\text { (relative to Lebesgue measure)} \end{aligned}$$
Table 1 Frequently used notation

Convergence in Hellinger distance

By a modification of Theorem 1 for integrals w.r.t. a weight \(\mathrm e^{-n\varPhi _n(x)}\) we may show a corresponding version of (4), i.e., for sufficiently smooth \(f\in L^1_{\mu _0}(\mathbb {R})\)

$$\begin{aligned} \left| \int _{\mathbb {R}^d} f(x)\ \mu _n(\mathrm {d}x) - \int _{\mathbb {R}^d} f(x)\ \mathcal L_{\mu _n}(\mathrm {d}x)\right| \in \mathcal O(n^{-1}). \end{aligned}$$

However, in this section we study a stronger notion of convergence of \(\mathcal L_{\mu _n}\) to \(\mu _n\), namely, w.r.t. the total variation distance\(d_\text {TV}\) and the Hellinger distance\(d_\mathrm {H}\). Given two probability measures \(\mu \), \(\nu \) on \(\mathbb {R}^d\) and another probability measure \(\rho \) dominating \(\mu \) and \(\nu \) the total variation distance of \(\mu \) and \(\nu \) is given by

$$\begin{aligned} d_\text {TV}(\mu ,\nu ) :=\sup _{A \in \mathcal B(\mathbb {R}^d)} \left| \mu (A) - \nu (A)\right| = \frac{1}{2} \int _{\mathbb {R}^d} \left| \frac{\mathrm {d}\mu }{\mathrm {d}\rho }(x) - \frac{\mathrm {d}\widetilde{\nu }}{\mathrm {d}\rho }(x)\right| \rho (\mathrm {d}x) \end{aligned}$$

and their Hellinger distance by

$$\begin{aligned} d^2_\text {H}(\mu ,\nu ) :=\int _{\mathbb {R}^d} \left| \sqrt{\frac{\mathrm {d}\mu }{\mathrm {d}\rho }(x)} - \sqrt{\frac{\mathrm {d}\nu }{\mathrm {d}\rho }(x)}\right| ^2 \rho (\mathrm {d}x). \end{aligned}$$

It holds true that

$$\begin{aligned} \frac{d^2_\text {H}(\mu ,\nu )}{2} \le d_\text {TV}(\mu ,\nu ) \le d_\mathrm {H}(\mu ,\nu ), \end{aligned}$$

see [17, Equation (8)]. Note that, \(d_\text {TV}(\mu _n,\mathcal L_{\mu _n})\rightarrow 0\) implies that \(|\int f \mathrm {d}\mu _n - \int f \mathrm {d}\mathcal L_{\mu _n}| \rightarrow 0\) for any bounded and continuous \(f:\mathbb {R}^d\rightarrow \mathbb {R}\). In order to establish our convergence result, we require almost the same assumptions as in Theorem 1, but now uniformly w.r.t. n:

Assumption 2

There holds \(\varPhi _n, \pi _0 \in C^3(\mathrm {S}_0,\mathbb {R})\) for all \(n\in \mathbb {N}\) and

  1. 1.

    there exist the limits

    $$\begin{aligned} x_\star :=\lim _{n\rightarrow \infty } x_n \qquad H_\star :=\lim _{n\rightarrow \infty } H_n, \qquad H_n :=\nabla ^2\varPhi _n(x_n) \end{aligned}$$

    in \(\mathbb {R}^d\) and \(\mathbb {R}^{d \times d}\), respectively, with \(H_\star \) being positive definite and \(x_\star \) belonging to the interior of \(\mathrm {S}_0\).

  2. 2.

    For each \(r > 0\) there exists an \(n_r\in \mathbb {N}\), \(\delta _r>0\) and \(K_r < \infty \) such that

    $$\begin{aligned} \delta _r \le \inf _{x \notin B_r(x_n)\cap \mathrm {S}_0} I_n(x) \qquad \forall n\ge n_r \end{aligned}$$

    as well as

    $$\begin{aligned} \max _{x\in B_r(0) \cap \mathrm {S}_0} \Vert \nabla ^3 \log \pi _0(x)\Vert \le K_r, \max _{x\in B_r(0)\cap \mathrm {S}_0} \Vert \nabla ^3 \varPhi _n(x)\Vert \le K_r \quad \forall n\ge n_r. \end{aligned}$$
  3. 3.

    There exists a uniformly bounding function \(q:\mathrm {S}_0\rightarrow [0,\infty )\) with

    $$\begin{aligned} \exp (-n I_n(x)) \le q(x), \qquad \forall x \in \mathrm {S}_0\ \forall n\ge n_0 \end{aligned}$$

    for an \(n_0\in \mathbb {N}\) such that \(q^{1-\epsilon }\) is integrable, i.e., \(\int _{\mathrm {S}_0} q^{1-\epsilon }(x)\ \mathrm {d}x < \infty \), for an \(\epsilon \in (0,1)\).

The only additional assumptions in comparison to the classical convergence theorem of the Laplace method are about the third derivatives of \(\pi _0\) and \(\varPhi _n\) and the convergence of \(x_n \rightarrow x_\star \). We remark that (12) implies

$$\begin{aligned} \lim _{n\rightarrow \infty } C^{-1}_n = \lim _{n\rightarrow \infty } \nabla ^2\left( \varPhi _n(x_n) - \frac{1}{n} \log \pi _0(x_n)\right) = H_\star \end{aligned}$$

and, thus, also \(\lim _{n\rightarrow \infty } C_n = H^{-1}_\star \). The uniform lower bound on \(I_n\) outside a ball around \(x_n\) as well as the integrable majorant of the unnormalized densities \(\mathrm e^{-nI_n}\le 1\) of \(\mu _n\) can be understood as uniform versions of the first and second assumption of Theorem 1. The third item of Assumption 2 implies the uniform integrabtility of the \(\mathrm e^{-nI_n}\le 1\) and is obviously satisfied for bounded supports \(\mathrm {S}_0\). However, in the unbounded case it seems to be crucialFootnote 1 for an increasing concentration of the \(\mu _n\).

We start our analysis with the following helpful lemma.

Lemma 1

Let Assumptions 1 and 2 be satisfied and let \(\pi _n, \widetilde{\pi }_n :\mathbb {R}^d\rightarrow [0,\infty )\) denote the unnormalized Lebesgue densities of \(\mu _n\) and \(\mathcal L_{\mu _n}\), respectively, given by

$$\begin{aligned} \pi _n(x) :={\left\{ \begin{array}{ll} \exp \left( - n \varPhi _n(x) \right) \pi _0(x), &{} x \in \mathrm {S}_0,\\ 0, &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$


$$\begin{aligned} \widetilde{\pi }_n(x) :=\exp \left( -\frac{n}{2} \Vert x-x_n\Vert ^2_{C^{-1}_n}\right) , \qquad x\in \mathbb {R}^d. \end{aligned}$$

Then, for any \(p\in \mathbb {N}\)

$$\begin{aligned} \int _{\mathbb {R}^d} \left| \left( \frac{\pi _n(x)}{\widetilde{\pi }_n(x)}\right) ^{1/p} - 1\right| ^p \mathcal L_{\mu _n}(\mathrm {d}x) \in \mathcal O(n^{-p/2}). \end{aligned}$$


We define the remainder term

$$\begin{aligned} R_n(x)&:=I_n(x) - {\widetilde{I}}_n(x) = I_n(x) - \frac{1}{2} \Vert x-x_n\Vert ^2_{C^{-1}_n}, \end{aligned}$$

i.e., for \(x\in \mathrm {S}_0\) we have \(\frac{\pi _n(x)}{\widetilde{\pi }_n(x)} = \exp (-nR_n(x))\). Moreover, note that for \(x\in \mathrm {S}_0^c\) there holds \(\pi _n(x) = 0\). Thus, we obtain

$$\begin{aligned} \int _{\mathbb {R}^d} \left| \left( \frac{\pi _n(x)}{\widetilde{\pi }_n(x)}\right) ^{1/p} - 1\right| ^p \mathcal L_{\mu _n}(\mathrm {d}x)&= \int _{\mathrm {S}_0^c} 1^p\ \mathcal L_{\mu _n}(\mathrm {d}x) \\&\qquad + \int _{\mathrm {S}_0} \left| \mathrm e^{-nR_n(x)/p} - 1\right| ^p\ \mathcal L_{\mu _n}(\mathrm {d}x)\\&= J_0(n) + J_1(n) + J_2(n) \end{aligned}$$

where we define for a given radius \(r>0\)

$$\begin{aligned} J_0(n)&:=\mathcal L_{\mu _n}(\mathrm {S}_0^c),\\ J_1(n)&:=\int _{B_r(x_n) \cap \mathrm {S}_0} \left| \mathrm e^{-nR_n(x)/p} -1 \right| ^{p}\ \mathcal L_{\mu _n}(\mathrm {d}x),\\ J_2(n)&:=\int _{B^c_r(x_n)\cap \mathrm {S}_0} \left| \mathrm e^{-n R_n(x)/p} -1 \right| ^p\ \mathcal L_{\mu _n}(\mathrm {d}x). \end{aligned}$$

In “Appendix B.1” we prove that

$$\begin{aligned} J_0(n) \in \mathcal O(\mathrm e^{-c_r n}), \qquad J_1(n) \in \mathcal O(n^{-p/2}), \qquad J_2(n) \in \mathcal O(\mathrm e^{- n c_{r,\epsilon }} n^{d/2} ), \end{aligned}$$

for \(c_r, c_{r,\epsilon } >0\), which then yields the statement. \(\square \)

Lemma 1 provides the basis for our main convergence theorem.

Theorem 2

Let the assumptions of Lemma 1 be satisfied. Then, there holds

$$\begin{aligned} d_\mathrm {H}(\mu _n, \mathcal L_{\mu _n}) \in \mathcal O(n^{-1/2}). \end{aligned}$$


We start with

$$\begin{aligned} d^2_\text {H}(\mu _n, \mathcal L_{\mu _n})&= \int _{\mathbb {R}^d} \left[ \frac{\sqrt{\pi _n(x)}}{\sqrt{Z_n}} - \frac{\sqrt{\widetilde{\pi }_n(x)}}{\sqrt{{\widetilde{Z}}_n}} \right] ^2\ \mathrm {d}x\\&\le \frac{2}{{\widetilde{Z}}_n} \int _{\mathbb {R}^d} \left[ \sqrt{\pi _n(x)} - \sqrt{\widetilde{\pi }_n(x)} \right] ^2 \mathrm {d}x + 2 \left( \frac{1}{\sqrt{Z_n}} - \frac{1}{\sqrt{{\widetilde{Z}}_n}}\right) ^2 Z_n\\&= 2 \int _{\mathbb {R}^d} \left[ \sqrt{\frac{\pi _n(x)}{\widetilde{\pi }_n(x)}} - 1\right] ^2\ \mathcal L_{\mu _n}(\mathrm {d}x) + 2 \left( \frac{1}{\sqrt{Z_n}} - \frac{1}{\sqrt{{\widetilde{Z}}_n}}\right) ^2 Z_n. \end{aligned}$$

For the first term there holds due to Lemma 1

$$\begin{aligned} \int _{\mathbb {R}^d} \left[ \sqrt{\frac{\pi _n(x)}{\widetilde{\pi }_n(x)}} - 1\right] ^2\ \mathcal L_{\mu _n}(\mathrm {d}x) \in \mathcal O(n^{-1}). \end{aligned}$$

For the second term on the right-hand side we obtain

$$\begin{aligned} 2\left( \frac{1}{\sqrt{Z_n}} - \frac{1}{\sqrt{{\widetilde{Z}}_n}}\right) ^2 Z_n&= \frac{2}{{\widetilde{Z}}_n} \left( \sqrt{Z_n} - \sqrt{{\widetilde{Z}}_n}\right) ^2 = \frac{2}{{\widetilde{Z}}_n} \left( \frac{Z_n - {\widetilde{Z}}_n}{\sqrt{Z_n} + \sqrt{{\widetilde{Z}}_n}}\right) ^2\\&\le 2\frac{\left| Z_n - {\widetilde{Z}}_n\right| ^2}{{\widetilde{Z}}^2_n}. \end{aligned}$$

Furthermore, due to Lemma 1 there exists a \(c<\infty \) such that

$$\begin{aligned} |Z_n - {\widetilde{Z}}_n|&\le \int _{\mathbb {R}^d} \left| \pi _n(x) - \widetilde{\pi }_n(x)\right| \ \mathrm {d}x = {\widetilde{Z}}_n \int _{\mathbb {R}^d} \left| \frac{\pi _n(x)}{\widetilde{\pi }_n(x)} - 1 \right| \ \mathcal L_{\mu _n}(\mathrm {d}x)\\&\le c n^{-1/2} {\widetilde{Z}}_n. \end{aligned}$$

This yields

$$\begin{aligned} 2\left( \frac{1}{\sqrt{Z_n}} - \frac{1}{\sqrt{{\widetilde{Z}}_n}}\right) ^2 Z_n \le 2\frac{\left| Z_n - {\widetilde{Z}}_n\right| ^2}{{\widetilde{Z}}^2_n} \le 2c^2 n^{-1} \in \mathcal O(n^{-1}), \end{aligned}$$

which concludes the proof. \(\square \)

Convergence of other Gaussian approximations Let us consider now a sequence of arbitrary Gaussian approximations \(\widetilde{\mu }_n = \mathcal N(a_n, \frac{1}{n} B_n)\) to the measures \(\mu _n\) in (1). Under which conditions on \(a_n \in \mathbb {R}^d\) and \(B_n \in \mathbb {R}^{d\times d}\) do we still obtain the convergence \(d_\text {H}(\mu _n, \widetilde{\mu }_n) \rightarrow 0\)? Of course, \(a_n\rightarrow x_\star \) seems to be necessary but how about the covariances \(B_n\)? Due to the particular scaling of 1/n appearing in the covariance of \(\mathcal L_{\mu _n}\), one might suppose that for example \(\widetilde{\mu }_n = \mathcal N(x_n, \frac{1}{n} I_d)\) or \(\widetilde{\mu }_n = \mathcal N(x_n, \frac{1}{n} B)\) with an arbitrary symmetric and positive definite (spd) \(B \in \mathbb {R}^{d\times d}\) should converge to \(\mu _n\) as \(n\rightarrow \infty \). However, since

$$\begin{aligned} \left| d_\text {H}(\mu _n, \mathcal L_{\mu _n}) - d_\text {H}(\mathcal L_{\mu _n}, \widetilde{\mu }_n)\right| \le d_\text {H}(\mu _n, \widetilde{\mu }_n) \le d_\text {H}(\mu _n, \mathcal L_{\mu _n}) + d_\text {H}(\mathcal L_{\mu _n}, \widetilde{\mu }_n) \end{aligned}$$

and \(d_\text {H}(\mu _n, \mathcal L_{\mu _n}) \rightarrow 0\), we have

$$\begin{aligned} d_\text {H}(\mu _n, \widetilde{\mu }_n) \rightarrow 0 \quad \text { iff } \quad d_\text {H}(\mathcal L_{\mu _n}, \widetilde{\mu }_n) \rightarrow 0. \end{aligned}$$

The following result shows that, in general, \(\widetilde{\mu }_n = \mathcal N(x_n, \frac{1}{n} I_d)\) or \(\widetilde{\mu }_n = \mathcal N(x_n, \frac{1}{n} B)\) do not converge to \(\mu _n\).

Theorem 3

Let the assumptions of Lemma 1 be satisfied.

  1. 1.

    For \(\widetilde{\mu }_n :=\mathcal N(x_n, \frac{1}{n} B_n)\), \(n\in \mathbb {N}\), with spd \(B_n\), we have that

    $$\begin{aligned} \lim _{n\rightarrow \infty } d_\mathrm {H}(\mu _n, \widetilde{\mu }_n) = 0 \quad \text { iff } \quad \lim _{n\rightarrow \infty } \det \left( \frac{1}{2} (H^{1/2}_\star B_n^{1/2} + H_\star ^{-1/2} B_n^{-1/2})\right) = 1.\nonumber \\ \end{aligned}$$

    If so and if \(\Vert C_n - B_n\Vert \in \mathcal O(n^{-1})\), then we even have \(d_\mathrm {H}(\mu _n, \widetilde{\mu }_n) \in \mathcal O(n^{-1/2})\).

  2. 2.

    For \(\widetilde{\mu }_n :=\mathcal N(a_n, \frac{1}{n} B_n)\), \(n\in \mathbb {N}\), with \(B_n\) satisfying (14) and \(\Vert x_n - a_n\Vert \in \mathcal O(n^{-1})\), we have that \(d_\mathrm {H}(\mu _n, {\widetilde{\mu }_n}) \in \mathcal O(n^{-1/2})\).

The proof is straightforward given the exact formula for the Hellinger distance of Gaussian measures and can be found in “Appendix B.2”. Thus, Theorem 3 tells us that, in general, the Gaussian measures \(\widetilde{\mu }_n = \mathcal N(x_n, \frac{1}{n} I_d)\) do not converge to \(\mu _n\) as \(n\rightarrow \infty \) whereas it is easily seen that \(\widetilde{\mu }_n = \mathcal N(x_n, \frac{1}{n} H_\star )\), indeed, do converge.

Relation to the Bernstein–von Mises theorem in Bayesian inference The Bernstein–von Mises (BvM) theorem is a classical result in Bayesian inference and asymptotic statistics in \(\mathbb {R}^d\) stating the posterior consistency under mild assumptions [41]. Its extension to infinite-dimensional situations does not hold in general [9, 15], but can be shown under additional assumptions [3, 4, 16, 28]. In order to state the theorem we introduce the following setting: let \(Y_i \sim \nu _{x_0}\), \(i\in \mathbb {N}\), be i.i.d. random variables on \(\mathbb {R}^D\), \(d\le D\), following a distribution \(\nu _{x_0}(\mathrm {d}y) = \exp (-\ell (y, x_0)) {\textit{\textbf{1}}}_{\mathrm {S}_y}(y) \mathrm {d}y\) where \(\mathrm {S}_y \subset \mathbb {R}^D\) and where \(\ell :{\mathrm {S}_y}\times \mathbb {R}^d \rightarrow [- \ell _{\min }, \infty )\) represents the negative log-likelihood function for observing \(y \in \mathrm {S}_y\) given a parameter value \(x \in \mathbb {R}^d\). Assuming a prior measure \(\mu _0(\mathrm {d}x) = \pi _0(x) {\textit{\textbf{1}}}_{\mathrm {S}_0}(x) \ \mathrm {d}x\) for the unknown parameter, the resulting posterior after n observations \(y_i\) of the independent \(Y_i\), \(i=1,\ldots ,n\), is of the form (1) with

$$\begin{aligned} \varPhi _n(x) = \varPhi _n(x; y_1,\ldots , y_n) = \frac{1}{n} \sum _{i=1}^n \ell (y_i,x). \end{aligned}$$

We will denote the corresponding posterior measure by \(\mu _n^{y_1,\ldots ,y_n}\) in order to highlight the dependence of the particular data \(y_1,\ldots ,y_n\). The BvM theorem states now the convergence of this posterior to a sequence of Gaussian measures. This looks very similar to the statement of Theorem 2. However, the difference lies in the Gaussian measures as well as the kind of convergence. In its usual form the BvM theorem states under similar assumptions as for Theorem 2 that there holds in the large data limit

$$\begin{aligned} d_\text {TV}\left( \mu ^{Y_1,\ldots , Y_n}_n,\ \mathcal N({\hat{x}}_n, n^{-1} \mathcal I^{-1}_{x_0}) \right) \xrightarrow [n\rightarrow \infty ]{\mathbb {P}} 0 \end{aligned}$$

where \(\mu ^{Y_1,\ldots , Y_n}_n\) is now a random measure depending on the n independent random variables \(Y_1,\ldots , Y_n\) and where the convergence in probability is taken w.r.t. randomness of the \(Y_i\). Moreover, \({\hat{x}}_n = {\hat{x}}_n(Y_1,\ldots , Y_n)\) denotes an efficient estimator of the true parameter \(x_0 \in \mathrm {S}_0\)—e.g., the maximum-likelihood or MAP estimator—and \(\mathcal I_{x_0}\) denotes the Fisher information at the true parameter \(x_0\), i.e.,

$$\begin{aligned} \mathcal I_{x_0} = {\varvec{\mathbb E}} \left[ \nabla ^2_x \ell (Y_i, x_0) \right] = \int _{\mathbb {R}^D} \nabla ^2_x \ell (Y, x_0) \ \exp (- \ell (y, x_0) )\ \mathrm {d}y. \end{aligned}$$

Now both, the BvM theorem and Theorem 2, state the convergence of the posterior to a concentrating Gaussian measure where the rate of concentration of the latter (or better: of its covariance) is of order \(n^{-1}\). Furthermore, also the rate of convergence in the BvM theorem can be shown to be of order \(n^{-1/2}\) [19]. However, the main differences are:

  • The BvM states convergence in probability (w.r.t. the randomness of the \(Y_i\)) and takes as basic covariance the inverse expected Hessian of the negative log likelihood at the data generating parameter value \(x_0\). Working with this quantity requires the knowledge of the true value \(x_0\) and the covariance operator is obtained by marginalizing over all possible data outcomes Y. This Gaussian measure is not a practical tool to be used but rather a limiting distribution of a powerful theoretical result reconciling Bayesian and classical statistical theory. For this reason, the Gaussian approximation in the statement of the BvM theorem can be thought of as being a “prior” approximation (in the loosest meaning of the word). Usually, a crucial requirement is that the problem is well-specified meaning that \(x_0\) is an interior point of the prior support \(\mathrm {S}_0\)—although there exist results for misspecified models, see [22]. Here, a BvM theorem is proven without the assumption that \(x_0\) belongs to the interior of \(\mathrm {S}_0\). However, in this case the basic covariance is not the Fisher information but the Hessian of the mapping \(x \mapsto d_\text {KL}(\nu _0 || \nu _x)\) evaluated at its unique minimizer where \(d_\text {KL}(\nu _0 || \nu _x)\) denotes the Kullback–Leibler divergence of the data distribution \(\nu _x\) given parameter \(x \in \mathrm {S}_0\) w.r.t. the true data distribution \(\nu _0\).

  • Theorem 2 states the convergence for given realizations \(y_i\) and takes the Hessian of the negative log posterior density evaluated at the current MAP estimate \(x_n\) and the current data \(y_1,\ldots ,y_n\). This means that we do not need to know the true parameter value \(x_0\) and we employ the actual data realization at hand rather than averaging over all outcomes. Hence, we argue that the Laplace approximation (as stated in this context) provides a “posterior” approximation converging to the Bayesian posterior as \(n\rightarrow \infty \). Also, we require that the limit \(x_\star = \lim _{n\rightarrow \infty } x_n\) is an interior point of the prior support \(\mathrm {S}_0\).

  • From a numerical point of view, the Laplace approximation requires the computation of the MAP estimate and the corresponding Hessian at the MAP, whereas the BvM theorem employs the Fisher information, i.e. requires an expectation w.r.t. the observable data. Thus, the Laplace approximation is based on fixed and finite data in contrast to the BvM.

The following example illustrates the difference between the two Gaussian measures: Let \(x_0\in \mathbb R\) be an unknown parameter. Consider n measurements \(y_k \in \mathbb {R}\), \(k=1,\ldots ,n\), where \(y_k\) is a realization of

$$\begin{aligned} Y_k = x_0^3 + \eta _k \end{aligned}$$

with \(\eta _k\sim N(0, \sigma ^2)\) i.i.d.. For the Bayesian inference we assume a prior \(N(0,\tau ^2)\) on x. Then the Bayesian posterior is of the form \(\mu _n(\mathrm {d}x) \propto \exp (-nI_n(x))\) where

$$\begin{aligned} I_n(x) = \frac{x^2}{n\cdot 2\tau ^2} + \underbrace{\frac{1}{n\cdot 2\sigma ^2}\sum _{k=1}^n ({y_k} - x^3)^2}_{=\varPhi _n(x)}. \end{aligned}$$

The MAP estimator \(x_n\) is the Laplace approximation’s mean and can be computed numerically as a minimizer of \(I_n(x)\). It can be shown that \(x_n\) converges to \(x_\star = x_0\) for almost surely all realizations \(y_k \) of \(Y_k\) due to the strong law of large numbers. Now we take the Hessian (w.r.t. x) of \(I_n\),

$$\begin{aligned} \nabla ^2I_n(x) = \frac{1}{n\cdot \tau ^2} + \frac{15}{\sigma ^2}\cdot x^4 - 6x\cdot \frac{1}{n\cdot \sigma ^2}\sum _{k=1}^n y_k \end{aligned}$$

and evaluate it in \(x_n\) to obtain the covariance of the Laplace approximation, and, thus,

$$\begin{aligned} \mathcal L_{\mu _n}= {\mathcal {N}}\left( x_n, \frac{1}{\frac{1}{n\cdot \tau ^2} + \frac{15}{\sigma ^2}\cdot x_n^4 - 6x_n\cdot \frac{1}{n\cdot \sigma ^2}\sum _{k=1}^n {y_k}}\right) . \end{aligned}$$

On the other hand we compute the Gaussian BvM approximation: The Fisher information is given as (recall that \(\varPhi \) is the loglikelihood term as defined above)

$$\begin{aligned} {\mathbb E}^{x_0} [\nabla ^2 \varPhi (x_0)]= & {} \mathbb E \left[ \frac{15}{\sigma ^2}\cdot x_0^4 - 6x_0\cdot \frac{1}{n\cdot \sigma ^2}\sum _{k=1}^n Y_k \right] \\= & {} \frac{15}{\sigma ^2}\cdot x_0^4 - 6x_0\cdot \frac{1}{\sigma ^2} x_0^3 = \frac{9}{\sigma ^2}x_0^4 \end{aligned}$$

and hence we get the Gaussian approximation

$$\begin{aligned} \mu _{\text {BVM}} = {\mathcal {N}}\left( x_n, \frac{\sigma ^2}{9\cdot x_0^4}\right) . \end{aligned}$$

Now we clearly see the difference between the two measures and how they will be asymptotically identical, since \(x_n\rightarrow x_\star = x_0\) due to consistency, \(\frac{1}{n}\sum _{k=1}^n y_k\) converging a.s. to \(x_0^3\) due to the strong law of large numbers, and with the prior-dependent part vanishing for \(n\rightarrow 0\).

Remark 3

Having raised the issue whether the BvM approximation \(\mathcal N({\hat{x}}_n, n^{-1} \mathcal I^{-1}_{x_0})\) or the Laplace one \(\mathcal L_{\mu _n}\) is closer to a given posterior \(\mu _n\), one can of course ask for the best Gaussian approximation of \(\mu _n\) w.r.t. a certain distance or divergence. Thus, we mention [26, 30] where such a best approximation w.r.t. the Kullback–Leibler divergence is considered. The authors also treat the case of best Gaussian mixture approximations for multimodal distributions and state a BvM like convergence result for the large data (and small noise) limit. However, the computation of such a best approximation can become costly whereas the Laplace approximation can be obtained rather cheaply.

The case of singular Hessians

The assumption, that the Hessians \(H_n = \nabla ^2 \varPhi _n(x_n)\) as well as their limit \(H_\star \) are positive definite, is quite restrictive. For example, for Bayesian inference with more unknown parameters than observational information, this assumption is not satisfied. Hence, we discuss in this subsection the convergence of the Laplace approximation in case of singular Hessians \(H_n\) and \(H_\star \). Nonetheless, we assume throughout the section that Assumption 1 is satisfied. This yields that the Laplace approximation \(\mathcal L_{\mu _n}\) is well-defined. This means in particular that we suppose a regularizing effect of the log prior density \(\log \pi _0\) on the minimization of \(I_n(x) = \varPhi _n(x) - \frac{1}{n} \log \pi _0(x)\).

We first discuss necessary conditions for the convergence of the Laplace approximation and subsequently state a positive result for Gaussian prior measures \(\mu _0\).

Necessary conditions Let us consider the simple case of \(\varPhi _n\equiv \varPhi \), i.e., the probability measures \(\mu _n\) are given by

$$\begin{aligned} \mu _n(\mathrm {d}x) \propto \exp \left( -n \varPhi (x)\right) \, \mu _0(\mathrm {d}x), \end{aligned}$$

where we assume now that \(\varPhi :\mathrm {S}_0 \rightarrow [c, \infty )\) with \(c>-\infty \). Intuitively, \(\mu _n\) should converge weakly to the Dirac measure \(\delta _{\mathcal M_\varPhi }\) on the set

$$\begin{aligned} \mathcal M_\varPhi :=\mathop {\mathrm{argmin}}\nolimits _{x\in \mathrm {S}_0} \varPhi (x). \end{aligned}$$

On the other hand, the associated Laplace approximations \(\mathcal L_{\mu _n}\) will converge weakly to the Dirac measure \(\delta _{\mathcal M_{\mathcal L}}\) in the affine subspace

$$\begin{aligned} \mathcal M_{\mathcal L} :=\{x \in \mathbb {R}^d:(x-x_\star )^\top H_\star (x-x_\star ) = 0\}. \end{aligned}$$

Hence, it is necessary for the convergence \(\mathcal L_{\mu _n}\rightarrow \mu _n\) in total variation or Hellinger distance that \(\mathcal M_\varPhi = \mathcal M_{\mathcal L}\), i.e., that the set of minimizers of \(\varPhi \) is linear. In order to ensure the latter, we state the following.

Assumption 3

Let \(\mathcal X\subseteq \mathbb {R}^d\) be a linear subspace such that for a projection \(\mathrm {P}_{\mathcal X}\) onto \(\mathcal X\) there holds

$$\begin{aligned} \varPhi _n \equiv \varPhi _n \circ \mathrm {P}_{\mathcal X} \qquad \text { on } \mathrm {S}_0 \text { for each } n\in \mathbb {N}\end{aligned}$$

and let the restriction \(\varPhi _n :\mathcal X \rightarrow \mathbb {R}\) possess a unique and nondegenerate global minimum for each \(n\in \mathbb {N}\).

For the case \(\varPhi _n = \varPhi \) this assumption implies, that

$$\begin{aligned} \mathcal M_{\varPhi } = \mathop {\mathrm{argmin}}\nolimits _{x\in \mathrm {S}_0} \varPhi (x) = x_\star + \mathcal X^c \end{aligned}$$

where \(\mathcal X^c\) denotes a complementary subspace to \(\mathcal X\), i.e., \(\mathcal X \oplus \mathcal X^c = \mathbb {R}^d\) and \(x_\star \in \mathcal X\) the unique minimizer of \(\varPhi \) over \(\mathcal X\). Besides that, Assumption 3 also yields that \(x^\top H_n x = 0\) iff \(x \in \mathcal X^c\). Hence, this also holds for the limit \(H_\star = \lim _{n\rightarrow \infty } H_n\) and we obtain

$$\begin{aligned} \mathcal M_{\mathcal L} = x_{\star } + \mathcal X^c = \mathcal M_{\varPhi }. \end{aligned}$$

Moreover, since Assumption 3 yields

$$\begin{aligned} \mu _n(\mathrm {d}x) \propto \exp \left( - n \varPhi _n(x_{\mathcal X})\right) \mu _0(\mathrm {d}x_{\mathcal X} \mathrm {d}x_{c}), \end{aligned}$$

where \(x_{\mathcal X} :=\mathrm {P}_{\mathcal X}x\) and \(x_{c} :=\mathrm {P}_{\mathcal X^c}x = x - x_{\mathcal X}\), the marginal of \(\mu _n\) coincides with the marginal of \(\mu _0\) on \(\mathcal X^c\). Hence, the Laplace approximation can only converge to \(\mu _n\) in total variation or Hellinger distance if this marginal is Gaussian. We, therefore, consider the special case of Gaussian prior measures \(\mu _0\).

Remark 4

Please note that, despite this to some extent negative result for the Laplace approximation for singular Hessians, the preconditioning of sampling and quadrature methods via the Laplace approximation may still lead to efficient algorithms in the small noise setting. The analysis of Laplace approximation-based sampling methods, as introduced in the next section, in the underdetermined case will be subject to future work.

Convergence for Gaussian prior  \(\mu _0\). A useful feature of Gaussian prior measures \(\mu _0\) is that the Laplace approximation possesses a convenient representation via its density w.r.t. \(\mu _0\).

Proposition 1

(cf. [43, Proposition 1]) Let Assumption 1 be satisfied and \(\mu _0\) be Gaussian. Then there holds

$$\begin{aligned} \frac{\mathrm {d}\mathcal L_{\mu _n}}{\mathrm {d}\mu _0}(x) \propto \exp (- n T_2\varPhi _n(x; x_n)), \qquad x \in \mathbb {R}^d, \end{aligned}$$

where \(T_2\varPhi _n(\cdot ; x_n)\) denotes the Taylor polynomial of order 2 of \(\varPhi _n\) at the point \(x_n\in \mathbb {R}^d\).

In fact, the representation (17) does only hold for prior measures \(\mu _0\) with Lebesgue density \(\pi _0:\mathbb {R}^d \rightarrow [0,\infty )\) satisfying \(\nabla ^3 \log \pi _0 \equiv 0\).

Corollary 1

Let Assumption 1 be satisfied and \(\mu _0\) be Gaussian. Further, let Assumption 3 hold true and assume that the restriction \(\varPhi _n :\mathcal X \rightarrow \mathbb {R}\) and the marginal density \(\pi _0\) on \(\mathcal X\) satisfy Assumption 2 on \(\mathcal X\). Then the approximation result of Theorem 2 holds.


By using Proposition 1, we can express the Hellinger distance \(d_\text {H}(\mu _n,\mathcal L_{\mu _n})\) as follows

$$\begin{aligned} d^2_\text {H}(\mu _n,\mathcal L_{\mu _n})&= \int _{\mathbb {R}^{d}} \left( \sqrt{\frac{\mathrm {d}\mu _n}{\mathrm {d}\mu _0}(x)} - \sqrt{\frac{\mathrm {d}\mathcal L_{\mu _n}}{\mathrm {d}\mu _0}(x)} \right) ^2 \mu _0(\mathrm {d}x)\\&= \int _{\mathbb {R}^{d}} \left( \sqrt{\frac{\exp (-n \varPhi _n(x))}{Z_n}} - \sqrt{\frac{\exp (-n T_2\varPhi _n(x;x_n))}{{\widetilde{Z}}_n}} \right) ^2 \mu _0(\mathrm {d}x). \end{aligned}$$

We use now the decomposition \(\mathbb {R}^d = \mathcal X \oplus \mathcal X^c\) with \(x :=x_{\mathcal X} + x_{c}\) for \(x\in \mathbb {R}^d\) with \(x_{\mathcal X} \in \mathcal X\) and \(x_c \in \mathcal X^c\). We note, that due to Assumption 3, we have that

$$\begin{aligned} T_2\varPhi _n(x;x_n) = T_2\varPhi _n(x_{\mathcal X};x_n), \qquad x \in \mathbb {R}^d. \end{aligned}$$

We then obtain by disintegration and denoting \(\widetilde{\varPhi }_n(x) :=T_2\varPhi _n(x;x_n) = \widetilde{\varPhi }_n(x_{\mathcal X})\)

$$\begin{aligned} d^2_\text {H}(\mu _n,\mathcal L_{\mu _n})&= \int _{\mathbb {R}^{d}} \left( \sqrt{\frac{\mathrm e^{-n \varPhi _n(x_{\mathcal X})}}{Z_n}} - \sqrt{\frac{\mathrm e^{-n \widetilde{\varPhi }_n(x_{\mathcal X})}}{{\widetilde{Z}}_n}} \right) ^2 \mu _0(\mathrm {d}x_{\mathcal X} \mathrm {d}x_c)\\&= \int _{\mathcal X} \int _{\mathcal X^c} \left( \sqrt{\frac{\mathrm e^{-n \varPhi _n(x_{\mathcal X})}}{Z_n}} - \sqrt{\frac{\mathrm e^{-n \widetilde{\varPhi }_n(x_{\mathcal X})}}{{\widetilde{Z}}_n}} \right) ^2 \mu _0(\mathrm {d}x_{c} | x_{\mathcal X}) \ \mu _0(\mathrm {d}x_{\mathcal X})\\&= \int _{\mathcal X} \left( \sqrt{\frac{\mathrm e^{-n \varPhi _n(x_{\mathcal X})}}{Z_n}} - \sqrt{\frac{\mathrm e^{-n \widetilde{\varPhi }_n(x_{\mathcal X})}}{{\widetilde{Z}}_n}} \right) ^2 \mu _0(\mathrm {d}x_{\mathcal X}), \end{aligned}$$

where \(\mu _0(\mathrm {d}x_{\mathcal X})\) denotes the marginal of \(\mu _0\) on \(\mathcal X\). Since \(\varPhi _n\) and \(I_n(x_{\mathcal X}) = \varPhi _n(x_{\mathcal X}) - \frac{1}{n} \log \pi _0(x_{\mathcal X})\), where \(\pi _0(x_{\mathcal X})\) denotes the Lebesgue density of the marginal \(\mu _0(\mathrm {d}x_{\mathcal X})\), satisfy the assumptions of Theorem 2 on \(\mathrm {S}_0 \cap \mathcal X = \mathcal X\), the statement follows. \(\square \)

We provide some illustrative examples for the theoretical results stated in this subsection.

Example 1

(Divergence of the Laplace approximation in the singular case) We assume a Gaussian prior \(\mu _0 = N(0, I_2)\) on \(\mathbb {R}^2\) and \(\varPhi (x) = \Vert y - \mathcal G(x)\Vert ^2\) where

$$\begin{aligned} y = 0, \qquad \mathcal G(x) = x_2-x_1^2, \qquad x = (x_1,x_2) \in \mathbb {R}^2. \end{aligned}$$

We plot the Lebesgue densities of the resulting \(\mu _n\) and \(\mathcal L_{\mu _n}\) for \(n=128\) in the left and middle panel of Fig. 1. The red line in both plots indicate the different sets

$$\begin{aligned} \mathcal M_\varPhi = \{x \in \mathbb {R}^2:x_2 = x_1^2\}, \qquad \mathcal M_{\mathcal L} = \{x \in \mathbb {R}^2:x_2 = 0\}, \end{aligned}$$

around which \(\mu _n\) and \(\mathcal L_{\mu _n}\), respectively, concentrate as \(n\rightarrow \infty \). As \(\mathcal M_\varPhi \ne \mathcal M_{\mathcal L}\), we observe no convergence of the Laplace approximation as \(n\rightarrow \infty \), see the right panel of Fig. 1. Here, the Hellinger distance is computed numerically by applying a tensorized trapezoidal rule on a sufficiently large subdomain of \(\mathbb {R}^2\).

Fig. 1
figure 1

Plots of the Lebesgue densities of \(\mu _n\) (left) and \(\mathcal L_{\mu _n}\) (middle) for \(n=128\) as well as the Hellinger distance between \(\mu _n\) and \(\mathcal L_{\mu _n}\) for Example 1. The red line in the left and middle panel represents the set \(\mathcal M_\varPhi \) and \(\mathcal M_{\mathcal L}\) around which \(\mu _n\) and \(\mathcal L_{\mu _n}\), respectively, concentrate as \(n\rightarrow \infty \)

Example 2

(Convergence of the Laplace approximation in the singular case in the setting of Corollary1) Again, we suppose a Gaussain prior \(\mu _0 = N(0, I_2)\) and \(\varPhi \) in the form of \(\varPhi (x) = \Vert y - \mathcal G(x)\Vert ^2\) with

$$\begin{aligned} y = \begin{pmatrix} \frac{\pi }{2}\\ 0.5\end{pmatrix}, \qquad \mathcal G(x) = \begin{pmatrix} \exp ((x_2-x_1)/5)\\ \sin (x_2-x_1) \end{pmatrix}, \qquad x = (x_1,x_2) \in \mathbb {R}^2. \end{aligned}$$

Thus, the invariant subspace is \(\mathcal X^c = \{x \in \mathbb {R}^2 :x_1 = x_2\}\). In the left and middle panel of Fig. 2 we present the Lebesgue densities of \(\mu _n\) and its Laplace approximation \(\mathcal L_{\mu _n}\) for \(n=25\) and by the red line the sets \(\mathcal M_\varPhi = \mathcal M_{\mathcal L} = x_\star + \mathcal X^c\). We observe the convergence guaranteed by Corollary 1 in the right panel of Fig. 2 where we can also notice a preasymptotic phase with a shortly increasing Hellinger distance. Such a preasmyptotic phase is to be expected due to \(d_\text {H}(\mu _n, \mathcal L_{\mu _n}) \in \mathcal O(n^{-1/2}) + \mathcal O(\mathrm e^{- n \delta _r} n^{d/2})\) as shown in the proof of Theorem 2.

Fig. 2
figure 2

Same as in Fig. 1 but for Example 2

Robustness of Laplace-based Monte Carlo methods

In practice, we are often interested in expectations or integrals of quantities of interest \(f:\mathbb {R}^d\rightarrow \mathbb {R}\) w.r.t. \(\mu _n\) such as

$$\begin{aligned} \int _{\mathbb {R}^d} f(x) \ \mu _n(\mathrm {d}x). \end{aligned}$$

For example, in Bayesian statistics the posterior mean (\(f(x) = x\)) or posterior probabilities (\(f(x) = {\mathbf {1}}_A(x)\), \(A \in \mathcal B(\mathbb {R}^d)\)) are desirable quantities. Since \(\mu _n\) is seldom given in explicit form, numerical integration must be applied for approximating such integrals. To this end, since the prior measure \(\mu _0\) is typically a well-known measure for which efficient numerical quadrature methods are available, the integral w.r.t. \(\mu _n\) is rewritten as two integrals w.r.t. \(\mu _0\)

$$\begin{aligned} \int _{\mathbb {R}^d} f(x) \ \mu _n(\mathrm {d}x) = \frac{\int _{\mathbb {R}^d} f(x) \ \exp (-n\varPhi _n(x))\ \mu _0(\mathrm {d}x)}{\int _{\mathbb {R}^d} \exp (-n\varPhi _n(x))\ \mu _0(\mathrm {d}x)}=:\frac{Z'_n}{Z_n}. \end{aligned}$$

If then a quadrature rule such as \(\int _{\mathbb {R}^d} g(x) \ \mu _0(\mathrm {d}x) \approx \frac{1}{N} \sum _{i=1}^N w_i\ g(x_i)\) is used, we end up with an approximation

$$\begin{aligned} \int _{\mathbb {R}^d} f(x) \ \mu _n(\mathrm {d}x) \approx \frac{\sum _{i=1}^N w_i\ f(x_i) \exp (-n\varPhi _n(x_i))}{\sum _{i=1}^N w_i\ \exp (-n\varPhi _n(x_i))}. \end{aligned}$$

This might be a good approximation for small \(n \in \mathbb {N}\). However, as soon as \(n\rightarrow \infty \) the likelihood term \(\exp (-n\varPhi _n(x_i))\) will deteriorate and this will be reflected by a deteriorating efficiency of the quadrature scheme—not in terms of the convergence rate w.r.t. N, but w.r.t. the constant in the error estimate, as we will display later in examples.

If the Gaussian Laplace approximation \(\mathcal L_{\mu _n}\) of \(\mu _n\) is used as the prior measure for numerical integration instead of \(\mu _0\), we get the following approximation

$$\begin{aligned} \int _{\mathbb {R}^d} f(x) \ \mu _n(\mathrm {d}x) \approx \frac{\sum _{i=1}^N w_i\ f(x_i) \frac{\pi _n(x_i)}{\widetilde{\pi }_n(x_i)}}{\sum _{i=1}^N w_i\ \frac{\pi _n(x_i)}{\widetilde{\pi }_n(x_i)}}, \end{aligned}$$

where \(\pi _n\) and \(\widetilde{\pi }_n\) denote the unnormalized Lebesgue density of \(\mu _n\) and \(\mathcal L_{\mu _n}\), respectively. This time, we can not only apply well-known quadrature and sampling rules for Gaussian measures, but moreover, we also know due to Lemma 1, that the ratio \(\frac{\pi _n(x)}{\widetilde{\pi }_n(x)}\) converges in mean w.r.t. \(\mathcal L_{\mu _n}\) to 1. Hence, we do not expect a deteriorating efficiency of the numerical integration as \(n\rightarrow \infty \). On the contrary, as we subsequently discuss for several numerical integration methods, their efficiency for a finite number of samples \(N\in \mathbb {N}\) will even improve as \(n\rightarrow \infty \) if they are based on the Laplace approximation \(\mathcal L_{\mu _n}\).

For the sake of simplicity, we consider the simple case of \(\varPhi _n \equiv \varPhi + \text {const}\) in the following presentation—nonetheless, the presented results can be extended to the general case given appropriate modifications of the assumptions. Thus, we consider probability measures \(\mu _n\) of the form

$$\begin{aligned} \mu _n(\mathrm {d}x) \propto \mathrm e^{-n \varPhi (x)} \ \mu _0(\mathrm {d}x) \end{aligned}$$

where we assume that \(\varPhi \) satisfies the assumptions of Theorem 1. However, when dealing with the Laplace approximation of \(\mu _n\) and, particularly, with the ratios of the corresponding normalizing constants, it is helpful to use the following representation

$$\begin{aligned} \mu _n(\mathrm {d}x) = \frac{1}{Z_n} \mathrm e^{-n \varPhi _n(x)} \mu _0(\mathrm {d}x), \qquad \varPhi _n(x) :=\varPhi (x) - \iota _n, \end{aligned}$$

where \(\iota _n :=\min _{x \in \mathrm {S}_0} \varPhi (x) - \frac{1}{n} \log \pi _0(x)\) and \( Z_n = \mathrm e^{n \iota _n} \int _{\mathbb {R}^d} \mathrm e^{-n\varPhi (x)} \pi _0(x)\ \mathrm {d}x\). By this construction the resulting \(I_n(x) :=\varPhi _n(x) - \frac{1}{n} \log \pi _0(x)\) satisfies \(I_n(x_n) = 0\) as required in Assumption 1 for the construction of the Laplace approximation \(\mathcal L_{\mu _n}\). Note, that for \(\varPhi _n = \varPhi - \iota _n\) the Assumptions 1 and 2 imply the assumptions of Theorem 1 for \(f = \pi _0\) and \(p=0\).

Preliminaries Before we start analyzing numerical methods based on the Laplace approximation as their reference measure, we take a closer look at the details of the asymptotic expansion for integrals provided in Theorem 1 and their implications for expectations w.r.t. \(\mu _n\) given in (22).

  1. 1.

    The coefficients: The proof of Theorem 1 in [44, Section IX.5] provides explicit expressionsFootnote 2 for the coefficients \(c_k \in \mathbb {R}\) in the asymptotic expansion

    $$\begin{aligned} \int _D f(x) \exp (-n \varPhi (x)) \mathrm {d}x = \mathrm e^{-n \varPhi (x_\star )} n^{-d/2} \left( \sum _{k=0}^p c_k(f) n^{- k} + \mathcal O\left( n^{-p-1}\right) \right) , \end{aligned}$$

    namely—given that \(f \in C^{2p+2}(D, \mathbb R)\) and \(\varPhi \in C^{2p+3}(D, \mathbb R)\)—that

    $$\begin{aligned} c_k(f) = \sum _{\varvec{\alpha }\in \mathbb {N}_ 0^d :|\varvec{\alpha }| = 2k} \frac{\kappa _{\varvec{\alpha }}}{\varvec{\alpha }!} D^{\varvec{\alpha }} F(0) \end{aligned}$$

    where for \(\varvec{\alpha }= (\alpha _1,\ldots ,\alpha _d)\) we have \(|\varvec{\alpha }| = \alpha _1+\cdots +\alpha _d\), \(\varvec{\alpha }! = \alpha _1 ! \cdots \alpha _d!\), \(D^{\varvec{\alpha }} = D^{\alpha _1}_{x_1} \cdots D^{\alpha _d}_{x_d}\) and

    $$\begin{aligned} F(x) :=f(h(x))\ \det (\nabla h(x)) \end{aligned}$$

    with \(h:\varOmega \rightarrow U(x_\star )\) being a diffeomorphism between \(0 \in \varOmega \subset \mathbb {R}^d\) and a particular neighborhood \(U(x_\star )\) of \(x_\star \) mapping \(h(0) = x_\star \) and such that \(\det (\nabla h(0)) = 1\). The diffeomorphism h is specified by the well-known Morse’s Lemma and depends only on \(\varPhi \). In particular, if \(\varPhi \in C^{2p+3}(D, \mathbb R)\), then \(h\in C^{2p+1}(\varOmega , U(x_\star ))\). For the constants \(\kappa _{\varvec{\alpha }} = \kappa _{\alpha _1}\cdots \kappa _{\alpha _d} \in \mathbb {R}\) we have \(\kappa _{\alpha _i} = 0\) if \(\alpha _i\) is odd and \(\kappa _{\alpha _i} = (2/\lambda _i)^{(\alpha _i +1)/2} \varGamma ((\alpha _i+1)/2)\) otherwise with \(\lambda _i>0\) denoting the ith eigenvalue of \(H_\star = \nabla ^2 \varPhi (x_\star )\). Hence, we get

    $$\begin{aligned} c_k(f) = \sum _{\varvec{\alpha }\in \mathbb {N}_ 0^d :|\varvec{\alpha }| = k} \frac{\kappa _{2\varvec{\alpha }}}{(2\varvec{\alpha })!} D^{2\varvec{\alpha }} F(0). \end{aligned}$$
  2. 2.

    The normalization constant of\(\mu _n\): Theorem 1 implies that if \(\pi _0\in C^2(\mathbb {R}^d; \mathbb {R})\) and \(\varPhi \in C^3(\mathbb {R}^d, \mathbb {R})\), then

    $$\begin{aligned} \int _{\mathbb {R}^d} \pi _0(x) \ \exp (-n\varPhi (x)) \ \mathrm {d}x = \mathrm e^{-n\varPhi (x_\star )} n^{-d/2} \left( \frac{(2\pi )^{d/2}\, \pi _0(x_\star )}{\sqrt{\det (H_\star )}} + \mathcal O(n^{-1})\right) . \end{aligned}$$

    Hence, we obtain for the normalizing constant \(Z_n\) in (22) that

    $$\begin{aligned} Z_n = \mathrm e^{n (\iota _n -\varPhi (x_\star ))} n^{-d/2} \left( \frac{(2\pi )^{d/2}\, \pi _0(x_\star )}{\sqrt{\det (H_\star )}} + \mathcal O(n^{-1})\right) . \end{aligned}$$

    If we compare this to the normalizing constant \({\widetilde{Z}}_n = n^{-d/2} \sqrt{\det (2\pi C_n)}\) of its Laplace approximation we get

    $$\begin{aligned} \frac{Z_n}{{\widetilde{Z}}_n} = \mathrm e^{n(\iota _n-\varPhi (x_\star ))} \frac{\frac{\pi _0(x_\star )}{\sqrt{\det (H_\star )}} + \mathcal O(n^{-1})}{\sqrt{\det (C_n)}}. \end{aligned}$$

    We now show that

    $$\begin{aligned} \frac{Z_n}{{\widetilde{Z}}_n} = 1 + \mathcal O(n^{-1}). \end{aligned}$$

    First, we get due to \(C_n \rightarrow H_\star ^{-1}\) that \(\sqrt{\det (C_n)}\rightarrow \frac{1}{\sqrt{\det (H_\star )}}\) as \(n\rightarrow \infty \). Moreover,

    $$\begin{aligned} \mathrm e^{n(\iota _n-\varPhi (x_\star ))} = \frac{\exp (n (\varPhi (x_n) - \varPhi (x_\star )))}{\pi _0(x_n)}. \end{aligned}$$

    Since \(x_n \rightarrow x_\star \) continuity implies \(\pi _0(x_n) \rightarrow \pi _0(x_\star )\) as \(n\rightarrow \infty \). Besides that, the strong convexity of \(\varPhi \) in a neighborhood of \(x_\star \)—due to \(\nabla ^2 \varPhi (X_\star ) >0\) and \(\varPhi \in C^3(\mathbb {R}^d,\mathbb {R})\)—implies that for a \(c>0\)

    $$\begin{aligned} \varPhi (x_n) - \varPhi (x_\star ) \le \frac{1}{2c} \Vert \nabla \varPhi (x_n)\Vert ^2, \end{aligned}$$

    also known as Polyak–Łojasiewicz condition. Because of

    $$\begin{aligned} \nabla \varPhi (x_n) = \frac{1}{n} \nabla \log \pi _0(x_n), \end{aligned}$$

    since \(\nabla I_n(x_n) = 0\), we have that \(|\varPhi (x_n) - \varPhi (x_\star )| \in \mathcal O(n^{-2})\), and hence,

    $$\begin{aligned} \lim _{n\rightarrow \infty } \mathrm e^{n(\iota _n-\varPhi (x_\star ))} = 1/\pi _0(x_\star ). \end{aligned}$$

    This yields (26).

  3. 3.

    The expectation w.r.t. \(\mu _n\): The expectation of a \(f \in L^1_{\mu _0}(\mathbb {R})\) w.r.t. \(\mu _n\) is given by

    $$\begin{aligned} {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] = \frac{ \int _{\mathrm {S}_0} f(x) \pi _0(x) \exp (-n\varPhi (x))\ \mathrm {d}x}{ \int _{\mathrm {S}_0} \pi _0(x) \exp (-n\varPhi (x))\ \mathrm {d}x}. \end{aligned}$$

    If \(f, \pi _0 \in C^2(\mathbb {R}^d, \mathbb {R})\) and and \(\varPhi \in C^3(\mathbb {R}^d, \mathbb {R})\), then we can apply the asymptotic expansion above to both integrals and obtain

    $$\begin{aligned} {\varvec{\mathbb E}}_{\mu _n} \left[ f \right]&= \frac{\mathrm e^{-n \varPhi (x_\star )} n^{-d/2}\ ( c_0(f\pi _0) + \mathcal O(n^{-1}))}{\mathrm e^{-n \varPhi (x_\star )} n^{-d/2}\ ( c_0(\pi _0) + \mathcal O(n^{-1}))} = f(x_\star ) + \mathcal O(n^{-1}). \end{aligned}$$

    If \(f, \pi _0 \in C^4(\mathbb {R}^d; \mathbb {R})\) and \(\varPhi \in C^5(\mathbb {R}^d, \mathbb {R})\), then we can make this more precise by using the next explicit terms in the asymptotic expansions of both integrals, apply the rule for the division of asymptotic expansions and obtain \( {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] = f(x_\star ) + {\widetilde{c}}_1(f,\pi _0) n^{-1} + \mathcal O(n^{-2})\) where \({\widetilde{c}}_1(f,\pi _0) = \frac{1}{c_0(\pi _0)} c_1(f\pi _0) - \frac{c_1(\pi _0)}{c^2_0(\pi _0)}c_0(f\pi _0)\).

  4. 4.

    The variance w.r.t. \(\mu _n\): The variance of a \(f \in L^2_{\mu _0}(\mathbb {R})\) w.r.t. \(\mu _n\) is given by

    $$\begin{aligned} \mathrm {Var}_{\mu _n}(f)&= {\varvec{\mathbb E}}_{\mu _n} \left[ f^2 \right] - {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] ^2. \end{aligned}$$

    If \(f, \pi _0 \in C^2(\mathbb {R}^d;\mathbb {R})\) and \(\varPhi \in C^3(\mathbb {R}^d, \mathbb {R})\), then we can exploit the result for the expectation w.r.t. \(\mu _n\) from above and obtain

    $$\begin{aligned} \mathrm {Var}_{\mu _n}(f)&= f^2(x_\star ) + \mathcal O(n^{-1}) - \left( f(x_\star ) + \mathcal O(n^{-1})\right) ^2 \in \mathcal O(n^{-1}). \end{aligned}$$

    If \(f,\pi _0\in C^4(\mathbb {R}^d, \mathbb {R})\) and \(\varPhi \in C^5(\mathbb {R}^d, \mathbb {R})\), then a straightforward calculation using the explicit formulas for \(c_1(f^2\pi _0)\) and \(c_1(f\pi _0)\) as well as \(\nabla h(0) = I\) yields

    $$\begin{aligned} \mathrm {Var}_{\mu _n}(f) = n^{-1} \Vert \nabla f(x_\star )\Vert ^2_{H_\star ^{-1}} + \mathcal O(n^{-2}). \end{aligned}$$

    Hence, the variance \(\mathrm {Var}_{\mu _n}(f)\) decays like \(n^{-1}\) provided that \(\nabla f(x_\star ) \ne 0\)—otherwise it decays (at least) like \(n^{-2}\).

Remark 5

As already exploited above, the assumptions of Theorem 1 imply that \(\varPhi \) is strongly convex in a neighborhood of \(x_\star = \lim _{n\rightarrow \infty } x_n\), where \(x_n = \mathop {\mathrm{argmin}}\nolimits _{x\in \mathrm {S}_0} \varPhi (x) - \frac{1}{n} \log \pi _0(x)\). This yields \(|\varPhi (x_n) - \varPhi (x_\star )| \in \mathcal O(n^{-2})\), and thus

$$\begin{aligned} \Vert x_n - x_\star \Vert \in \mathcal O(n^{-1}). \end{aligned}$$

Importance sampling

Importance sampling is a variant of Monte Carlo integration where an integral w.r.t. \(\mu \) is rewritten as an integral w.r.t. a dominating importance distribution\(\mu \ll \nu \), i.e.,

$$\begin{aligned} \int _{\mathbb {R}^d} f(x) \ \mu (\mathrm {d}x) = \int _{\mathbb {R}^d} f(x) \ \frac{\mathrm {d}\mu }{\mathrm {d}\nu }(x) \ \nu (\mathrm {d}x). \end{aligned}$$

The integral appearing on the righthand side is then approximated by Monte Carlo integration w.r.t. \(\nu \): given N independent draws \(x_i\), \(i=1,\ldots ,N\), according to \(\nu \) we estimate

$$\begin{aligned} \int _{\mathbb {R}^d} f(x) \ \mu (\mathrm {d}x) \approx \frac{1}{N} \sum _{i=1}^N w(x_i) f(x_i), \qquad w(x_i) :=\frac{\mathrm {d}\mu }{\mathrm {d}\nu }(x_i). \end{aligned}$$

Often the density or importance weight function\(w = \frac{\mathrm {d}\mu }{\mathrm {d}\nu }:\mathbb {R}^d \rightarrow [0,\infty )\) is only known up to a normalizing constant \({\widetilde{w}} \propto \frac{\mathrm {d}\mu }{\mathrm {d}\nu }\). In this case, we can use self-normalized importance sampling

$$\begin{aligned} \int _{\mathbb {R}^d} f(x) \ \mu (\mathrm {d}x) \approx \frac{\sum _{i=1}^N {\widetilde{w}}(x_i) \ f(x_i)}{\sum _{i=1}^N {\widetilde{w}}(x_i)} =:\mathrm {IS}^{(N)}_{\mu ,\nu }(f). \end{aligned}$$

As for Monte Carlo, there holds a strong law of large numbers (SLLN) for self-normalized importance sampling, i.e.,

$$\begin{aligned} \frac{\sum _{i=1}^N {\widetilde{w}}(X_i) \ f(X_i)}{\sum _{i=1}^N {\widetilde{w}}(X_i)} \xrightarrow [N\rightarrow \infty ]{\text {a.s.}} {\varvec{\mathbb E}}_{\mu } \left[ f \right] , \end{aligned}$$

where \(X_i\sim \nu \) are i.i.d., which follows from the ususal SLLN and the continuous mapping theorem. Moreover, by the classical central limit theorem (CLT) and Slutsky’s theorem also a similar statement holds for self-normalized importance sampling: given that

$$\begin{aligned} \sigma ^2_{\mu ,\nu }(f) := {\varvec{\mathbb E}}_{\nu } \left[ \left( \frac{\mathrm {d}\mu }{\mathrm {d}\nu }\right) ^2 (f- {\varvec{\mathbb E}}_{\mu } \left[ f \right] )^2 \right] < \infty \end{aligned}$$

we have

$$\begin{aligned} \sqrt{N} \left( \frac{\sum _{i=1}^N {\widetilde{w}}(X_i) \ f(X_i)}{\sum _{i=1}^N {\widetilde{w}}(X_i)} - {\varvec{\mathbb E}}_{\mu } \left[ f \right] \right) \xrightarrow [N\rightarrow \infty ]{\mathcal D} \mathcal N(0, \sigma ^2_{\mu ,\nu }(f)). \end{aligned}$$

Thus, the asymptotic variance \(\sigma ^2_{\mu ,\nu }(f)\) serves as a measure of efficiency for self-normalized importance sampling. To ensure a finite \(\sigma ^2_{\mu ,\nu }(f)\) for many functions of interest f, e.g., bounded f, the importance distribution \(\nu \) has to have heavier tails than \(\mu \) such that the ratio \(\frac{\mathrm {d}\mu }{\mathrm {d}\nu }\) belongs to \(L^2_\nu (\mathbb {R})\), see also [31, Section 3.3]. Moreover, if we even have \(\frac{\mathrm {d}\mu }{\mathrm {d}\nu } \in L^\infty _\nu (\mathbb {R})\) we can bound

$$\begin{aligned} \sigma ^2_{\mu ,\nu }(f) \le \left\| \frac{\mathrm {d}\mu }{\mathrm {d}\nu }\right\| _{L^\infty _\nu }\ {\varvec{\mathbb E}}_{\mu } \left[ (f- {\varvec{\mathbb E}}_{\mu } \left[ f \right] )^2 \right] \qquad \Leftrightarrow \qquad \frac{\sigma ^2_{\mu ,\nu }(f)}{\mathrm {Var}_\mu (f)} \le \left\| \frac{\mathrm {d}\mu }{\mathrm {d}\nu }\right\| _{L^\infty _\nu }, \end{aligned}$$

i.e., the ratio between the asymptotic variance of importance sampling w.r.t. \(\nu \) and plain Monte Carlo w.r.t. \(\mu \) can be bounded by the \(L^\infty _\nu \)- or supremum norm of the importance weight \(\frac{\mathrm {d}\mu }{\mathrm {d}\nu }\).

For the measures \(\mu _n\) a natural importance distribution (called \(\nu \) above) which allows for direct sampling are the prior measure \(\mu _0\) and the Gaussian Laplace approximation \(\mathcal L_{\mu _n}\). We study the behaviour of the resulting asymptotic variances \(\sigma ^2_{\mu _n,\mu _0}(f)\) and \(\sigma ^2_{\mu _n,\mathcal L_{\mu _n}}(f)\) in the following.

Prior importance sampling First, we consider \(\mu _0\) as importance distribution. For this choice the importance weight function \(w_n :=\frac{\mathrm {d}\mu _n}{\mathrm {d}\mu _0}\) is given by

$$\begin{aligned} w_n(x) = \frac{1}{Z_n} \exp (-n \varPhi _n(x)), \qquad x \in \mathrm {S}_0, \end{aligned}$$

with \(\varPhi _n(x) = \varPhi (x)-\iota _n\), see (22). Concerning the bound in (31) we immediately obtain for sufficiently smooth \(\pi _0\) and \(\varPhi \) by (25), assuming w.l.o.g. \(\min _x \varPhi (x) = \varPhi (x_\star )=0\), that

$$\begin{aligned} \Vert w_n\Vert _{L^\infty } = Z^{-1}_n \mathrm e^{n\iota _n} = {\widetilde{c}} n^{d/2}, \qquad {\widetilde{c}} > 0, \end{aligned}$$

explodes as \(n\rightarrow \infty \). Of course, this is just the deterioration of an upper bound, but in fact we can prove the following rather negative result where we use the notation \(g(n) \sim h(n)\) for the asymptotic equivalence of functions of n, i.e., \(g(n) \sim h(n)\) iff \(\lim _{n\rightarrow \infty } \frac{g(n)}{h(n)} =1\).

Lemma 2

Given \(\mu _n\) as in (22) with \(\varPhi \) satisfying the assumptions of Theorem 1 for \(p=1\) and \(\pi _0 \in C^4(\mathbb {R}^d,\mathbb {R})\) with \(\pi _0(x_\star ) \ne 0\), we have for any \(f\in C^4(\mathbb {R}^d, \mathbb {R})\cap L^1_{\mu _0}(\mathbb {R})\) with \(\nabla f(x_\star ) \ne 0\) that

$$\begin{aligned} \sigma ^2_{\mu _n,\mu _0}(f) \sim {\widetilde{c}}_fn^{d/2 -1}, \qquad {\widetilde{c}}_f > 0, \end{aligned}$$

which yields \(\frac{\sigma ^2_{\mu _n,\mu _0}(f)}{\mathrm {Var}_{\mu _n}(f)} \sim {\widetilde{c}}_f n^{d/2}\) for another \({\widetilde{c}}_f > 0\).


W.l.o.g. we may assume that \(f(x_\star ) = 0\), since \(\sigma ^2_{\mu _n,\mu _0}(f) = \sigma ^2_{\mu _n,\mu _0}(f-c)\) for any \(c\in \mathbb {R}\). Moreover, for simplicity we assume w.l.o.g. that \(\varPhi (x_\star ) = 0\). We study

$$\begin{aligned} \sigma ^2_{\mu _n,\mu _0}(f)&= \frac{1}{Z_n^2} \int _{\mathrm {S}_0} \mathrm e^{- 2n\varPhi _n(x)} \ (f(x) - {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] )^2 \ \mu _0(\mathrm {d}x)\\&= \frac{1}{\mathrm e^{-2n\iota _n} Z_n^2} \int _{\mathrm {S}_0} \mathrm e^{- 2n\varPhi (x)} \ (f(x) - {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] )^2 \ \mu _0(\mathrm {d}x) \end{aligned}$$

by analyzing the growth of the numerator and denominator w.r.t. n. Due to the preliminaries presented above we know that \(\mathrm e^{-2n\iota _n}Z^2_n = c_0^2 n^{-d} + \mathcal O(n^{-d-1})\) with \(c_0 = (2\pi )^{d/2}\, \pi _0(x_\star )/\sqrt{\det (H_\star )} >0\). Concerning the numerator we start with decomposing

$$\begin{aligned} \int _{\mathrm {S}_0} \mathrm e^{- 2n\varPhi (x)} \ (f(x) - {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] )^2 \ \mu _0(\mathrm {d}x) = J_1(n) - 2 J_2(n) + J_3(n) \end{aligned}$$

where this time

$$\begin{aligned} J_1(n)&:=\int _{\mathrm {S}_0} f^2(x) \mathrm e^{- 2n\varPhi (x)} \ \mu _0(\mathrm {d}x),\\ J_2(n)&:= {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] \int _{\mathrm {S}_0} f(x) \mathrm e^{- 2n\varPhi (x)} \ \mu _0(\mathrm {d}x),\\ J_3(n)&:= {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] ^2 \int _{\mathrm {S}_0} \mathrm e^{- 2n\varPhi (x)} \ \mu _0(\mathrm {d}x). \end{aligned}$$

We derive now asymptotic expansions of these terms based on Theorem 1. It is easy to see that the assumptions of Theorem 1 are also fulfilled when considering integrals w.r.t. \(\mathrm e^{- 2n\varPhi }\). We start with \(J_1\) and obtain due to \(f(x_\star ) = 0\) that

$$\begin{aligned} J_1(n)&= \int _{\mathrm {S}_0} f^2(x) \mathrm e^{- 2n\varPhi (x)} \ \mu _0(\mathrm {d}x) = c'_{1}(f^2\pi _0) n^{-d/2-1} + \mathcal O(n^{-d/2-2}) \end{aligned}$$

where \(c'_{1}(f^2\pi _0) \in \mathbb {R}\) is the same as \(c_1(f^2\pi _0)\) in (23) but for \(2\varPhi \) instead of \(\varPhi \).

Next, we consider \(J_2\) and recall that due to \(f(x_\star ) = 0\) we have \( {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] \in \mathcal O(n^{-1})\), see (27). Furthermore, \(f(x_\star ) = 0\) also implies \(\int _{\mathrm {S}_0} f(x) \pi _0(x)\, \mathrm e^{- 2n\varPhi (x)}\ \mathrm {d}x \in \mathcal O(n^{-1-d/2})\), see Theorem 1. Thus, we have

$$\begin{aligned} J_2(n) = {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] \int _{\mathrm {S}_0} f(x) \mathrm e^{- 2n\varPhi (x)} \ \mu _0(\mathrm {d}x)&\in \mathcal O\left( n^{-d/2-2} \right) . \end{aligned}$$

Finally, we take a look at \(J_3\). By Theorem 1 we have \(\int _{\mathrm {S}_0} \exp (- 2n\varPhi (x)) \ \mu _0(\mathrm {d}x) \in \mathcal O(n^{-d/2})\) and, hence, obtain

$$\begin{aligned} J_3(n) = {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] ^2 \int _{\mathrm {S}_0} \exp (- 2n\varPhi (x)) \ \mu _0(\mathrm {d}x) \, \in \mathcal O(n^{-2-d/2}). \end{aligned}$$

Hence, \(J_1\) has the dominating power w.r.t. n and we have that

$$\begin{aligned} \int _{\mathrm {S}_0} \mathrm e^{- 2n\varPhi (x)} \ (f(x) - {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] )^2 \ \mu _0(\mathrm {d}x)&\sim c'_{1}(f^2\pi _0) n^{-d/2-1}. \end{aligned}$$

At this point, we remark that due to the assumption \(\nabla f(x_\star ) \ne 0\) we have \(c'_{1}(f^2\pi _0) \ne 0\): we know by (24) that \(c'_{1}(f^2\pi _0) =\frac{1}{2} \sum _{j=1}^d \kappa _{2\textit{\textbf{e}}_j} D^{2\textit{\textbf{e}}_j} F(0)\) where \(F(x) = \pi _0(h'(x)) f^2(h'(x)) \det (\nabla h'(x))\) and \(h'\) denotes the diffeomorphism for \(2\varPhi \) appearing in Morse’s lemma and mapping 0 to \(x_\star \); applying the product formula and using \(f(x_\star )=0\) as well as \(\det (\nabla h'(0)) = 1\) we get that \(D^{2\textit{\textbf{e}}_j} F(x_\star ) = \pi _0(x_\star ) D^{2\textit{\textbf{e}}_j} ( f^2(h'(x_\star )) )\); similarly, we get using \(f(x_\star )=0\) that \(D^{2\textit{\textbf{e}}_j} ( f^2(h'(x_\star ))) = 2 | \textit{\textbf{e}}_j^\top \nabla h'(0) \nabla f(x_\star )|^2\); since \(h'\) is a diffeomorphishm \(\nabla h'(0)\) is regular and, thus, \(c'_{1}(f^2\pi _0) \ne 0\). The statement follows now by

$$\begin{aligned} \sigma ^2_{\mu _n,\mu _0}(f) = \frac{c'_{1}(f^2\pi _0) n^{-d/2-1} + \mathcal O(n^{-d/2-2})}{c_0^2 n^{-d} + \mathcal O(n^{-d-1})} \sim \frac{c'_{1}(f^2\pi _0)}{c_0^2} \ n^{d/2-1} \end{aligned}$$

and by recalling that \(\mathrm {Var}_{\mu _n}(f) \sim c n^{-1}\) because of \(\nabla f(x_\star ) \ne 0\), see (29). \(\square \)

Thus, Lemma 2 tells us that the asymptotic variance of importance sampling for \(\mu _n\) with the prior \(\mu _0\) as importance distribution grows like \(n^{d/2-1}\) as \(n\rightarrow \infty \) for a large class of integrands. Hence, its efficiency deteriorates like \(n^{d/2-1}\) for \(d\ge 3\) as the target measures \(\mu _n\) become more concentrated.

Laplace-based importance sampling We now consider the Laplace approximation \(\mathcal L_{\mu _n}\) as importance distribution which yields the following importance weight function

$$\begin{aligned} w_n(x) :=\frac{\mathrm {d}\mu _n}{\mathrm {d}\mathcal L_{\mu _n}}(x) = \frac{{\widetilde{Z}}_n}{Z_n} \exp (-n R_n(x)) \mathbf{1 }_{\mathrm {S}_0}(x), \qquad x \in \mathbb {R}^d, \end{aligned}$$

with \(R_n(x) = I_n(x) - {\widetilde{I}}_n(x) = I_n(x_n) - \frac{1}{2} \Vert x-x_n\Vert ^2_{C_n^{-1}}\) for \(x\in \mathrm {S}_0\). In order to ensure \(w_n \in L^2_{\mathcal L_{\mu _n}}(\mathbb {R})\) we need that

$$\begin{aligned} {\varvec{\mathbb E}}_{\mathcal L_{\mu _n}} \left[ \exp (-2n R_n) \right] = \frac{1}{{\widetilde{Z}}_n} \int _{\mathrm {S}_0} \exp (- n[2I_n(x) - {\widetilde{I}}_n(x)])\ \mathrm {d}x < \infty . \end{aligned}$$

Despite pathological counterexamples a sound requirement for \(w_n \in L^2_{\mathcal L_{\mu _n}}(\mathbb {R})\) is that

$$\begin{aligned} {\lim _{\Vert x\Vert \rightarrow \infty } 2I_n(x) - {\widetilde{I}}_n(x) = +\infty ,} \end{aligned}$$

for example by assuming that there exist \(\delta , c_1 > 0\), \(c_0 > 0\), and \(n_0 \in \mathbb {N}\) such that

$$\begin{aligned} {I_n(x) \ge c_1 \Vert x\Vert ^{2+\delta } + c_0,} \qquad \forall x \in \mathrm {S}_0\ \forall n\ge n_0. \end{aligned}$$

If the Lebesgue density \(\pi _0\) of \(\mu _0\) is bounded, then (33) is equivalent to the existence of \(n_0\) and a \({\widetilde{c}}_0\) such that

$$\begin{aligned} {\varPhi _n(x) \ge c_1 \Vert x\Vert ^{2+\delta } + {\widetilde{c}}_0,} \qquad \forall x \in \mathrm {S}_0\ \forall n\ge n_0. \end{aligned}$$

Unfortunately, condition (33) is not enough to ensure a well-behaved asymptotic variance \(\sigma ^2_{\mu _n,\mathcal L_{\mu _n}}(f)\) as \(n\rightarrow \infty \), since

$$\begin{aligned} \Vert w_n\Vert _{L^\infty } = \frac{{\widetilde{Z}}_n}{Z_n} \exp (- n \min _{x\in \mathrm {S}_0} R_n(x)). \end{aligned}$$

Although, we know due to (26) that \(\frac{{\widetilde{Z}}_n}{Z_n} \rightarrow 1\) as \(n\rightarrow \infty \), the supremum norm of the importance weight \(w_n\) of Laplace-based importance sampling will explode exponentially with n if \(\min _x R_n(x) < 0\). This can be sharpened to proving that even the asymptotic variance of Laplace-based importance sampling w.r.t. \(\mu _n\) as in (22) deteriorates exponentially as \(n\rightarrow \infty \) for many functions \(f:\mathbb {R}^d\rightarrow \mathbb {R}\) if

$$\begin{aligned} \exists x \in \mathrm {S}_0 :\varPhi (x) < \frac{1}{2} \varPhi (x_\star ) + \frac{1}{4} \Vert x - x_\star \Vert ^2_{H^{-1}_\star } \end{aligned}$$

by means of Theorem 1 applied to

$$\begin{aligned} \int _{\mathbb {R}^d} (f(x) - {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] )^2 \ \exp (- n[2I_n(x) - {\widetilde{I}}_n(x)]) \ \mathrm {d}x. \end{aligned}$$

This means, except when \(\varPhi \) is basically strongly convex, the asymptotic variance of Laplace-based important sampling can explode exponentially or not even exist as n increases. However, in the good case, so to speak, we obtain the following.

Proposition 2

Consider the measures \(\mu _n\) as in (22) with \(\varPhi _n = \varPhi - \iota _n\) and \(\pi _0\) satisfying Assumptions 1 and 2. If there exist an \(n_0\in \mathbb {N}\) such that for all \(n\ge n_0\) we have

$$\begin{aligned} I_n(x) \ge I_n(x_n) + \frac{1}{2} (x-x_n)^\top \nabla ^2 I_n(x_n) (x-x_n) \qquad \forall x \in \mathrm {S}_0, \end{aligned}$$

then for any \(f \in L^2_{\mu _0}(f)\)

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{\sigma ^2_{\mu _n,\mathcal L_{\mu _n}}(f)}{\mathrm {Var}_{\mu _n}(f)} = 1. \end{aligned}$$


The assumption (34) ensures that \(R_n(x) = I_n(x) - {\widetilde{I}}_n(x) \ge 0\) for each \(x\in \mathrm {S}_0\). Thus,

$$\begin{aligned} \Vert w_n\Vert _{L^\infty } = \frac{{\widetilde{Z}}_n}{Z_n} \end{aligned}$$

and the assertion follows by (31) and the fact that \(\lim _{n\rightarrow \infty } \frac{{\widetilde{Z}}_n}{Z_n} = 1\) due to (26). \(\square \)

Condition (34) is for instance satisfied, if \(I_n\) is strongly convex with a constant \(\gamma \ge \lambda _{\min }(\nabla ^2 I_n(x_n))\) where the latter denotes the smallest eigenvalue of the positive definite Hessian \(\nabla ^2 I_n(x_n)\). However, this assumption or even (34) is quite restrictive and, probably, hardly fulfilled for many interesting applications. Moreover, the success in practice of Laplace-based importance sampling is well-documented. How come that despite a possible infinite asymptotic variance Laplace-based importance sampling performs that well? In the following we refine our analysis and exploit the fact that the Laplace approximation concentrates around the minimizer of \(I_n\). Hence, with an increasing probability samples drawn from the Laplace approximation are in a small neighborhood of the minimizer. Thus, if \(I_n\) is, e.g., only locally strongly convex—which the assumptions of Theorem 2 actually imply—then with a high probability the mean squared error might be small.

We clarify these arguments in the following and present a positive result for Laplace-based importance sampling under mild assumptions but for a weaker error criterion than the decay of the mean squared error.

First we state a concentration result for N samples drawn from \(\mathcal L_{\mu _n}\) which is an immediate consequence of Proposition 4.

Proposition 3

Let \(N\in \mathbb {N}\) be arbitrary and let \(X^{(n)}_i \sim \mathcal L_{\mu _n}\) be i.i.d. where \(i = 1,\ldots ,N\). Then, for a sequence of radii \(r_n \ge r_0 n^{-q}>0\), \(n\in \mathbb {N}\), with \(q \in (0, 1/2)\) we have

$$\begin{aligned} \mathbb {P}\left( \max _{i = 1,\ldots , N} \Vert X^{(n)}_i - x_n\Vert \le r_n \right) = 1 - \mathrm e^{- c_0 N n^{1-2q}} \xrightarrow [n\rightarrow \infty ]{} 1. \end{aligned}$$

Remark 6

In the following we require expectations w.r.t. restrictions of the measures \(\mu _n\) in (22) to shrinking balls \(B_{r_n}(x_n)\). To this end, we note that the statements of Theorem 1 also hold true for shrinking domains \(D_n = B_{r_n}(x_\star )\) with \(r_n = r_0n^{-q}\) as long as \(q <1/2\). This can be seen from the proof of Theorem 1 in [44, Section IX.5]. In particular, all coefficients in the asymptotic expansion for \(\int _{D_n} f(x) \exp (-n \varPhi (x)) \mathrm {d}x\) with sufficiently smooth f are the same as for \(\int _{D} f(x) \exp (-n \varPhi (x)) \mathrm {d}x\) and the difference between both integrals decays for increasing n like \(\exp (-c n^\epsilon )\) for an \(\epsilon >0\) and \(c>0\). Concerning the balls \(B_{r_n}(x_n)\) with decaying radii \(r_n = r_0 n^{-q}\), \(q\in [0,1/2)\), we have due to \(\Vert x_n - x_\star \Vert \in \mathcal O(n^{-1})\)—see Remark 5—that \(B_{r_n/2}(x_\star ) \subset B_{r_n}(x_n) \subset B_{2r_n}(x_\star )\) for sufficiently large n. Thus, the facts for \(\mu _n\) as in (22) stated in the preliminaries before Sect. 3.1 do also apply to the restrictions of \(\mu _n\) to \(B_{r_n}(x_n)\) with \(r_n = r_0 n^{-q}\), \(q\in [0,1/2)\). In particular, the difference between \( {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] \) and \( {\varvec{\mathbb E}}_{\mu _n} \left[ f\ | \ B_{r_n}(x_n) \right] \) decays faster than any negative power of n as \(n\rightarrow \infty \).

The next result shows that the mean absolute error of the Laplace-based importance sampling behaves like \(n^{-(3q-1)}\) conditional on all N samples belonging to shrinking balls \(B_{r_n}(x_n)\) with \(r_n = r_0 n^{-q}\), \(q \in (1/3, 1/2)\).

Lemma 3

Consider the measures \(\mu _n\) in (22) and suppose they satisfy the assumptions of Theorem 2. Then, for any \(f \in C^2(\mathbb {R}^d, \mathbb {R}) \cap L^2_{\mu _0}(\mathbb {R})\) there holds for the error

$$\begin{aligned} e_{n,N}(f) :=\left| \mathrm {IS}^{(N)}_{\mu _n, \mathcal L_{\mu _n}}(f) - {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] \right| , \end{aligned}$$

of the Laplace-based importance sampling with \(N\in \mathbb {N}\) samples that

$$\begin{aligned} {\varvec{\mathbb E}} \left[ e_{n,N}(f) \ \big | \ X^{(n)}_1, \ldots , X^{(n)}_N \in B_{r_n}(x_n) \right] \in \mathcal O(n^{-(3q-1)}), \end{aligned}$$

where \(r_n = r_0 n^{-q}\) with \(q \in (1/3,1/2)\).


We start with

$$\begin{aligned} e_{n,N}(f)&:=\left| \mathrm {IS}^{(N)}_{\mu _n, \mathcal L_{\mu _n}}(f) - {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] \right| \\&\le \left| \mathrm {IS}^{(N)}_{\mu _n, \mathcal L_{\mu _n}}(f) - {\varvec{\mathbb E}}_{\mu _n} \left[ f\ | \ B_{r_n}(x_n) \right] \right| + \left| {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] - {\varvec{\mathbb E}}_{\mu _n} \left[ f\ | \ B_{r_n}(x_n) \right] \right| . \end{aligned}$$

The second term decays subexponentially w.r.t. n, see Remark 6. Hence, it remains to prove that

$$\begin{aligned} {\varvec{\mathbb E}} \left[ \left. \left| \mathrm {IS}^{(N)}_{\mu _n, \mathcal L_{\mu _n}}(f) - {\varvec{\mathbb E}}_{\mu _n} \left[ f\ | \ B_{r_n}(x_n) \right] \right| \ \right| \ X^{(n)}_1, \ldots , X^{(n)}_N \in B_{r_n}(x_n) \right] \in \mathcal O(n^{- (3q-1)}). \end{aligned}$$

To this end, we write the self-normalizing Laplace-based importance sampling estimator as

$$\begin{aligned} \mathrm {IS}^{(N)}_{\mu _n, \mathcal L_{\mu _n}}(f)&= \frac{\frac{1}{N} \sum _{i=1}^N {\widetilde{w}}_n (X^{(n)}_i) f(X^{(n)}_i) }{\frac{1}{N} \sum _{i=1}^N {\widetilde{w}}_n (X^{(n)}_i)} = H_{n,N} \ S_{n,N} \end{aligned}$$

where we define

$$\begin{aligned} H_{n,N} :=\frac{Z_n}{{\widetilde{Z}}_n} \ \frac{1}{\frac{1}{N} \sum _{i=1}^N {\widetilde{w}}_n (X^{(n)}_i)}, \qquad S_{n,N} = \frac{1}{N} \sum _{i=1}^N w_n (X^{(n)}_i) f(X^{(n)}_i), \end{aligned}$$

and recall that \(w_n\) is as in (32) and \({\widetilde{w}}_n(x) = \exp (-n R_n(x))\). Notice that

$$\begin{aligned} {\varvec{\mathbb E}} \left[ S_{n,N} \right] = {\varvec{\mathbb E}}_{\mu _n} \left[ f \right] , \quad {\varvec{\mathbb E}} \left[ S_{n,N}\ | \ X^{(n)}_1, \ldots , X^{(n)}_N \in B_{r_n}(x_n) \right] = {\varvec{\mathbb E}}_{\mu _n} \left[ f \ | \ B_{r_n}(x_n) \right] . \end{aligned}$$

Let us denote the event that \(X^{(n)}_1, \ldots , X^{(n)}_N \in B_{r_n}(x_n)\) by \(A_{n,N}\) for brevity. Then,

$$\begin{aligned}&{\varvec{\mathbb E}} \left[ \left. \left| \mathrm {IS}^{(N)}_{\mu _n, \mathcal L_{\mu _n}}(f) - {\varvec{\mathbb E}}_{\mu _n} \left[ f\ | \ B_{r_n}(x_n) \right] \right| \ \right| \ A_{n,N} \right] \\&\quad = \quad {\varvec{\mathbb E}} \left[ \left. \left| H_{n,N} S_{n,N} - {\varvec{\mathbb E}} \left[ S_{n,N} \ | \ A_{n,N} \right] \right| \ \right| \ A_{n,N} \right] \\&\quad \le {\varvec{\mathbb E}} \left[ \left. \left| S_{n,N} - {\varvec{\mathbb E}} \left[ S_{n,N} \ | \ A_{n,N} \right] \right| \ \right| \ A_{n,N} \right] \\&\quad + {\varvec{\mathbb E}} \left[ \left. \left| \left( H_{n,N} -1\right) S_{n,N} \right| \ \right| \ A_{n,N} \right] \end{aligned}$$

The first term in the last line can be bounded by the conditional variance of \(S_{n,N}\) given \(X^{(n)}_1, \ldots , X^{(n)}_N \in B_{r_n}(x_n)\), i.e., by Jensen’s inequality we obtain

$$\begin{aligned} {\varvec{\mathbb E}} \left[ \left. \left| S_{n,N} - {\varvec{\mathbb E}} \left[ S_{n,N} \ | \ A_{n,N} \right] \right| \ \right| \ A_{n,N} \right] ^2&\le \mathrm {Var}( \left. S_{n,N} \ \right| \ A_{n,N} )\\&= \frac{1}{N} \mathrm {Var}_{\mu _n}\left( \left. f \ \right| \ B_{r_n}(x_n) \right) \in \mathcal O(n^{-1}), \end{aligned}$$

see Remark 6 and the preliminaries before Sect. 3.1. Thus,

$$\begin{aligned} {\varvec{\mathbb E}} \left[ \left. \left| S_{n,N} - {\varvec{\mathbb E}} \left[ S_{n,N} \ | \ A_{n,N} \right] \right| \ \right| \ A_{n,N} \right] \in \mathcal O(n^{-1/2}) \, \subset \mathcal O(n^{- (3q-1)}) \end{aligned}$$

and it remains to study if \( {\varvec{\mathbb E}} \left[ \left. \left| \left( H_{n,N} -1\right) S_{n,N} \right| \ \right| \ A_{n,N} \right] \in \mathcal O(n^{- (3q-1)})\). Given that \(X^{(n)}_1, \ldots , X^{(n)}_N \in B_{r_n}(x_n)\) we can bound the values of the random variable \(H_{n,N}\) for sufficiently large n: first, we have \(Z_n / {\widetilde{Z}}_n = 1 + \mathcal O(n^{-1})\), see (26), and second

$$\begin{aligned} \exp \left( - n \max _{|x-x_n| \le r_n} |R_n(x)| \right) \le \frac{1}{\frac{1}{N} \sum _{i=1}^N {\widetilde{w}}_n (X^{(n)}_i)} \le \exp \left( n \max _{|x-x_n| \le r_n} |R_n(x)| \right) . \end{aligned}$$

Since \(|R_n(x)| \le c_3 \Vert x-x_n\Vert ^3\) for \(|x-x_n| \le r_n\) due to the local boundedness of the third derivative of \(I_n\) and \(r_n = r_0n^{-q}\), we have that

$$\begin{aligned} \exp \left( -c n^{1-3q} \right) \le \frac{1}{\frac{1}{N} \sum _{i=1}^N {\widetilde{w}}_n (X^{(n)}_i)} \le \exp \left( c n^{1-3q} \right) , \end{aligned}$$

where \(c>0\). Thus, there exist \(\alpha _n \le 1 \le \beta _n\) with \(\alpha _n = \mathrm e^{- c n^{1-3q} } (1 + \mathcal O(n^{-1}))\) and \(\beta _n \sim \mathrm e^{cn^{1-3q}} (1 + \mathcal O(n^{-1}))\) such that

$$\begin{aligned} \mathbb {P}\left( \alpha _n \le H_{n,N} \le \beta _n \ | \ X^{(n)}_1, \ldots , X^{(n)}_N \in B_{r_n}(x_n) \right) = 1. \end{aligned}$$

Since \(\mathrm e^{\pm c n^{1-3q} } (1 + \mathcal O(n^{-1})) = 1 \pm cn^{1-3q} + \mathcal O(n^{-1})\) we get that for sufficiently large n there exists a \({\widetilde{c}} >0\) such that

$$\begin{aligned} \mathbb {P}\left( \left| H_{n,N} -1 \right| \le c n^{1-3q} + {\widetilde{c}} n^{-1} \ | \ X^{(n)}_1, \ldots , X^{(n)}_N \in B_{r_n}(x_n)\right) = 1. \end{aligned}$$


$$\begin{aligned} {\varvec{\mathbb E}} \left[ \left. \left| \left( H_{n,N} -1\right) S_{n,N} \right| \ \right| \ A_{n,N} \right]&\le \left( c n^{1-3q} + {\widetilde{c}} n^{-1} \right) {\varvec{\mathbb E}} \left[ \left. |S_{n,N}| \ \right| \ A_{n,N} \right] \\&\in \mathcal O(n^{- (3q-1)} ), \end{aligned}$$

since \( {\varvec{\mathbb E}} \left[ |S_{n,N}| \ | \ A_{n,N} \right] \le {\varvec{\mathbb E}}_{\mu _n} \left[ |f| \ | \ B_{r_n(x_n)} \right] \) is uniformly bounded w.r.t. n. This concludes the proof. \(\square \)

We now present our main result for the Laplace-based importance sampling which states that the corresponding error decays in probability to zero as \(n\rightarrow \infty \) and the order of decay is arbitrary close to \(n^{-1/2}\).

Theorem 4

Let the assumptions of Lemma 3 be satisfied. Then, for any \(f \in C^2(\mathbb {R}^d, \mathbb {R}) \cap L^2_{\mu _0}(\mathbb {R})\) and each sample size \(N\in \mathbb {N}\) the error \(e_{n,N}(f)\) of Laplace-based importance sampling satisfies

$$\begin{aligned} n^{\delta } e_{n,N}(f) \xrightarrow [n\rightarrow \infty ]{\mathbb {P}} 0, \qquad \delta \in [0,1/2). \end{aligned}$$


Let \(0\le \delta < 1/2\) and \(\epsilon > 0\) be arbitrary. We need to show that

$$\begin{aligned} \lim _{n\rightarrow \infty } \mathbb {P}\left( n^{\delta } e_{n,N}(f) > \epsilon \right) = 0. \end{aligned}$$

Again, let us denote the event that \(X^{(n)}_1, \ldots , X^{(n)}_N \in B_{r_n}(x_n)\) by \(A_{n,N}\) for brevity. By Proposition 3 we obtain for radii \(r_n = r_0 n^{-q}\) with \(q \in (1/3, 1/2)\) that

$$\begin{aligned} \mathbb {P}\left( n^{\delta } \ e_{n,N}(f) \le \epsilon \right)&\ge \mathbb {P}\left( n^{\delta } e_{n,N}(f) \le \epsilon \text { and } X_1,\ldots , X_N \in B_{r_n}(x_n)\right) \\&= \mathbb {P}\left( n^{\delta }e_{n,N}(f) \le \epsilon \ | \ A_{n,N} \right) \ \mathbb {P}(A_{n,N}) \\&\ge \mathbb {P}\left( n^{\delta }e_{n,N}(f) \le \epsilon \ | \ A_{n,N} \right) \ \left( 1 - C_N \mathrm e^{- c_0 N n^{1-2q}}\right) . \end{aligned}$$

The second term on the righthand side in the last line obviously tends to 1 exponentially as \(n\rightarrow \infty \). Thus, it remains to prove that

$$\begin{aligned} \lim _{n\rightarrow \infty } \mathbb {P}\left( n^{\delta } e_{n,N}(f) \le \epsilon \ | \ X_1,\ldots , X_N \in B_{r_n}(x_n) \right) = 1 \end{aligned}$$

To this end, we apply a conditional Markov inequality for the positive random variable \(e_{n,N}(f)\), i.e.,

$$\begin{aligned} \mathbb {P}\left( n^{\delta } e_{n,N}(f) > \epsilon \ | \ A_{n,N}\right) \le \frac{n^{\delta }}{\epsilon } {\varvec{\mathbb E}} \left[ e_{n,N}(f) \ | \ A_{n,N} \right] \in \mathcal O\left( n^{\delta - \min (3q-1, 1/2)} \right) \end{aligned}$$

where we used Lemma 3. Choosing \(q \in (1/3, 1/2)\) such that \(q > \frac{1+\delta }{3} \in [1/3, 1/2) \) yields the statement. \(\square \)

Quasi-Monte Carlo integration

We now want to approximate integrals as in (20) w.r.t. measures \(\mu _n(\mathrm {d}x) \propto \exp (-n\varPhi (x)) \mu _0(\mathrm {d}x)\) as in (22) by Quasi-Monte Carlo methods.

These will be used to estimate the ratio \(Z'_n/Z_n\) by separately approximating the two integrals \(Z'_n\) and \(Z_n\) in (20). The preconditioning strategy using the Laplace approximation will be explained exemplarily for Gaussian and uniform priors, two popular choices for Bayesian inverse problems.

We start the discussion by first focusing on a uniform prior distribution \(\mu _0={\mathcal {U}}([-\frac{1}{2}, \frac{1}{2}]^d)\). The integrals \(Z'_n\) and \(Z_n\) are then

$$\begin{aligned} Z'_n=\int _{[-\frac{1}{2}, \frac{1}{2}]^d} f(x)\varTheta _n(x)\mathrm {d}x, \qquad Z_n=\int _{[-\frac{1}{2}, \frac{1}{2}]^d} \varTheta _n(x)\mathrm {d}x, \end{aligned}$$

where we set \(\varTheta _n(x) :=\exp (-n\varPhi (x))\) for brevity.

We consider Quasi-Monte Carlo integration based on shifted Lattice rules: an N-point Lattice rule in the cube \([-\frac{1}{2}, \frac{1}{2}]^d\) is based on points

$$\begin{aligned} x_i=\mathrm {frac}\Big (\frac{iz}{N}+\varDelta \Big ) -\frac{1}{2}, \quad i=1,\ldots ,N, \end{aligned}$$

where \(z \in \{1,\ldots , N-1\}^d\) denotes the so-called generating vector, \(\varDelta \) is a uniformly distributed random shift on \([-\frac{1}{2}, \frac{1}{2}]^d\) and \(\mathrm {frac}\) denotes the fractional part (component-wise). These randomly shifted points provide unbiased estimators

$$\begin{aligned} Z'_{n,QMC} :=\frac{1}{N}\sum _{i=1}^N f(x_i)\varTheta (x_i), \qquad Z_{n,QMC} :=\frac{1}{N}\sum _{i=1}^N \varTheta (x_i) \end{aligned}$$

of the two integrals \(Z'_n\) and \(Z_n\) in (35). Under the assumption that the quantity of interest \(f:\mathbb {R}^d\rightarrow \mathbb {R}\) is linear and bounded, we can focus in the following on the estimation of the normalization constant \(Z_n\), the results can be then straightforwardly generalized to the estimation of \(Z'_n\). For the estimator \(Z_{n,QMC}\) we have the following well-known error bound.

Theorem 5

[12, Thm. 5.10] Let \(\gamma = \{\gamma _{\varvec{\nu }}\}_{{\varvec{\nu }}\subset \{1,\ldots ,d\}}\) denote POD (product and order dependent) weights of the form \(\gamma _{\varvec{\nu }}=\alpha _{|{\varvec{\nu }}|}\prod _{j\in {\varvec{\nu }}}\beta _j\) specified by two sequences \(\alpha _0=\alpha _1=1, \alpha _2,\ldots \ge 0\) and \(\beta _1\ge \beta _2\ge \ldots >0\) for \({\varvec{\nu }}\subset \{1,\ldots ,d\}\) and \(|{\varvec{\nu }}|=\# {\varvec{\nu }}\). Then, a randomly shifted Lattice rule with \(N=2^m, m\in \mathbb N\), can be constructed via a component-by-component algorithm with POD weights at costs of \({\mathcal {O}}(dN\log N+d^2 N)\) operations, such that for sufficiently smooth \(\varTheta :[-\frac{1}{2}, \frac{1}{2}]^d \rightarrow [0,\infty )\)

$$\begin{aligned} \mathbb E_{\varDelta }[(Z_n-Z_{n,QMC})^2]^{1/2}\le \left( 2\sum _{\emptyset \ne {\varvec{\nu }}\subset \{1,\ldots ,d\}}\gamma _{\varvec{\nu }}^\kappa \left( \frac{2\zeta (2\kappa )}{(2\pi ^2)^\kappa }\right) ^{|{\varvec{\nu }}|} \right) ^{\frac{1}{2\kappa }} \Vert \varTheta _n\Vert _\gamma \ \ N^{- \frac{1}{2\kappa }} \end{aligned}$$

for \(\kappa \in (1/2,1]\) with

$$\begin{aligned} \Vert \varTheta _n\Vert _\gamma ^2 :=\sum _{{\varvec{\nu }}\subset \{1,\ldots ,d\}} \frac{1}{\gamma _{\varvec{\nu }}}\int _{[-\frac{1}{2}, \frac{1}{2}]^{|{\varvec{\nu }}|}}\left( \int _{[-\frac{1}{2}, \frac{1}{2}]^{d-|{\varvec{\nu }}|}}\frac{\partial ^{|{\varvec{\nu }}|}\varTheta _n}{\partial x_{\varvec{\nu }}}(x)\mathrm d x_{{1:d}{\setminus } {\varvec{\nu }}}\right) ^2\mathrm d x_{\varvec{\nu }} \end{aligned}$$

and \(\zeta (a):=\sum _{k=1}^\infty k^{-a}\).

The norm \(\Vert \varTheta _n\Vert _\gamma \) in the convergence analysis depends on n, in particular, it can grow polynomially w.r.t. the concentration level n of the measures \(\mu _n\) as we state in the next result.

Lemma 4

Let \(\varPhi :\mathbb {R}^d\rightarrow [0,\infty )\) satisfy the assumptions of Theorem 1 for \(p=2d\). Then, for the norm \(\Vert \varTheta _n\Vert _\gamma \) in the error bound in Theorem 5 there holds

$$\begin{aligned} \lim _{n\rightarrow \infty } n^{-d/4} \Vert \varTheta _n\Vert _\gamma > 0. \end{aligned}$$

The proof of Lemma 4 is rather technical and can be found in “Appendix B.3”. We remark that Lemma 4 just tells us that the root mean squared error estimate for QMC integration based on the prior measure explodes like \(n^{d/4}\). This does in general not indicate that the error itself explodes; in fact the QMC integration error for the normalization constant is bounded by 1 in our setting. Nonetheless, Lemma 4 indicates that a naive Quasi-Monte Carlo integration based on the uniform prior \(\mu _0\) is not suitable for highly concentrated target or posterior measures \(\mu _n\). We subsequently propose and study a Quasi-Monte Carlo integration based on the Laplace approximation \(\mathcal L_{\mu _n}\).

Laplace-based Quasi-Monte Carlo To stabilize the numerical integration for concentrated \(\mu _n\), we propose a preconditioning based on the Laplace approximation, i.e., an affine rescaling according to the mean and covariance of \(\mathcal L_{\mu _n}\). In the uniform case, the functionals \(I_n\) are independent of n. The computation of the Laplace approximation requires therefore only one optimization to solve for \(x_n = x_\star = \mathop {\mathrm{argmin}}\nolimits _{x\in [-\frac{1}{2}, \frac{1}{2}]^d} \varPhi (x)\). In particular, the Laplace approximation of \(\mu _n\) is given by \(\mathcal L_{\mu _n}= {\mathcal {N}}(x_\star , \frac{1}{n} H_{\star }^{-1})\) where \(H_{\star }\) denotes the positive definite Hessian \(\nabla ^2 \varPhi (x_\star )\). Hence, \(H_\star \) allows for an orthogonal diagonalization \(H_\star = QDQ^\top \) with orthogonal matrix \(Q\in \mathbb R^{d\times d}\) and diagonal matrix \(D={{\,\mathrm{diag}\,}}(\lambda _1,\ldots \lambda _d)\in \mathbb R^{d\times d}\), \(\lambda _1\ge \cdots \ge \lambda _d>0\).

We now use this decomposition in order to construct an affine transformation which reverses the increasing concentration of \(\mu _n\) and yields a QMC approach robust w.r.t. n. This transformation is given by

$$\begin{aligned} g_n(x) :=x_\star + \sqrt{\frac{2|\ln \tau |}{n}} QD^{-\frac{1}{2}}x, \qquad x \in [-\frac{1}{2}, \frac{1}{2}]^d, \end{aligned}$$

where \(\tau \in (0,1)\) is a truncation parameter. The idea of the transformation \(g_n\) is to zoom into the parameter domain and thus, to counter the concentration effect. The domain will then be truncated to \(G_n :=g_n([-\frac{1}{2}, \frac{1}{2}]^d) \subset [-\frac{1}{2}, \frac{1}{2}]^d\) and we consider

$$\begin{aligned} {\hat{Z}}_n :=\int _{G_n} \varTheta _n(x)\ \mathrm {d}x = C_{\text {trans}, n} \int _{[-\frac{1}{2}, \frac{1}{2}]^d} \varTheta _n(g_n(x))\ \mathrm {d}x. \end{aligned}$$

The determinant of the Jacobian of the transformation \(g_n\) is given by \( \det (\nabla g_n(x)) \equiv C_{\text {trans}, n} = \left( \frac{2|\ln \tau |}{n}\right) ^{\frac{d}{2}} \sqrt{\det (H_\star )} \sim c_\tau n^{-d/2}\). We will now explain how the parameter \(\tau \) effects the truncation error. For given \(\tau \in (0,1)\), the Laplace approximation is used to determine the truncation effect:

$$\begin{aligned} \int _{G_n} \ \mathcal L_{\mu _n}(\mathrm {d}x)&= \frac{C_{\text {trans}, n}}{{\widetilde{Z}}_n} \int _{[-\frac{1}{2}, \frac{1}{2}]^d} \exp \left( -\frac{n}{2} \Vert g_n(x) - x_\star \Vert ^2_{H_\star } \right) \ \mathrm {d}x\\&= \left( \frac{|\ln \tau |}{\pi }\right) ^{d/2} \int _{[-\frac{1}{2}, \frac{1}{2}]^d} \exp \left( - |\ln \tau | x^2 \right) \ \mathrm {d}x\\&= \left( \frac{|\ln \tau |}{\pi }\right) ^{d/2} \left( \frac{\sqrt{\pi }\mathrm {erf}(0.5 \sqrt{|\ln \tau }|)}{\sqrt{|\ln \tau }|)}\right) ^d = \mathrm {erf}(0.5 \sqrt{|\ln \tau }|)^d. \end{aligned}$$

Thus, since due to the concentration effect of the Laplace approximation we have \(\int _{\mathrm {S}_0} \ \mathcal L_{\mu _n}(\mathrm {d}x) \rightarrow 1\) exponentially with n, we get

$$\begin{aligned} \int _{\mathrm {S}_0{\setminus } G_n} \ \mathcal L_{\mu _n}(\mathrm {d}x) \le 1 - \mathrm {erf}(0.5 \sqrt{|\ln \tau }|)^d, \end{aligned}$$

thus, the truncation error \(\int _{\mathrm {S}_0{\setminus } G_n} \ \mathcal L_{\mu _n}(\mathrm {d}x)\) becomes arbitrarily small for sufficiently small \(\tau \ll 1\), since \(\mathrm {erf}(t) \rightarrow 1\) as \(t\rightarrow 1\). If we apply now QMC integration using shifted Lattice rule in order to compute the integral over \([-\frac{1}{2}, \frac{1}{2}]^d\) on the righthand side of (38), we obtain the following estimator for \({\hat{Z}}_n\) in (38):

$$\begin{aligned} {\hat{Z}}_{n,QMC} :=\frac{C_{\text {trans}, n}}{N}\sum _{i=1}^N \varTheta (g_n(x_i)) \end{aligned}$$

with \(x_i\) as in (36). Concerning the norm \(\Vert \varTheta _n\circ g_n \Vert _\gamma \) appearing in the error bound for \(|{\hat{Z}}_n - {\hat{Z}}_{n,QMC}|\) we have now the following result.

Lemma 5

Let \(\varPhi :\mathbb {R}^d\rightarrow [0,\infty )\) satisfy the assumptions of Theorem 1 for \(p=2d\). Then, for the norm \(\Vert \varTheta _n\circ g_n\Vert _\gamma \) with \(g_n\) as above there holds

$$\begin{aligned} \Vert \varTheta _n\circ g_n \Vert _\gamma \in \mathcal O(1) \qquad \text { as } n\rightarrow \infty . \end{aligned}$$

Again, the proof is rather technical and can be found in “Appendix B.4”. This proposition yields now our main result.

Corollary 2

Given the assumptions of Lemma 5, a randomly shifted lattice rule with \(N=2^m, m\in \mathbb N\), can be constructed via a component-by-component algorithm with product and order dependent weights at costs of \({\mathcal {O}}(dN\log N+d^2 N)\) operations, such that for \(\kappa \in (1/2,1]\)

$$\begin{aligned} \mathbb E_{\varDelta }[(Z_n-{\hat{Z}}_{n,QMC})^2]^{1/2} \le n^{-\frac{d}{2}} \left( c_1 h(\tau )+ c_2 n^{-\frac{1}{2}}+ c_3 N^{- \frac{1}{2\kappa }}\right) \end{aligned}$$

with constants \(c_1,c_2, c_3>0\) independent of n and \(h(\tau ) = 1 - \mathrm {erf}(0.5 \sqrt{|\ln \tau }|)^d\).


The triangle inequality leads to a separate estimation of the domain truncation error of the integral w.r.t. the posterior and the QMC approximation error, i.e.

$$\begin{aligned} \mathbb E_{\varDelta }[(Z_n-{\hat{Z}}_{n,QMC})^2]^{1/2}\le & {} |Z_n - {\hat{Z}}_n| + \mathbb E_{\varDelta }[({\hat{Z}}_n-{\hat{Z}}_{n,QMC})^2]^{1/2}. \end{aligned}$$

The second term on the right hand side corresponds to the QMC approximation error. Thus, Theorem 5 and Lemma 5 imply

$$\begin{aligned} \mathbb E_{\varDelta }[({\hat{Z}}_n-{\hat{Z}}_{n,QMC})^2]^{1/2}\le c_3 n^{-\frac{d}{2}} N^{- \frac{1}{2\kappa }} , \quad \kappa \in (1/2,1], \end{aligned}$$

where the term \(n^{-\frac{d}{2}}\) is due to \(C_{\text {trans}, n} \sim c_\lambda n^{-\frac{d}{2}}\). The domain truncation error can be estimated similar to the proof of Lemma 1:

$$\begin{aligned} |Z_n -{\hat{Z}}_n|= & {} \big | \int _{\mathrm {S}_0{\setminus } G_n} \varTheta _n(x)\ \mathrm {d}x \big |\\= & {} \big |\int _{\mathrm {S}_0 {\setminus } G_n} \varTheta _n(x) \ \mathrm {d}x -\widetilde{Z_n} \int _{\mathrm {S}_0 {\setminus } G_n} \ \mathcal L_{\mu _n}(\mathrm {d}x) +\widetilde{Z}_n\int _{\mathrm {S}_0 {\setminus } G_n} \ \mathcal L_{\mu _n}(\mathrm {d}x) \Big |\\= & {} \big | \widetilde{Z}_n \int _{\mathrm {S}_0 {\setminus } G_n}e^{-n \varPhi (x)} e^{n \widetilde{\varPhi }(x)} \mathcal L_{\mu _n}(\mathrm {d}x)-\widetilde{Z}_n \int _{\mathrm {S}_0 {\setminus } G_n} \ \mathcal L_{\mu _n}(\mathrm {d}x)\\&\qquad +\widetilde{Z}_n\int _{\mathrm {S}_0 {\setminus } G_n} \ \mathcal L_{\mu _n}(\mathrm {d}x)\Big |\\\le & {} \widetilde{Z}_n\int _{\mathrm {S}_0 {\setminus } G_n} \big | e^{-n {\widetilde{R}}_n(x)} -1 \big | \mathcal L_{\mu _n}(\mathrm {d}x) +\widetilde{Z}_n\big |\int _{\mathrm {S}_0 {\setminus } G_n} \ \mathcal L_{\mu _n}(\mathrm {d}x)\Big |\\\le & {} {\widetilde{Z}}_n\int _{\mathrm {S}_0 } \big | e^{-n {\widetilde{R}}(x)} -1 \big | \mathcal L_{\mu _n}(\mathrm {d}x) + \widetilde{Z}_n \left( 1 - \mathrm {erf}(0.5 \sqrt{|\ln \tau }|)^d\right) \end{aligned}$$

where \(\widetilde{Z}_n=n^{-\frac{d}{2}}\sqrt{\det (2\pi H_\star ^{-1})}\). The result follows by the proof of Lemma 1. \(\square \)

Remark 7

In the case of Gaussian priors, the transformation simplifies to \(w=g_n(x)=x_\star + n^{\frac{1}{2}}QD^{-\frac{1}{2}}x\) due to the unboundedness of the prior support. However, to show an analogous result to Corollary 2, uniform bounds w.r. to n on the norm of the mixed first order derivatives of the preconditioned posterior density \(\varTheta _n(g_n(T^{-1}x))\) in a weighted Sobolev space, where \(T^{-1}\) denotes the inverse cumulative distribution function of the normal distribution, need to be proven. See [23] for more details on the weighted space setting in the Gaussian case. Then, a similar result to Corollary 2 follows straightforwardly from [23, Thm 5.2]. The numerical experiments shown in Sect. 3.3 suggest that we can expect a noise robust behavior of Laplace-based QMC methods also in the Gaussian case. This will be subject to future work.

Remark 8

Note that the QMC analysis in Theorem 5 can be extended to an infinite dimensional setting, cp. [23] and the references therein for more details. This opens up the interesting possibility to generalize the above results to the infinite dimensional setting and to develop methods with convergence independent of the number of parameters and independent of the measurement noise. Furthermore, higher order QMC methods can be used for cases with smooth integrands, cp. [10, 11, 13], leading to higher convergence rates than the first order methods discussed here. In the uniform setting, it has been shown in [38] that the assumptions on the first order derivatives (and also higher order derivatives) of the transformed integrand are satisfied for Bayesian inverse problems related to a class of parametric operator equations, i.e., the proposed approach leads to a robust performance w.r.t. the size of the measurement noise for integrating w.r.t. posterior measure resulting from this class of forward problems. The theoretical analysis of this setting will be subject to future work.

Remark 9

(Numerical quadrature) Higher regularity of the integrand allows to use higher order methods such as sparse quadrature and higher order QMC methods, leading to faster convergence rates. In the infinite dimensional Bayesian setting with uniform priors, we refer to [35, 36] for more details on sparse quadrature for smooth integrands. In the case of uniform priors, the methodology introduced above can be used to bound the quadrature error for the preconditioned integrand by the truncation error and the sparse grid error.


In this subsection we present two examples illustrating our previous theoretical results for importance sampling and quasi-Monte Carlo integration based on the prior measure \(\mu _0\) and the Laplace approximation \(\mathcal L_{\mu _n}\) of the target measure \(\mu _n\). Both examples are Bayesian inference or inverse problems where the first one uses a toy forward map and the second one is related to inference for a PDE model.

Algebraic Bayesian inverse problem

We consider inferring \(x \in [-\frac{1}{2}, \frac{1}{2}]^d\) for \(d=1,2,3,4\) based on a uniform prior \(\mu _0 = {\mathcal {U}}([-\frac{1}{2}, \frac{1}{2}]^d)\) and a realisation of the noisy observable of \(Y = \mathcal G(X) + \eta _n\) where \(X\sim \mu _0\) and the noise \(\eta _n \sim N(0, n^{-1} \varGamma _d)\), \(\varGamma _d = 0.1I_d\), are independent, and \(\mathcal G(x) = (\mathcal G_1(x),\ldots ,\mathcal G_d(x))\) with

$$\begin{aligned} \mathcal {G}_1(x) = \exp (x_1/5), \quad \mathcal {G}_2(x) = x_2 - x_1^2, \quad \mathcal {G}_3(x) = x_3, \quad \mathcal {G}_4(x) = 2x_4 + x_1^2, \end{aligned}$$

for \(x = (x_1,\ldots ,x_d)\). The resulting posterior measure \(\mu _n\) on \([-\frac{1}{2}, \frac{1}{2}]^d\) are of the form (22) with

$$\begin{aligned} \varPhi (x) = \frac{1}{2} \Vert y -\mathcal {G}(x)\Vert ^2_{\varGamma _d^{-1}}. \end{aligned}$$

We used \(y = \mathcal {G}(0.25\cdot {\textit{\textbf{1}}}_d)\) throughout where \({{\textit{\textbf{1}}}}_d = (1,\ldots ,1) \in \mathbb {R}^d\). We then compute the posterior expectation of the quantity of interest \(f(x) = x_1+\cdots +x_d\). To this end, we employ importance sampling and quasi-Monte Carlo integration based on \(\mu _0\) and the Laplace approximation \(\mathcal L_{\mu _n}\) as outlined in the precious subsections. We compare the output of these methods to a reference solution obtained by a brute-force tensor grid trapezoidal rule for integration. In particular, we estimate the root mean squared error (RMSE) of the methods and how it evolves as n increases.

Results for importance sampling In order to be sufficiently close to the asymptotic limit, we use \(N = 10^5\) samples for self-noramlized importance sampling. We run 1000 independent simulations of the importance sampling integration and compute the resulting empirical RMSE. In Fig. 3 we present the results for increasing n and various d for prior-based and Laplace-based importance sampling. We obtain a good match to the theoretical results, i.e., the RMSE for choosing the prior measure as importance distribution behaves like \(n^{d/4 - 1/2}\) in accordance to Lemma 2. Besides that the RMSE for choosing the Laplace approximation as importance distribution decays like \(n^{1/2}\) after a preasymptotic phase. This is relates to the statement of Theorem 4 where we have shown that the absolute errorFootnote 3 decays in probability like \(n^{1/2}\). Note that the assumptions of Proposition 2 are not satisfied for this example for all \(d=1,2,3,4\).

Fig. 3
figure 3

Growth and decay of the empirical RMSE of prior-based (left) and Laplace-based (right) importance sampling for the example in Sect. 3.3.1 for decaying noise level \(n^{-1}\) and various dimensions d

Results for quasi-Monte Carlo We use \(N = 2^{10}\) quasi-Monte Carlo points for prior- and Laplace-based QMC. For the Laplace-case we employ a truncation parameter of \(\tau = 10^{-6}\) and discard all transformed points outside of the domain \([-\frac{1}{2}, \frac{1}{2}]^d\). Again, we run 1000 random shift simulations for both QMC methods and estimate the empirical RMSE. However, for QMC we report the relative RMSE, since, for example, the decay of the normalization constant \(Z_n\in \mathcal O(n^{-d/2})\) dominates the growth of the absolute error of prior QMC integration for the normalization constant. In Fig. 4 and 5 we display the resulting relative RMSE for the quantity related integral \(Z_n'\), the normalization constant \(Z_n\), i.e.,

$$\begin{aligned} Z'_n = \int _{[-\frac{1}{2}, \frac{1}{2}]^d} f(x) \ \exp (-n \varPhi (x))\ \mu _0(\mathrm {d}x), \; Z_n = \int _{[-\frac{1}{2}, \frac{1}{2}]^d} 1 \ \exp (-n \varPhi (x))\ \mu _0(\mathrm {d}x), \end{aligned}$$

and the resulting ratio \(\frac{Z'_n}{Z_n}\) for increasing n and various d for prior-based and Laplace-based QMC. For prior-based QMC we observe for dimensions \(d \ge 2\) a algebraic growth of the relative error. In the previous subsection we have proven that the corresponding classical error bound will explode which does not necessary imply that the error itself explodes—as we can see for \(d=1\). However, this simple example shows that also the error will often grow algebraically with increasing n. For the Laplace-based QMC we observe on the other hand in Fig. 5 a decay of the relative empirical RMSE. By Corollary 2 we can expect that the relative errors stay bounded as \(n\rightarrow \infty \). This provides motivation for a further investigation. In particular, we will analysize the QMC ratio estimator for \(\frac{Z'_n}{Z_n}\) in a future work.

Fig. 4
figure 4

Empirical relative RMSE of prior-based quasi-Monte Carlo for the example in Sect. 3.3.1 for decaying noise level \(n^{-1}\) and various dimensions d

Fig. 5
figure 5

Empirical relative RMSE of Laplace-based quasi-Monte Carlo for the example in Sect. 3.3.1 for decaying noise level \(n^{-1}\) and various dimensions d

Bayesian inference for an elliptic PDE

In the following we illustrate the preconditioning ideas from the previous section by Bayesian inference with differential equations. To this end we consider the following model parametric elliptic problem

$$\begin{aligned} -\text{ div }({\hat{u}}_d \nabla q)=f \quad \text{ in } D:=[0,1], \ q=0 \quad \text{ in } \partial D, \end{aligned}$$

with \(f(t)=100\cdot t\), \(t\in [0,1]\), and diffusion coefficient

$$\begin{aligned} {\hat{u}}_d(t) = \exp \left( \sum _{k=1}^dx_k \ \psi _k(t) \right) , \qquad d\in \{1,2,3\}, \end{aligned}$$

where \(\psi _k(t)= \frac{0.1}{k} \sin (k\pi t)\) and the \(x_k \in \mathbb {R}\), \(k=1,\ldots ,d\), are to be inferred by noisy observations of the solution q at certain points \(t_j \in [0,1]\). For \(d=1,2\) these observations are taken at \(t_1 = 0.25\) and \(t_2 = 0.75\) and for \(d=3\) they are taken at \(t_j \in \{0.125,0.25,0.375,0.6125,0.75,0.875\}\). We suppose an additive Gaussian observational noise \(\eta \sim \mathcal N(0, \varGamma _n)\) with noise covariance \(\varGamma _n = n^{-1}\varGamma _{obs}\) and \(\varGamma _{obs}\in \mathbb {R}^{2\times 2}\) or \(\varGamma _{obs}\in \mathbb {R}^{7\times 7}\), respectively, specified later on. In the following we place a uniform and a Gaussian prior \(\mu _0\) on \(\mathbb {R}^d\) and would like to integrate w.r.t. the resulting posterior \(\mu _n\) on \(\mathbb {R}^d\) which is of the form (22) with

$$\begin{aligned} \varPhi (x) = \frac{1}{2} \Vert y -\mathcal G(x)\Vert ^2_{\varGamma ^{-1}_{obs}} \end{aligned}$$

where \(\mathcal {G}:\mathbb {R}^d \rightarrow \mathbb {R}^2\) for \(d=1,2\), and \(\mathcal {G}:\mathbb {R}^d \rightarrow \mathbb {R}^7\) for \(d=3\), respectively, denotes the mapping from the coefficients \(x:=(x_k)_{k=1}^d\) to the observations of the solution q of the elliptic problem above and the vector \(y \in \mathbb {R}^2\) or \(y\in \mathbb {R}^7\), respectively, denotes the observational data resulting from \(Y = \mathcal G(X) + \eta \) with \(\eta \) as above. Our goal is then to compute the posterior expectation (i.e., w.r.t. \(\mu _n\)) of the following quantity of interest \(f:\mathbb {R}^d\rightarrow \mathbb {R}\): f(u) is the value of the solution q of the elliptic problem at \(t=0.5\).

Uniform prior We place a uniform prior \(\mu _0 = {\mathcal {U}}([-\frac{1}{2}, \frac{1}{2}]^d)\) for \(d=1,2\) or 3 and choose \(\varGamma _{obs} = 0.01 I_2\) for \(d=1,2\) and \(\varGamma _{obs} = 0.01 I_7\) for \(d=3\). We display the resulting posteriors \(\mu _n\) for \(d=2\) in Fig. 6 illustrating the concentration effect of the posterior for various values of the noise scaling n and the resulting transformed posterior with \(\varPhi \circ g_n\) based on Laplace approximation. The truncation parameter is set to \(\tau =10^{-6}\). We observe the almost quadratic behavior of the preconditioned posterior, as expected from the theoretical results.

Fig. 6
figure 6

The first row shows the posterior distribution for various values of the noise scaling, the second row shows the corresponding preconditioned posteriors based on Laplace approximation, 2d test case, uniform prior distribution, \(\tau =10^{-6}\)

We are now interested in the numerical performance of the Importance Sampling and QMC method based on the prior distribution compared to the performance of the preconditioned versions based on Laplace approximation. The QMC estimators are constructed by an off-the-shelf generating vector (order-2 randomly shifted weighted Sobolev space), which can be downloaded from (exod2_base2_m20_CKN.txt). The reference solution used to estimate the error is based on a (tensor grid) trapezoidal rule with \(10^6\) points in 1D, \(4\times 10^6\) points in 2D in the original domain, i.e., the truncation error is quantified and in the transformed domain in 3D with \(10^6\) points. Figure 7 illustrates the robust behavior of the preconditioning strategy w.r.t. the scaling 1/n of the observational noise. Though we know from the theory that in the low dimensional case (1D, 2D), the importance sampling method based on the prior is expected to perform robust, we encounter numerical instabilities due to the finite number of samples used for the experiments. The importance sampling results are based on \(10^6\) sampling points, the QMC results on 8192 shifted lattice points with \(2^6\) random shifts.

Fig. 7
figure 7

The (estimated) root mean square error (RMSE) of the approximation of the quantity of interest for different noise levels (\(n=10^{2},\ldots ,10^{10}\)) using the Importance Sampling strategy and QMC method for \(d= 1,2,3\)

Figure 8 shows the RMSE of the normalization constant \(Z_n\) using the preconditioned QMC approach with respect to the noise scaling 1/n. We observe a numerical confirmation of the predicted dependence of the error w.r.t. the dimension (cp. Corollary 2).

Fig. 8
figure 8

The (estimated) root mean square error (RMSE) of the approximation of the normalization constant Z for different noise levels (\(n=10^{2},\ldots ,10^{10}\)) using the preconditioned QMC method for \(d=1,2,3\)

We remark that the numerical experiments for the ratio suggest a behavior \(n^{-1/2}\), i.e., a rate independent of the dimension d, of the RMSE for the preconditioned QMC approach, cp. Figure 7. This will be subject to future work.

Gaussian prior We choose as prior \(\mu _0 = {\mathcal {N}}(0,I_2)\) for the coefficients \(x=(x_1,x_2) \in \mathbb {R}^2\) for \({\hat{u}}_2\) in the elliptic problem above. For the noise covariance we set this time \(\varGamma _{obs}=I_2\). The performance of the prior based and preconditioned version of Importance Sampling is depicted in Fig. 9. Clearly, the Laplace approximation as a preconditioner improves the convergence behavior; we observe a robust behavior w.r.t. the noise level.

The convergence of the QMC approach is depicted in Fig. 10, showing a consistent behavior with the considerations in the previous section.

Fig. 9
figure 9

The (estimated) root mean square error (RMSE) of the approximation of the quantity of interest for different noise levels (\(n=10^{0},\ldots ,10^{8}\)) using the Importance Sampling strategy. The first row shows the result based on prior information (Gaussian prior), the second row the results using the Laplace approximation for preconditioning. The reference solution is based on a tensor grid Gauss–Hermite rule with \(10^4\) points for the preconditioned integrand using the Laplace approximation

Fig. 10
figure 10

The (estimated) root mean square error (RMSE) of the approximation of the quantity of interest for different noise levels (\(n=10^{0},\ldots ,10^{8}\)) using the QMC method (below). The first row shows the result based on prior information (Gaussian prior), the second row the results using the Laplace approximation for preconditioning. The reference solution is based on a tensor grid Gauss–Hermite rule with \(10^4\) points for the preconditioned integrand using the Laplace approximation

Conclusions and outlook to future work

This paper makes a number of contributions in the development of numerical methods for Bayesian inverse problems, which are robust w.r.t. the size of the measurement noise or the concentration of the posterior measure, respectively. We analyzed the convergence of the Laplace approximation to the posterior distribution in Hellinger distance. This forms the basis for the design of variance robust methods. In particular, we proved that Laplace-based importance sampling behaves well in the small noise or large data size limit, respectively. For uniform priors, Laplace-based QMC methods have been developed with theoretically and numerically proven errors decaying with the noise level or concentration of the measure, respectively. Some future directions of this work include the development of noise robust Markov chain Monte Carlo methods and the combination of dimension independent and noise robust strategies. This will require the study of the Laplace approximation in infinite dimensions in a suitable setting. Finally, we could study in more details the error in the ratio estimator using Laplace-based QMC methods. The use of higher order QMC methods has been proven to be a promising direction for a broad class of Bayesian inverse problems and the design of noise robust versions is an interesting and potentially fruitful research direction.