Abstract
Conditional particle filters (CPFs) are powerful smoothing algorithms for general nonlinear/nonGaussian hidden Markov models. However, CPFs can be inefficient or difficult to apply with diffuse initial distributions, which are common in statistical applications. We propose a simple but generally applicable auxiliary variable method, which can be used together with the CPF in order to perform efficient inference with diffuse initial distributions. The method only requires simulatable Markov transitions that are reversible with respect to the initial distribution, which can be improper. We focus in particular on random walk type transitions which are reversible with respect to a uniform initial distribution (on some domain), and autoregressive kernels for Gaussian initial distributions. We propose to use online adaptations within the methods. In the case of random walk transition, our adaptations use the estimated covariance and acceptance rate adaptation, and we detail their theoretical validity. We tested our methods with a linear Gaussian random walk model, a stochastic volatility model, and a stochastic epidemic compartment model with timevarying transmission rate. The experimental findings demonstrate that our method works reliably with little user specification and can be substantially better mixing than a direct particle Gibbs algorithm that treats initial states as parameters.
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1 Introduction
In statistical applications of general state space hidden Markov models (HMMs), commonly known also as state space models, it is often desirable to initialise the latent state of the model with a diffuse (uninformative) initial distribution (cf. Durbin and Koopman 2012). We mean by ‘diffuse’ the general scenario, where the first marginal of the smoothing distribution is highly concentrated relative to the prior of the latent Markov chain, which may also be improper.
The conditional particle filter (CPF) (Andrieu et al. 2010), and in particular its backward sampling variants (Whiteley 2010; Lindsten et al. 2014), has been found to provide efficient smoothing even with long data records, both empirically (e.g. Fearnhead and Künsch 2018) and theoretically (Lee et al. 2020). However, a direct application of the CPF to a model with a diffuse initial distribution will lead to poor performance, because most of the initial particles will ultimately be redundant, as they become drawn from highly unlikely regions of the state space.
There are a number of existing methods which can be used to mitigate this inefficiency. For simpler settings, it is often relatively straightforward to design proposal distributions that lead to an equivalent model, which no longer has a diffuse initial distribution. Indeed, if the first filtering distribution is already informative, its analytical approximation may be used directly as the first proposal distribution. The iteratively refined lookahead approach suggested by Guarniero et al. (2017) extends to more complicated settings, but can require careful tuning for each class of problems.
We aim here for a general approach, which does not rely on any problemspecific constructions. Such a general approach which allows for diffuse initial conditions with particle Markov chain Monte Carlo (MCMC) is to include the initial latent state of the HMM as a ‘parameter’. This was suggested by Murray et al. (2013) with the particle marginal Metropolis–Hastings (PMMH). The same approach is directly applicable also with the CPF (using particle Gibbs); see Fearnhead and Meligkotsidou (2016), who discuss general approaches based on augmentation schemes.
Our approach may be seen as an instance of the general ‘pseudoobservation’ framework of Fearnhead and Meligkotsidou (2016), but we are unaware of earlier works about the specific class of methods we focus on here. Indeed, instead of building the auxiliary variable from the conjugacy perspective as Fearnhead and Meligkotsidou (2016), our approach is based on Markov transitions that are reversible with respect to the initial measure of the HMM. This approach may be simpler to understand and implement in practice, and is very generally applicable. We focus here on two concrete cases: the ‘diffuse Gaussian‘ case, where the initial distribution is Gaussian with a relatively uninformative covariance matrix, and the ‘fully diffuse‘ case, where the initial distribution is uniform. We suggest online adaptation mechanisms for the parameters, which make the methods easy to apply in practice.
We start in Sect. 2 by describing the family of models we are concerned with, and the general auxiliary variable initialisation CPF that underlies all of our developments. We present the practical methods in Sect. 3. Section 4 reports experiments of the methods with three academic models and concludes with a realistic inference task related to modelling the COVID19 epidemic in Finland. We conclude with a discussion in Sect. 5.
2 The model and auxiliary variables
Our main interest is with HMMs having a joint smoothing distribution \(\pi \) of the following form:
where \(\ell \):u denotes the sequence of integers from \(\ell \) to u (inclusive), \(x_{1:T}\) denotes the latent state variables, and \(y_{1:T}\) the observations. Additionally, \(\pi \) may depend on (hyper)parameters \(\theta \), the dependence on which we omit for now, but return to later, in Sect. 3.4.
For the convenience of notation, and to allow for some generalisations, we focus on the Feynman–Kac form of the HMM smoothing problem (cf. Del Moral 2004), where the distribution of interest \(\pi \) is represented in terms of a \(\sigma \)finite measure \(M_1({\mathrm {d}}x_1)\) on the state space \({\mathsf {X}}\), Markov transitions \(M_2,\ldots ,M_T\) on \({\mathsf {X}}\) and potential functions \(G_k:{\mathsf {X}}^k\rightarrow [0,\infty )\) so that
The classical choice, the socalled ‘bootstrap filter’ (Gordon et al. 1993), corresponds to \(M_1({\mathrm {d}}x_1) = p(x_1) {\mathrm {d}}x_1\) and \(M_k(x_{k1}, {\mathrm {d}}x_k) = p(x_k\mid x_{k1}) {\mathrm {d}}x_k\), where ‘\({\mathrm {d}}x\)’ stands for the Lebesgue measure on \({\mathsf {X}}={\mathbb {R}}^d\), and \(G_k(x_{1:k}) = p(y_k\mid x_k)\), but other choices with other ‘proposal distributions’ \(M_k\) are also possible. Our main focus is when \(M_1\) is diffuse with respect to the first marginal of \(\pi \). We stress that our method accomodates also improper \(M_1\), such as the uniform distribution on \(\mathbb {R}^d\), as long as (2) defines a probability.
The key ingredient of our method is an auxiliary Markov transition, Q, which we can simulate from, and which satisfies the following:
Assumption 1
(\(M_1\)reversibility) The Markov transition probability Q is reversible with respect to the \(\sigma \)finite measure \(M_1\), or \(M_1\)reversible, if
for all measurable \(A,B\subset {\mathsf {X}}\).
We discuss practical ways to choose Q in Sect. 3. Assuming an \(M_1\)reversible Q, we define an augmented target distribution, involving a new ‘pseudostate’ \(x_0\) which is connected to \(x_1\) by Q:
It is clear by construction that \(\tilde{\pi }\) admits \(\pi \) as its marginal, and therefore, if we can sample \(x_{0:T}\) from \(\tilde{\pi }\), then \(x_{1:T}\sim \pi \).
Our method may be viewed as a particle Gibbs (Andrieu et al. 2010) which targets \(\tilde{\pi }\), regarding \(x_0\) as the ‘parameter’, and \(x_{1:T}\) the ‘latent state’, which are updated using the CPF. Algorithm 1 summarises the method, which we call the ‘auxiliary initialisation’ CPF (AICPF). Algorithm 1 determines a \(\pi \)invariant Markov transition \({\dot{x}}_{1:T} \rightarrow \tilde{X}_{1:T}^{(B_{1:T})}\); the latter output of the algorithm will be relevant later, when we discuss adaptation.
Line 1 of Algorithm 1 implements a Gibbs step sampling \(X_0\) conditional on \(X_{1:T}={\dot{x}}_{1:T}\), and lines 2–4 implement together a CPF targeting the conditional of \(X_{1:T}\) given \(X_0\). Line 3 runs what we call a ‘forward’ CPF, which is just a standard CPF conditional on the first state particles \(X_{1}^{(1:N)}\), detailed in Algorithm 2. Line 4 refers to a call of \(\textsc {PickPathAT}\) (Algorithm 3) for ancestor tracing as in the original work of Andrieu et al. (2010), or \(\textsc {PickPathBS}\) (Algorithm 4) for backward sampling (Whiteley 2010). \(\mathrm {Categ}(w^{(1:N)})\) stands for the categorical distribution, that is, \(A \sim \mathrm {Categ}(w^{(1:N)})\) if \(\Pr (A=i) = w^{(i)}\).
The ancestor tracing variant can be used when the transition densities are unavailable. However, our main interest here is with backward sampling, summarised in Algorithm 4 in the common case where the potentials only depend on two consecutive states, that is, \(G_k(x_{1:k}) = G_k(x_{k1:k})\), and the transitions admit densities \(M_k(x_{k1},{\mathrm {d}}x_k) = M_k(x_{k1},x_k) {\mathrm {d}}x_k\) with respect to some dominating \(\sigma \)finite measure ‘\({\mathrm {d}}x_k\)’.
We conclude with a brief discussion on the general method of Algorithm 1.

(i)
We recognise that Algorithm 1 is not new per se, in that it may be viewed just as a particle Gibbs applied for a specific auxiliary variable model. However, we are unaware of Algorithm 1 being presented with the present focus: with an \(M_1\)reversible Q, and allowing for an improper \(M_1\).

(ii)
Algorithm 1 may be viewed as a generalisation of the standard CPF. Indeed, taking \(Q(x_0,{\mathrm {d}}x_1) = M_1({\mathrm {d}}x_1)\) in Algorithm 1 leads to the standard CPF. Note that Line 1 is redundant in this case, but is necessary in the general case.

(iii)
In the case \(T=1\), Line 3 of Algorithm 1 is redundant, and the algorithm resembles certain multipletry Metropolis methods (cf. Martino 2018) and has been suggested earlier by Mendes et al. (2015).

(iv)
Algorithm 2 is formulated using multinomial resampling, for simplicity. We note that any other unbiased resampling may be used, as long as the conditional resampling is designed appropriately; see Chopin and Singh (2015).
The ‘CPF generalisation’ perspective of Algorithm 1 may lead to other useful developments; for instance, one could imagine the approach to be useful with the CPF applied for static (nonHMM) targets, as in sequential Monte Carlo samplers (Del Moral et al. 2006). The aim of the present paper is, however, to use Algorithm 1 with diffuse initial distributions.
3 Methods for diffuse initialisation of conditional particle filters
To illustrate the typical problem that arises with a diffuse initial distribution \(M_1\), we examine a simple noisy AR(1) model:
for \(k\ge 1\), \(x_1 \sim N(0, \sigma _1^2)\), \(M_1({\mathrm {d}}x_1) = p(x_1) {\mathrm {d}}x_1\), \(M_k(x_{k1}, {\mathrm {d}}x_k) = p(x_k\mid x_{k1}) {\mathrm {d}}x_k\) and \(G_k(x_{1:k}) = p(y_k\mid x_k)\).
We simulated a dataset of length \(T=50\) from this model with \(x_1 = 0\), \(\rho = 0.8\) and \(\sigma _x = \sigma _y = 0.5\). We then ran 6000 iterations of the CPF with backward sampling (CPFBS) with \(\sigma _1 \in \{10, 100, 1000\}\); that is, Algorithm 1 with \(Q(x_0,\,\cdot \,) = M_1(\,\cdot \,)\) together with Algorithm 4, and discarded the first 1000 iterations as burnin. For each value of \(\sigma _1\), we monitored the efficiency of sampling \(x_1\). Figure 1 displays the resulting traceplots. The estimated integrated autocorrelation times (\({\mathrm {IACT}}\)) were approximately 3.75, 28.92 and 136.64, leading to effective sample sizes (\({\mathrm {n}}_{\mathrm {eff}}\)) of 1600, 207 and 44, respectively. This demonstrates how the performance of the CPFBS deteriorates as the initial distribution of the latent state becomes more diffuse.
3.1 Diffuse Gaussian initialisation
In the case that \(M_1\) in (2) is Gaussian with mean \(\mu \) and covariance \(\varSigma \), we can construct a Markov transition function that satisfies (3) using an autoregressive proposal similar to ‘preconditioning’ in the CrankNicolson algorithm (cf. Cotter et al. 2013). This proposal comes with a parameter \(\beta \in (0, 1]\), so we denote this kernel by \(Q_{\beta }^{\mathrm {AR}}\). A variate \(Z \sim Q_{\beta }^{\mathrm {AR}}(x, \,\cdot \,)\) can be drawn simply by setting
where \(W \sim N(0, \varSigma )\). We refer to Algorithm 1 with \(Q = Q_{\beta }^{\mathrm {AR}}\) as the diffuse Gaussian initialisation CPF (DGICPF). In the special case \(\beta = 1\), we have \(Q_{1}^{\mathrm {AR}} = M_1\), and so the DGICPF is equivalent with the standard CPF.
3.2 Fully diffuse initialisation
Suppose that \(M_1({\mathrm {d}}x) = M_1(x) {\mathrm {d}}x\) where \(M_1(x)\equiv 1\) is a uniform density on \({\mathsf {X}}={\mathbb {R}}^d\). Then, any symmetric transition Q satisfies \(M_1\)reversibility. In this case, we suggest to use \(Q_{C}^{\mathrm {RW}}(x,{\mathrm {d}}y) = q_{C}^{\mathrm {RW}}(x,y){\mathrm {d}}y\) with a multivariate normal density \(q_{C}^{\mathrm {RW}}(x,y) = N(y; x, C)\), with covariance \(C\in {\mathbb {R}}^{d\times d}\). In case of constraints, that is, a nontrivial domain \(D\subset {\mathbb {R}}^d\), we have \(M_1 = 1(x\in D)\). Then, we suggest to use a Metropolis–Hastings type transition probability:
where \(r(x)\in [0,1]\) is the rejection probability. This method works, of course, with arbitrary \(M_1\), but our focus is with a diffuse case, where the domain D is regular and large enough, so that rejections are rare. We stress that also in this case, \(M_1(x) = 1(x\in D)\) may be improper. We refer to Algorithm 1 with \(Q_{C}^{\mathrm {RW}}\) as the ‘fully diffuse initialisation’ CPF (FDICPF).
We note that whenever \(M_1\) can be evaluated pointwise, the FDICPF can always be applied, by considering the modified Feynman–Kac model \(\tilde{M}_1\equiv 1\) and \(\tilde{G}_1(x) = M_1(x) G_1(x)\). However, when \(M_1\) is Gaussian, the DGICPF can often lead to a more efficient method. As with standard random walk Metropolis algorithms, choosing the covariance \(C\in {\mathbb {R}}^{d\times d}\) is important for the efficiency of the FDICPF.
3.3 Adaptive proposals
Finding a good autoregressive parameter of \(Q_{\beta }^{\mathrm {AR}}\) or the covariance parameter of \(Q_{C}^{\mathrm {RW}}\) may be timeconsuming in practice. Inspired by the recent advances in adaptive MCMC (cf. Andrieu and Thoms 2008; Vihola 2020), it is natural to apply adaptation also with the (iterated) AICPF. Algorithm 5 summarises a generic adaptive AICPF (AAICPF) using a parameterised family \(\{Q_\zeta \}_{\zeta \in {\mathsf {Z}}}\) of \(M_1\)reversible proposals, with parameter \(\zeta \).
The function \(\textsc {Adapt}\) implements the adaptation, which typically leads to \(\zeta ^{(j)} \rightarrow \zeta ^*\), corresponding to a wellmixing configuration. We refer to the instances of the AAICPF with the AICPF step corresponding to the DGICPF and the FDICPF as the adaptive DGICPF and FDICPF, respectively.
We next focus on concrete adaptations which may be used within our framework. In the case of the FDICPF, Algorithm 6 implements a stochastic approximation variant (Andrieu and Moulines 2006) of the adaptive Metropolis covariance adaptation of Haario et al. (2001).
Here, \(\eta _j\) are step sizes that decay to zero, \(\zeta _j = (\mu _j,\varSigma _j)\) the estimated mean and covariance of the smoothing distribution, respectively, and \(Q_\zeta = Q_{c \varSigma }^{\mathrm {RW}}\) where \(c>0\) is a scaling factor of the covariance \(\varSigma \). In the case of random walk Metropolis, this scaling factor is usually taken as \(2.38^2/d\) (Gelman et al. 1996), where d is the state dimension of the model. In the present context, however, the optimal value of \(c > 0\) appears to depend on the model and on the number of particles N. This adaptation mechanism can be used both with PickPathAT and with PickPathBS, but may require some manual tuning to find a suitable \(c>0\).
Algorithm 7 details another adaptation for the FDICPF, which is intended to be used together with PickPathBS only. Here, \(\zeta _j = (\mu _j,\varSigma _j, \delta _j)\) contains the estimated mean, covariance and the scaling factor, and \(Q_\zeta = Q_{C(\zeta )}^{\mathrm {RW}}\), where \(C(\zeta ) = e^\delta \varSigma \).
This algorithm is inspired by a Rao–Blackwellised variant of the adaptive Metropolis within adaptive scaling method (cf. Andrieu and Thoms 2008), which is applied with standard random walk Metropolis. We use all particles with their backward sampling weights to update the mean \(\mu \) and covariance \(\varSigma \), and an ‘acceptance rate’ \(\alpha \), that is, the probability that the first coordinate of the reference trajectory is not chosen. Recall that after the AI–CPF in Algorithm 5 has been run, the first coordinate of the reference trajectory and its associated weight reside in the first index of the particle and weight vectors contained in \(\xi ^{(j)}\).
The optimal value of the acceptance rate parameter \({\alpha _{*}}\) is typically close to one, in contrast with random walk Metropolis, where \({\alpha _{*}}\in [0.234,0.44]\) are common (Gelman et al. 1996). Even though the optimal value appears to be problemdependent, we have found empirically that \(0.7\le {\alpha _{*}}\le 0.9\) often leads to reasonable mixing. We will show empirical evidence for this finding in Sect. 4.
Algorithm 8 describes a similar adaptive scaling type mechanism for tuning \(\beta = {\mathrm {logit}}^{1}(\zeta )\) in the DGICPF, with \(Q_\zeta = Q_{\beta }^{\mathrm {AR}}\). The algorithm is most practical with PickPathBS.
We conclude this section with a consistency result for Algorithm 5, using the adaptation mechanisms in Algorithms 6 and 7. In Theorem 1, we denote \((\mu _j,\varSigma _j) = \zeta _j\) in the case of Algorithm 6, and \((\mu _j,\varSigma _j,\delta _j) = \zeta _j\) with Algorithm 7.
Theorem 1
Suppose D is a compact set, a uniform mixing condition (Assumption 2 in Appendix A) holds, and there exists an \(\epsilon >0\) such that for all \(j\ge 1\), the smallest eigenvalue \(\lambda _{\min }(\varSigma _j)\ge \epsilon \), and with Algorithm 7 also \(\delta _j\in [\epsilon ,\epsilon ^{1}]\). Then, for any bounded function \(f:{\mathsf {X}}\rightarrow \infty \),
The proof of Theorem 1 is given in Appendix A. The proof is slightly more general, and accomodates for instance tdistributed instead of Gaussian proposals for the FDICPF. We note that the latter stability condition, that is, existence of the constant \(\epsilon >0\), may be enforced by introducing a ‘rejection’ mechanism in the adaptation; see the end of Appendix A. However, we have found empirically that the adaptation is stable also without such a stabilisation mechanism.
3.4 Use within particle Gibbs
Typical application of HMMs in statistics involves not only smoothing, but also inference of a number of ‘hyperparameters’ \(\theta \), with prior density \(\mathrm {pr}(\theta )\), and with
The full posterior, \(\check{\pi }(\theta , x_{1:T}) \propto \mathrm {pr}(\theta ) \gamma _\theta (x_{1:T})\) may be inferred with the particle Gibbs (PG) algorithm of Andrieu et al. (2010). (We assume here that \(M_1\) is diffuse, and thereby independent of \(\theta \).)
The PG alternates between (Metropoliswithin)Gibbs updates for \(\theta \) conditional on \(x_{1:T}\), and CPF updates for \(x_{1:T}\) conditional on \(\theta \). The (A)AICPF applied with \(M_{2:T}^{(\theta )}\) and \(G_{1:T}^{(\theta )}\) may be used as a replacement of the CPF steps in a PG. Another adaptation, independent of the AAICPF, may be used for the hyperparameter updates (cf. Vihola 2020).
Algorithm 9 summarises a generic adaptive PG with the AAICPF. Line 2 involves an update of \(\theta ^{(j1)}\) to \(\theta ^{(j)}\) using transition probabilities \(K_{\zeta _\theta }(\,\cdot \,, \,\cdot \,\mid x_{1:T})\) which leave \(\check{\pi }(\theta \mid x_{1:T})\) invariant, and Line 3 is (optional) adaptation. This could, for instance, correspond to the robust adaptive Metropolis algorithm (RAM) (Vihola 2012). Lines 4 and 5 implement the AAICPF. Note that without Lines 3 and 5, Algorithm 9 determines a \(\check{\pi }\)invariant transition rule.
4 Experiments
In this section, we study the application of the methods presented in Sect. 3 in practice. Our focus will be on the case of the bootstrap filter, that is, \(M_1({\mathrm {d}}x_1) = p(x_1) {\mathrm {d}}x_1\), \(M_k(x_{k1}, {\mathrm {d}}x_k) = p(x_k\mid x_{k1}) {\mathrm {d}}x_k\) and \(G_k(x_{1:k}) = p(y_k\mid x_k)\).
We start by investigating two simple HMMs: the noisy random walk model (RW), that is, (4) with \(\rho = 1\), and the following stochastic volatility (SV) model:
with \(x_1 \sim N(0, \sigma _1^2)\), \(\eta _k \sim N(0, \sigma _x^2)\) and \(\epsilon _k \sim N(0, \sigma _y^2)\). In Sect. 4.3, we study the dependence of the method with varying dimension, with a static multivariate normal model. We conclude in Sect. 4.4 by applying our methods in a realistic inference problem related to modelling the COVID19 epidemic in Finland.
4.1 Comparing DGICPF and CPFBS
We first studied how the DGICPF performs in comparison to the CPFBS when the initial distributions of the RW and SV model are diffuse. Since the efficiency of sampling is affected by both the values of the model parameters (cf. Fig. 1) and the number of particles N, we experimented with a range of values \(N \in \{8, 16, 32, 64, 128, 256, 512\}\) for which we applied both methods with \(n = 10000\) iterations plus 500 burnin. We simulated data from both the RW and SV models with \(T = 50\), \(x_{1} = 0\), \(\sigma _y = 1\) and varying \(\sigma _x \in \{0.01, 0.05, 0.1, 0.5, 1, 2, 5, 10, 20, 50, 100, 200\}\). We then applied both methods for each dataset with the corresponding \(\sigma _x\), but with varying \(\sigma _1 \in \{10, 50, 100, 200, 500, 1000\}\), to study the sampling efficiency under different parameter configurations (\(\sigma _x\) and \(\sigma _1\)). For the DGICPF, we varied the parameter \(\beta \in \{0.01, 0.02, \ldots , 0.99\}\). We computed the estimated integrated autocorrelation time (\({\mathrm {IACT}}\)) of the simulated values of \(x_1\) and scaled this by the number of particles N. The resulting quantity, the inverse relative efficiency (\({\mathrm {IRE}}\)), measures the asymptotic efficiencies of estimators with varying computational costs (Glynn and Whitt 1992).
Figure 2 shows the comparison of the CPFBS with the best DGICPF, that is, the DGICPF with the \(\beta \) that resulted in the lowest \({\mathrm {IACT}}\) for each parameter configuration and N.
The results indicate that with N fixed, a successful tuning of \(\beta \) can result in greatly improved mixing in comparison with the CPFBS. While the performance of the CPFBS approaches that of the best DGICPF with increasing N, the difference in performance remains substantial with parameter configurations that are challenging for the CPFBS.
The optimal N which minimizes the \({\mathrm {IRE}}\) depends on the parameter configuration. For ‘easy’ configurations (where \({\mathrm {IRE}}\) is small), even \(N=8\) can be enough, but more ‘difficult’ configurations (where \({\mathrm {IRE}}\) is large), higher values of N can be optimal. Similar results for the SV model are shown in Online Resource 1 (Fig. 1), and lead to similar conclusions.
The varying ‘difficulty’ of the parameter configurations is further illustrated in Fig. 3, which shows the \(\log {({\mathrm {IACT}})}\) for the SV model with \(N = 256\) particles. The CPFBS performed the worst when the initial distribution was very diffuse with respect to the state noise \(\sigma _x\), as expected. In contrast, the welltuned DGICPF appears rather robust with respect to changing parameter configuration. The observations were similar with other N, and for the RW model; see Online Resource 1 (Fig. 2).
The results in Figs. 2 and 3 illustrate the potential of the DGICPF, but are overly optimistic because in practice, the \(\beta \) parameter of the DGICPF cannot be chosen optimally. Indeed, the choice of \(\beta \) can have a substantial effect on the mixing. Figure 4 illustrates this in the case of the SV model by showing the logarithm of the mean \({\mathrm {IACT}}\) over replicate runs of the DGICPF, for a range of \(\beta \). Here, a \(\beta \) of approximately 0.125 seems to yield close to optimal performance, but if the \(\beta \) is chosen too low, the sampling efficiency is greatly reduced, rendering the CPFBS more effective.
This highlights the importance of choosing an appropriate value for \(\beta \), and motivates our adaptive DGICPF, that is, Algorithm 5 together with Algorithm 8. We explored the effect of the target acceptance rate \({\alpha _{*}}\in \{0.01, 0.02, \ldots , 1\}\), with the same datasets and parameter configurations as before. Figure 5 summarises the results for both the SV and RW models, in comparison with the CPFBS. The results indicate that with a wide range of target acceptance rates, the adaptive DGICPF exhibits improved mixing over the CPFBS. When N increases, the optimal values for \({\alpha _{*}}\) appear to tend to one. However, in practice, we are interested in a moderate N, for which the results suggest that the best candidates for values of \({\alpha _{*}}\) might often be found in the range from 0.7 to 0.9.
For the CPFBS, the mean \({\mathrm {IRE}}\) is approximately constant, which might suggest that the optimal number of particles is more than 512. In contrast, for an appropriately tuned DGICPF, the mean \({\mathrm {IRE}}\) is optimised by \(N = 32\) in this experiment.
4.2 Comparing FDICPF and particle Gibbs
Next, we turn to study a fully diffuse initialisation. In this case, \(M_1\) is improper, and we cannot use the CPF directly. Instead, we compare the performance of the adaptive FDICPF with what we call the diffuse particle Gibbs (DPGBS) algorithm. The DPGBS is a standard particle Gibbs algorithm, where the first latent state \(x_1\) is regarded as a ‘parameter’, that is, the algorithm alternates between the update of \(x_1\) conditional on \(x_{2:T}\) using a random walk MetropoliswithinGibbs step, and the update of the latent state variables \(x_{2:T}\) conditional on \(x_1\) using the CPFBS. We also adapt the MetropoliswithinGibbs proposal distribution \(Q_{\mathrm {DPG}}\) of the DPGBS, using the RAM algorithm (cf. Vihola 2020). For further details regarding our implementation of the DPGBS, see Appendix B.
We used a similar simulation experiment as with the adaptive DGICPF in Sect. 4.1, but excluding \(\sigma _1\), since the initial distribution was now fully diffuse. The target acceptance rates in the FDICPF with the ASWAM adaptation were again varied in \({\alpha _{*}}\in \{0.01, 0.02, \ldots , 1\}\) and the scaling factor in the AM adaptation was set to \(c = 2.38^2\). In the DPGBS, the target acceptance rate for updates of the initial state using the RAM algorithm was fixed to 0.441 following Gelman et al. (1996).
Figure 6 shows results with the RW model for the DPGBS, the FDICPF with the AM adaptation, and the FDICPF with the ASWAM adaptation using the best value for \({\alpha _{*}}\). The FDICPF variants appear to perform better and improve upon the performance of the DPGBS especially with small \(\sigma _x\). Similar to Figs. 2 and 3, the optimal N minimizing the \({\mathrm {IRE}}\) depends on the value of \(\sigma _x\): smaller values of \(\sigma _x\) call for higher number of particles.
The performance of the adaptive FDICPF appears similar regardless of the adaptation used, because the chosen scaling factor \(c = 2.38^2\) for a univariate model was close to the optimal value found by the ASWAM variant in this example. We experimented also with \(c = 1\), which led to less efficient AM, in the middle ground between the ASWAM and the DPGBS.
The \({\mathrm {IACT}}\) for the DPGBS stays approximately constant with increasing N, which results in a \(\log {({\mathrm {IRE}})}\) that increases roughly by a constant as N increases. This is understandable, because in the limit as \(N\rightarrow \infty \), the CPFBS (within the DPGBS) will correspond to a Gibbs step, that is, a perfect sample of \(x_{2:T}\) conditional on \(x_1\). Because of the strong correlation between \(x_1\) and \(x_2\), even an ‘ideal’ Gibbs sampler remains inefficient, and the small variation seen in the panels for the DPGBS is due to sampling variability. The results for the SV model, with similar findings, are shown in Online Resource 1 (Fig. 3).
Figure 7 shows the logarithm of the mean \({\mathrm {IRE}}\) of the FDICPF with the ASWAM adaptation with respect to varying target acceptance rate \({\alpha _{*}}\). The results are reminiscent of Fig. 5 and show that with a moderate fixed N, the FDICPF with the ASWAM adaptation outperforms the DPGBS with a wide range of values for \({\alpha _{*}}\). The optimal value of \({\alpha _{*}}\) seems to tend to one as N increases, but again, we are mostly concerned with moderate N. For a welltuned FDICPF the minimum mean \({\mathrm {IRE}}\) is found when N is roughly between 32 and 64.
4.3 The relationship between state dimension, number of particles and optimal target acceptance rate
A well chosen value for the target acceptance rate \({\alpha _{*}}\) appears to be key for obtaining good performance with the adaptive DGICPF and the FDICPF with the ASWAM adaptation. In Sects. 4.1–4.2, we observed a relationship between N and the optimal target acceptance rate, denoted here by \(\alpha _{\mathrm {opt}}\), with two univariate HMMs. It is expected that \(\alpha _{\mathrm {opt}}\) is generally somewhat modeldependent, but in particular, we suspected that the methods might behave differently with models of different state dimension d.
In order to study the relationship between N, d and \(\alpha _{\mathrm {opt}}\) in more detail, we considered a simple multivariate normal model with \(T = 1\), \(M_1(x) \propto 1\), and \(G_1(x_1) = N(x_1; 0, \sigma I_d)\), the density of d independent normals. We conducted a simulation experiment with 6000 iterations plus 500 burnin. We applied the FDICPF with the ASWAM adaptation with all combinations of \(N \in \{2^4, 2^5, \ldots , 2^{11}\}\), \({\alpha _{*}}\in \{0.01, 0.02, \ldots , 1\}\), \(\sigma \in \{1, 5, 10, 50, 100\}\), and with dimension \(d \in \{1, 2, \ldots , 10\}\). Unlike before, we monitor the \({\mathrm {IACT}}\) over the samples of \(x_1\) as an efficiency measure.
Figure 8 summarises the results of this experiment. With a fixed state dimension, \(\alpha _{\mathrm {opt}}\) tended towards 1 with increasing numbers of particles N, as observed with the RW and SV models above. With a fixed number of particles N, \(\alpha _{\mathrm {opt}}\) appears to get smaller with increasing state dimension d, but the change rate appears slower with higher d. Again, with moderate values for N and d, the values in the range 0.7–0.9 seem to yield good performance.
Figure 9 shows a different view of the same data: \({\mathrm {logit}}{(\alpha _{\mathrm {opt}})}\) is plotted with respect to \(\log {(N)}\) and d. Here, we computed \(\alpha _{\mathrm {opt}}\) by taking the target acceptance rate that produced the lowest \({\mathrm {IACT}}\) in the simulation experiment, for each value of \(\sigma \), N and d. At least with moderate \(\alpha _{\mathrm {opt}}\) and N, there appears to be a roughly linear relationship between \({\mathrm {logit}}(\alpha _{\mathrm {opt}})\) and \(\log (N)\), when d is fixed. However, because of the lack of theoretical backing, we do not suggest to use such a simple model for choosing \(\alpha _{\mathrm {opt}}\) in practice.
4.4 Modelling the COVID19 epidemic in Finland
Our final experiment is a realistic inference problem arising from the modelling of the progress of the COVID19 epidemic in Uusimaa, the capital region of Finland. Our main interest is in estimating the timevarying transmission rate, or the basic reproduction number \({{\mathscr {R}}_{0}}\), which is expected to change over time, because of a number of mitigation actions and social distancing. The model consists of a discretetime ‘SEIR’ stochastic compartment model, and a dynamic model for \({{\mathscr {R}}_{0}}\); such epidemic models have been used earlier in different contexts (e.g. Shubin et al. 2016).
We use a simple SEIR without age/regional stratification. That is, we divide the whole population \({N}_{\mathrm {pop}}\) to four separate states: susceptible (S), exposed (E), infected (I) and removed (R), so that \({N}_{\mathrm {pop}}= S + E + I + R\), and assume that \({N}_{\mathrm {pop}}\) is constant. We model the transformed \({{\mathscr {R}}_{0}}\), denoted by \(\rho \), such that \({{\mathscr {R}}_{0}}= {{\mathscr {R}}_{0}}^{\mathrm {max}}{\mathrm {logit}}^{1}(\rho )\), where \({{\mathscr {R}}_{0}}^{\mathrm {max}}\) is the maximal value for \({{\mathscr {R}}_{0}}\). The state vector of the model at time k is, therefore, \(X_k = (S_k, E_k, I_k, R_k, \rho _k)\). One step of the SEIR is:
where the increments are as distributed as follows:
Here, \(\beta _k = {{\mathscr {R}}_{0}}^{\mathrm {max}}{\mathrm {logit}}^{1}(\rho _k) p_\gamma \) is the timevarying infection rate, and \(a^{1}\) and \(\gamma ^{1}\) are the mean incubation period and recovery time, respectively. Finally, the random walk parameter \(\sigma \) controls how fast \((\rho _k)_{k \ge 2}\) can change.
The data we use in the modelling consist of the daily number of individuals tested positive for COVID19 in Uusimaa (Finnish Institute for Health and Welfare 2020). We model the counts with a negative binomial distribution dependent on the number of infected individuals:
Here, the parameter e denotes sampling effort, that is, the average proportion of infected individuals that are observed, and p is the failure probability of the negative binomial distribution, which controls the variability of the distribution.
In the beginning of the epidemic, there is little information available regarding the initial states, rendering the diffuse initialisation a convenient strategy. We set
where the number of removed \(R_1 = 0\) is justified because we assume all were susceptible to COVID19, and that the epidemic has started very recently.
In addition to the state estimation, we are interested in estimating the parameters \(\sigma \) and p. We assign the prior \(N(2.0, (0.3)^2)\) to \(\log {(\sigma )}\) to promote gradual changes in \({{\mathscr {R}}_{0}}\), and an uninformative prior, \(N(0, 10^2)\), for \({\mathrm {logit}}(p)\). The remaining parameters are fixed to \({N}_{\mathrm {pop}}= 1638469\), \({{\mathscr {R}}_{0}}^{\mathrm {max}} = 10\), \(a = 1/3\), \(\gamma = 1/7\) and \(e = 0.15\), which are in part inspired by the values reported by the Finnish Institute for Health and Welfare.
We used the AAIPG (Algorithm 9) with the FDICPF with the ASWAM adaptation, and a RAM adaptation (Vihola 2012) for \(\sigma \) and p, (i.e. in the Lines 2–3 of Algorithm 9). The form of (9) leads to the version of the FDICPF discussed in Sect. 3.2 where the initial distribution is uniform with constraints. We use a random walk proposal to generate proposals \((\rho _1,E_1,I_1)\rightarrow (\rho _1^*,E_1^*,I_1^*)\), round \(E_1^*\) and \(I_1^*\) to the nearest integer, and then set \(R_1^* = 0\) and \(S_1^* = {N}_{\mathrm {pop}} E_1^*  I_1^*  R_1^{*}\). We refer to this variant of the AAIPG as the FDIPG algorithm. Motivated by our findings in Sects. 4.1–4.3, we set the target acceptance rate \({\alpha _{*}}\) in the FDICPF (within the FDIPG) to 0.8.
As an alternative to the FDIPG we also used a particle Gibbs algorithm that treats \(\sigma \), p as well as the initial states \(E_1\), \(I_1\) and \(\rho _1\) as parameters, using the RAM to adapt the random walk proposal (Vihola 2012). This algorithm is the DPGBS detailed in Appendix B with the difference that the parameters \(\sigma \) and p are updated together with the initial state, and \(p^{\mathrm {DPG}}\) additionally contains all terms of (6) which depend on \(\sigma \) and p.
We ran both the FDIPG and the DPGBS with \(N = 64\) a total of \(n=500,000\) iterations plus 10, 000 burnin, and thinning of 10. Figures 10 and 11 show the first 50 autocorrelations and traceplots of \(E_1\), \(I_1\), \(({{\mathscr {R}}_{0}})_1\), \(\sigma \) and p, for both methods, respectively. The corresponding \({\mathrm {IACT}}\) and \({\mathrm {n}}_{\mathrm {eff}}\) as well as credible intervals for the means of these variables are shown in Table 1. The FDIPG outperformed the DPGBS with each variable. However, as is seen from Online Resource 1 (Fig. 4), the difference is most notable with the initial states, and the relative performance of the DPGBS approaches that of the FDIPG with increasing state index. The slow improvement in the mixing of the state variable R occurs because of the cumulative nature of the variable in the model, and the slow mixing of early values of I. We note that even though the mixing with the DPGBS was worse, the inference with 500, 000 iterations leads in practice to similar findings. However, the FDIPG could provide reliable inference with much less iterations than the DPGBS. The marginal density estimates of the initial states and parameters are shown in Online Resource 1 (Fig. 5). The slight discrepancies in the density estimates of \(E_1\) and \(I_1\) between the methods are likely because of the poor mixing of these variables with the DPGBS.
We conclude with a few words about our findings regarding the changing transmission rate, which may be of some independent interest. Figure 12 displays the data and a posterior predictive simulation, and the estimated distribution of \({{\mathscr {R}}_{0}}\) computed by the FDIPG with respect to time, with annotations about events that may have had an effect on the spread of the epidemic, and/or the data. The initial \({{\mathscr {R}}_{0}}\) is likely somewhat overestimated, because of the influx of infections from abroad, which were not explicitly modelled. There is an overall decreasing trend since the beginning of ‘lockdown’, that is, when the government introduced the first mitigation actions, including school closures. Changes in the testing criteria likely cause some bias soon after the change, but no single action or event stands out.
Interestingly, if we look at our analysis, but restrict our focus up to the end of April, we might be tempted to quantify how much certain mitigation actions contribute to the suppression of the transmission rate in order to build projections using scenario models (cf. Anderson et al. 2020). However, when the mitigation measures have been gradually lifted by opening the schools and restaurants, the openings do not appear to have had notable consequences, at least until now. It is possible that at this point, the number of infections was already so low, that it has been possible to test all suspected cases and trace contacts so efficiently, and that nearly all transmission chains have been contained. Also, the public may have changed their behaviour, and are now following the hygiene and social distancing recommendations voluntarily. Such a behaviour is, however, subject to change over time.
5 Discussion
We presented a simple general auxiliary variable method for the CPF for HMMs with diffuse initial distributions and focused on two concrete instances of it: the FDICPF for a uniform initial density \(M_1\) and the DGICPF for a Gaussian \(M_1\). We introduced two mechanisms to adapt the FDICPF automatically: the adaptive Metropolis (AM) of Haario et al. (2001) and a method similar to a Rao–Blackwellised adaptive scaling within adaptive Metropolis (ASWAM) (cf. Andrieu and Thoms 2008), and provided a proof of their consistency. We also suggested an adaptation for the DGICPF, based on an acceptance rate optimisation. The FDICPF or the DGICPF, including their adaptive variants, may be used directly within a particle Gibbs as a replacement for the standard CPF.
Our experiments with a noisy random walk model and a stochastic volatility model demonstrated that the DGICPF and the FDICPF can provide orders of magnitude speedups relative to a direct application of the CPF and to diffuse initialisation using particle Gibbs, respectively. Improvement was substantial also in our motivating practical example, where we applied the adaptive FDICPF (within particle Gibbs) in the analysis of the COVID19 epidemic in Finland, using a stochastic ‘SEIR’ compartment model with changing transmission rate. Latent compartment models are, more generally, a good example where our approach can be useful: there is substantial uncertainty in the initial states, and it is difficult to design directly a modified model that leads to efficient inference.
Our adaptation schemes are based on the estimated covariance matrix and a scaling factor which can be adapted using acceptance rate optimisation. For the latter, we found empirically that with a moderate number of particles, good performance was often reached with a target acceptance rate ranging in 0.7–0.9. We emphasise that even though we found this ‘0.8 rule’ to work well in practice, it is only a heuristic, and the optimal target acceptance rate may depend on the model of interest. Related to this, we investigated how the optimal target acceptance rate varied as a function of the number of particles and state dimension in a multivariate normal model, but did not find a clear pattern. Theoretical verification of the acceptance rate heuristic, and/or development of more refined adaptation rules, is left for future research. We note that while the AM adaptation performed well in our limited experiments, the ASWAM may be more appropriate when used within particle Gibbs (cf. Vihola 2020). The scaling of the AM remains similarly challenging, due to the lack of theory for tuning.
Data Availability Statement
All data analysed in this work are either freely available or available at https://nextcloud.jyu.fi/index.php/s/zjeiwDoxaegGcRe.
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Acknowledgements
We wish to acknowledge CSC, IT Center for Science, Finland, for computational resources, and thank Arto Luoma for inspiring discussions that led to the COVID19 example.
Funding
This work was supported by Academy of Finland Grant 315619. Open access funding provided by University of Jyväskylä (JYU).
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Appendices
Appendix
Proof of Theorem 1
For a finite signed measure \(\xi \), the total variation of \(\xi \) is defined as \(\Vert \xi \Vert _{\mathrm {tv}} = \sup _{\Vert f\Vert _\infty \le 1} \xi (f)\), where \(\Vert f\Vert _\infty = \sup _x f(x)\), and the supremum is over measurable realvalued functions f, and \(\xi (f) = \int f {\mathrm {d}}\xi \). For Markov transitions P and \(P'\), define \(d(P,P') = \sup _x \Vert P(x,\,\cdot \,)  P'(x,\,\cdot \,) \Vert _{\mathrm {tv}}\).
In what follows, we adopt the following definitions:
Definition 1
Consider Lines 3 and 4 of Algorithm 1 with \(\tilde{X}_1^{(1:N)}=\tilde{x}_1^{(1:N)}\) and \({\dot{x}}_{2:T}\), and define:

(i)
\(P_{\mathrm {CPF}}(\tilde{x}_1^{(1:N)},{\dot{x}}_{2:T}; \,\cdot \,)\) as the law of \(\tilde{X}_{1:T}^{(B_{1:T})}\), and

(ii)
(In case PickPathBS is used:) \(\tilde{P}_{\mathrm {CPF}}(\tilde{x}_1^{(1:N)},{\dot{x}}_{2:T}; \,\cdot \,)\) as the law of \(\big (\tilde{X}_{1:T}^{(B_{1:T})}, (B_{1},V^{(1:N)}, \tilde{X}_{1}^{(1:N)})\big )\).
Consider then Algorithm 1 with parameterised \(Q=Q_\zeta \), and define, analogously:

(iii)
\(P_\zeta \) is the Markov transition from \({\dot{x}}_{1:T}\) to \(\tilde{X}_{1:T}^{(B_{1:T})}\).

(iv)
\(\tilde{P}_\zeta \) is the Markov transition from from \(({\dot{x}}_{1:T},\,\cdot \,)\) to \(\big (\tilde{X}_{1:T}^{(B_{1:T})}, (B_{1},V^{(1:N)}, \tilde{X}_{1}^{(1:N)})\big )\).
Lemma 1
We have \(d(P_\zeta , P_{\zeta '}) \le N d(Q_\zeta , Q_{\zeta '})\) and \(d(\tilde{P}_\zeta , \tilde{P}_{\zeta '}) \le N d(Q_\zeta , Q_{\zeta '})\).
Proof
Let \((\hat{P}_\mathrm {CPF}, \hat{P}_\zeta ) \in \{(P_\mathrm {CPF}, P_\zeta ),(\tilde{P}_\mathrm {CPF},\tilde{P}_\zeta )\}\) and take measurable realvalued function f on the state space of \(\hat{P}_\zeta \) with \(\Vert f\Vert _\infty =1\).
We may write
and therefore, upper bound
with functions defined below, which satisfy \(\Vert g_0^{({\dot{x}}_{1:T})}\Vert _\infty \le 1\) and \(\Vert g_i^{({\dot{x}}_{1:T},x_0)}\Vert _\infty \le 1\):
\(\square \)
The following result is direct:
Lemma 2
Let \(Q_\varSigma \) stand for the random walk Metropolis type kernel with increment proposal distribution \(q_\varSigma \), and with target function \(M_1\ge 0\), that is, a transition probability of the form:
Then, \(\Vert Q_\varSigma (x,\,\cdot \,)  Q_{\varSigma '}(x,\,\cdot \,)\Vert _\mathrm {tv} \le 2 \Vert q_\varSigma  q_{\varSigma '}\Vert _\mathrm {tv}\).
The following result is from (Vihola 2011, proof of Proposition 26):
Lemma 3
Let \(q_\varSigma (x,{\mathrm {d}}y)\) stand for the centred Gaussian distribution with covariance \(\varSigma \), or the centred multivariate tdistribution with shape \(\varSigma \) and some constant degrees of freedom \(\nu >0\). Then, for any \(0<b_\ell<b_u<\infty \) there exists a constant \(c=c(b_\ell ,b_u)<\infty \) such that for all \(\varSigma ,\varSigma '\) with all eigenvalues within \([b_\ell ,b_u]\),
where the latter stands for the Frobenius norm in \(\mathbb {R}^d\).
Assumption 2
(Mixing) The potentials are bounded:

(i)
\(\Vert G_k\Vert _\infty <\infty \) for all \(k=1,\ldots ,T\).
Furthermore, there exists \(\epsilon >0\) and probability measures \(\nu _{\zeta }\) such that for all \(\zeta \in {\mathsf {Z}}\):

(ii)
\(Q_\zeta (x_0,A) \ge \epsilon \nu _{\zeta }(A)\) for all \(x_0\in \mathsf {X}\) and measurable \(A\subset \mathsf {X}\).

(iii)
\(\int \nu _{\zeta }({\mathrm {d}}x_0) Q_\zeta (x_0, {\mathrm {d}}x_1) G_1(x_1) \prod _{k=2}^T M_k(x_{k1}, {\mathrm {d}}x_k) G_k(x_{k1},x_k) {\mathrm {d}}x_{1:T}\) \(\ge \epsilon \).
Lemma 4
Suppose that Assumption 2 holds, then the kernels \(P_{\zeta }\) and \(\tilde{P}_\zeta \) satisfy simultaneous minorisation conditions, that is, there exists \(\delta >0\) and probability measures \(\nu _\zeta ,\tilde{\nu }_\zeta \), such that
for all \(x_{1:T}\in {\mathsf {X}}\), \(\tilde{x}_1^{(1:N)}\in {\mathsf {X}}^N\), and \(\zeta \in {\mathsf {Z}}\).
Proof
For \(\hat{P}_\zeta \in \{P_\zeta , \tilde{P}_\zeta \}\), we may write as in the proof of Lemma1
where the latter term refers to the term in brackets in (10) — the transition probability of a conditional particle filter, with reference \(x_{1:T}\), and the Feynman–Kac model \(\check{M}_1^{(\zeta ,x_0)}({\mathrm {d}}x_1) = Q_\zeta (x_0, {\mathrm {d}}x_1)\), \(M_{2:T}\) and \(G_{1:T}\), whose normalised probability we call \(\pi ^*_{\zeta ,x_0}\). Assumption 2, 2 and 2 guarantee that \(P^*_{\text {CPF},\zeta ,x_0}(x_{1:T},{\mathrm {d}}x'_{1:T}) \ge \varepsilon \pi ^*_{\zeta ,x_0}({\mathrm {d}}x'_{1:T})\), where \(\hat{\epsilon }>0\) is independent of \(x_0\) and \(\zeta \) (Andrieu et al. 2018, Corollary 12). Note that the same conclusion holds also with backward sampling, because it is only a further Gibbs step to the standard CPF. Likewise, in case of \(\tilde{P}_\zeta \), the result holds because we may regard \(\tilde{P}_\zeta \) as an augmented version of \(P_\zeta \) (e.g. Franks and Vihola 2020). We conclude that
where the integral defines a probability measure independent of \(x_{1:T}\). \(\square \)
We may write the k:th step of Algorithm 5 as:

(i)
\((X_k,\xi _k) \sim \tilde{P}_{\zeta _{k1}}(X_{k1},\,\cdot \,)\),

(ii)
\(\zeta _k^* = \zeta _{k1} + \eta _k H(\zeta _{k1}, X_k, \xi _k )\),
where H correspond to Algorithm 6 or , respectively. The stability may be enforced by introducing the following optional step:

(iii)
\(\zeta _k = \zeta _k^* 1(\zeta _k\in {\mathsf {Z}}) + \zeta _{k1} 1(\zeta _k^*\notin {\mathsf {Z}})\),
which ensures that \(\zeta \in {\mathsf {Z}}\), the feasible set for adaptation.
Proof
(Proof of Theorem 1) The result follows by (Saksman and Vihola 2010, Theorem 2), as (A1) is direct, Lemma 4 implies (A2) with \(V\equiv 1\), \(\lambda _n=0\), \(b_n=1\), \(\delta _n=\delta \) and \(\epsilon =0\), Lemmas 2 and 3 imply (A3), and (A4) holds trivially, as \(\Vert H (\,\cdot \,)\Vert _\infty < \infty \), thanks to the compactness of D. \(\square \)
Details of the DPGBS algorithm
The diffuse particle Gibbs algorithm targets (2) by alternating the sampling of \(x_{2:T}\) given \(x_1\), and \(x_1\) given \(x_{2:T}\). Hence, the algorithm is simply particle Gibbs where the initial state is treated as a parameter. Define
With this definition, the DPGBS algorithm can be written as in Algorithm 10. Lines 3–5 constitute a CPFBS update for \(x_{2:T}\), and Line 6 updates \(x_1\). A version of the RAM algorithm (Vihola 2012) (Algorithm 11) is used for adapting the normal proposal used in sampling \(x_1\) from \(p^{\mathrm {DPG}}\).
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Karppinen, S., Vihola, M. Conditional particle filters with diffuse initial distributions. Stat Comput 31, 24 (2021). https://doi.org/10.1007/s11222020099751
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DOI: https://doi.org/10.1007/s11222020099751