Bayesian parameter inference for partially observed stopped processes

Abstract

We consider Bayesian parameter inference associated to partially-observed stochastic processes that start from a set B ₀ and are stopped or killed at the first hitting time of a known set A. Such processes occur naturally within the context of a wide variety of applications. The associated posterior distributions are highly complex and posterior parameter inference requires the use of advanced Markov chain Monte Carlo (MCMC) techniques. Our approach uses a recently introduced simulation methodology, particle Markov chain Monte Carlo (PMCMC) (Andrieu et al. 2010), where sequential Monte Carlo (SMC) (Doucet et al. 2001; Liu 2001) approximations are embedded within MCMC. However, when the parameter of interest is fixed, standard SMC algorithms are not always appropriate for many stopped processes. In Chen et al. (2005), Del Moral (2004), the authors introduce SMC approximations of multi-level Feynman-Kac formulae, which can lead to more efficient algorithms. This is achieved by devising a sequence of sets from B ₀ to A and then performing the resampling step only when the samples of the process reach intermediate sets in the sequence. The choice of the intermediate sets is critical to the performance of such a scheme. In this paper, we demonstrate that multi-level SMC algorithms can be used as a proposal in PMCMC. In addition, we introduce a flexible strategy that adapts the sets for different parameter proposals. Our methodology is illustrated on the coalescent model with migration.

Appendix

Proof of Proposition 4.1

The result is a straight forward application of Theorem 6 of Roberts and Rosenthal (2011), which adapted to our notation states:

where the conditional expectation is the expectation w.r.t. the SMC algorithm (i.e. Ξ∼ψ _θ ) and the outer expectation is w.r.t. the PIMH target (i.e. $\xi\sim\pi_{p}^{N}$). We also denote the estimate of the normalising constant as $\hat{Z}_{p}(\cdot )$ with ⋅ denoting which random variables generate the estimate.

Now, clearly via (A1),

with the convention that τ ₀=0. Thus, it follows that

and we obtain:

Note that by assumption, $Z_{p}(\varXi) (\rho\varphi )^{2\sum_{n=1}^{p}\bar{\tau}_{n}}\leq1$, and thus we have

Given Del Moral (2004, Theorem 7.4.2, Eq. (7.17), p. 239) and the fact that γ _θ is defined to be strictly positive in (A1), we have that the SMC approximation $\hat {Z}_{p}(\cdot)$ is an unbiased estimate of the normalising constant Z _p :

(13)

and we can easily conclude. □

Proof of Proposition 4.2

The proof of parts 1 and 2 follows the line of arguments used in Theorem 4 of Andrieu et al. (2010), which we will adapt to our set-up. The main difference lies in the multi-level construction and second statement regarding the marginal of $\bar{\pi}^{N}$. For the validity of the multi-level set-up, we will rely on Proposition 3.1.

Suppose we design a Metropolis-Hastings kernel with invariant density $\bar{\pi}^{N}$ and use a proposal $q^{N}(\theta,k,\bar {\mathcal{X}}_{1:p}, \bar{\mathbf{a}}_{1:p-1})=\psi_{\theta}(\bar {\mathcal{X}}_{1:p},\bar{\mathbf{a}}_{1:p-1})f(k\mid W_{p})\bar {q}(\theta(i-1)\mid \theta^{\prime})=\psi_{\theta}(\bar{\mathcal {X}}_{1:p},\bar{\mathbf{a}}_{1:p-1})\bar{W}_{p}^{(k)}\bar{q}(\theta \mid \theta^{\prime})$. Then

where we denote the normalising constant of the posterior in (1) as:

Therefore, the Metropolis-Hastings procedure to sample from $\bar{\pi }_{p}^{N}$ will be as in Algorithm 4.

Alternatively using similar arguments, one may write

Summing over k and using the unbiased property of the SMC algorithm in Eq. (13), it follows that $\bar{\pi }_{p}^{N}(\cdot)$ admits $\bar{\pi}(\theta)$ as a marginal, so the proof of part 1 is complete.

Part 2 is a direct consequence of Theorem 1 in Andrieu and Roberts (2009) and Assumption (A2). □

Proof of Proposition 4.3

The proof is similar to that of Proposition 4.2. For the proof of the first statement of part 1, one repeats the same arguments as for Proposition 4.2, with the difference being in the inclusion of $\bar{\varLambda}_{\theta}(v)$ for $\bar {\pi}^{N}$ and $\bar{q}^{N}$. For the second statement, to get the marginal of $\bar{\pi}^{N}$, one re-writes the target as

Let $\bar{\pi}_{p}^{N}(\theta)$ denote the marginal of $\bar {\pi}_{p}^{N}(\cdot)$ obtained in Proposition 4.2. Using (11) and the conditional independence of v and $\bar{\mathcal{X}}_{1:p(v)},\bar{\mathbf {a}}_{1:p(v)-1}$, then for the marginal of $\bar{\pi}^{N}(\cdot)$ w.r.t. v, $\bar{\mathcal{X}}_{1:p(v)},\bar{\mathbf {a}}_{1:p(v)-1}$, k we have that

where the summing over k and integrating w.r.t. $\bar{\mathcal {X}}_{1:p(v)}, \bar{\mathbf{a}}_{1:p(v)-1}$ is as in Proposition 4.2.

For part 2 note that the conditional density given k and v and θ of $\mathcal{X}_{1:p(v)}^{(k)}$ is

Hence the sequence $(\theta(i),\mathcal {X}_{1:p(v)}^{(k)}(i) )_{i\geq0}$ satisfies the required property as a direct consequence of Theorem 1 in Andrieu and Roberts (2009) and Assumption (A2). □