Efficient Sequential Monte-Carlo Samplers for Bayesian Inference

Abstract

International audienceIn many problems, complex non-Gaussian and/or nonlinear models are required to accurately describe a physical system of interest. In such cases, Monte Carlo algorithms are remarkably flexible and extremely powerful approaches to solve such inference problems. However, in the presence of a high-dimensional and/or multimodal posterior distribution, it is widely documented that standard Monte-Carlo techniques could lead to poor performance. In this paper, the study is focused on a Sequential Monte- Carlo (SMC) sampler framework, a more robust and efficient Monte Carlo algorithm. Although this approach presents many advantages over traditional Monte-Carlo methods, the potential of this emergent technique is however largely underexploited in signal processing. In this work, we aim at proposing some novel strategies that will improve the efficiency and facilitate practical implementation of the SMC sampler specifically for signal processing applications. Firstly, we propose an automatic and adaptive strategy that selects the sequence of distributions within the SMC sampler that minimizes the asymptotic variance of the estimator of the posterior normalization constant. This is critical for performing model selection in modelling applications in Bayesian signal processing. The second original contribution we present improves the global efficiency of the SMC sampler by introducing a novel correction mechanism that allows the use of the particles generated through all the iterations of the algorithm (instead of only particles from the last iteration). This is a significant contribution as it removes the need to discard a large portion of the samples obtained, as is standard in standard SMC methods. This will improve estimation performance in practical settings where computational budget is important to consider

A sequential Monte Carlo approach to gene expression deconvolution

“…Before we introduce the SMC sampler algorithm for gene expression decomposition, we will succinctly describe the general principle of SMC samplers in Bayesian inference settings [ 28 – 30 ]. Denote the prior distribution, the likelihood function and the posterior distribution in a Bayesian inference setup as p ( θ ), p ( Y | θ ) and p ( θ | Y ), respectively.…”

“…Using the Bayes rule, the posterior distribution can be written as a function of the prior distribution and the likelihood function as follows: where Z = ∫ Θ p ( θ ) p ( Y | θ ) d θ , a constant with respect to θ , is referred to as the evidence. With SMC samplers, rather than sampling from the posterior distribution p ( θ | Y ) in (5) , a sequence of intermediate target distributions, , are designed, that transitions smoothly from the prior distribution, i.e., π 1 = p ( θ ), which is usually easier to sample from, and gradually introduce the effect of the likelihood so that in the end, we have π T = p ( θ | Y ) which is the posterior distribution of interest [ 28 , 29 ]. For such sequence of intermediate distributions, a natural choice is the likelihood tempered target sequence [ 28 , 41 ]: where is a non-decreasing temperature schedule with ϵ 1 = 0 and ϵ T = 1, is the unnormalized target distribution and is the evidence at time t .…”

“…In this paper, we propose a new probabilistic method, sequential Monte Carlo (SMC) sampler [ 28 – 31 ] for static models to estimate the cell type proportions and the cell-type specific expression profiles, given the heterogeneous gene expression data. Specifically, we model the heterogeneous gene expression data using a Bayesian framework where the cell-type specific expression profiles and the cell type proportions are the unknown model parameters.…”

“…It is easy to implement and applicable to very general settings. As noted in [ 28 ], SMC algorithms address some of the major shortcomings of the MCMC-based algorithms: (i) diagnosing convergence of a Markov chain (ii) requirement of burn-in period, and (iii) MCMC algorithms getting trapped in local modes if the target distribution is highly multi-modal. In addition, in big data analyses, unlike the MCMC approach, SMC algorithms can be parallelized to reduce the computational time [ 28 ].…”

Peer Review #2 of "Characterization of tumor heterogeneity by latent haplotypes: a sequential Monte Carlo approach (v0.1)"

“…However, if more VAFs are available for newly called SNV(s), both algorithms have to be restarted in order to incorporate the newly called SNV(s). Moreover, MCMC approach in general as previously shown in (Nguyen et al, 2016;Jasra et al, 2007), is plagued with some inherent issues which often limit its performance: (i) sometimes, it is difficult to assess when the Markov chain has reached its stationary regime of interest (ii) requirement of burn-in period and thinning interval, and most importantly, (iii) if the target distribution is highly multi-modal, MCMC algorithms can easily become trapped in local modes.…”