Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers

Differentiable samplers for deep latent variable models

Latent variable models are a popular class of models in statistics. Combined with neural networks to improve their expressivity, the resulting deep latent variable models have also found numerous applications in machine learning. A drawback of these models is that their likelihood function is intractable so approximations have to be carried out to perform inference. A standard approach consists of maximizing instead an evidence lower bound (ELBO) obtained based on a variational approximation of the posterior distribution of the latent variables. The standard ELBO can, however, be a very loose bound if the variational family is not rich enough. A generic strategy to tighten such bounds is to rely on an unbiased low-variance Monte Carlo estimate of the evidence. We review here some recent importance sampling, Markov chain Monte Carlo and sequential Monte Carlo strategies that have been proposed to achieve this. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'.

Unbiased estimation of the Hessian for partially observed diffusions

In this article, we consider the development of unbiased estimators of the Hessian, of the log-likelihood function with respect to parameters, for partially observed diffusion processes. These processes arise in numerous applications, where such diffusions require derivative information, either through the Jacobian or Hessian matrix. As time-discretizations of diffusions induce a bias, we provide an unbiased estimator of the Hessian. This is based on using Girsanov’s Theorem and randomization schemes developed through Mcleish (2011 Monte Carlo Methods Appl.17, 301–315 (doi:10.1515/mcma.2011.013)) and Rhee & Glynn (2016 Op. Res.63, 1026–1043). We demonstrate our developed estimator of the Hessian is unbiased, and one of finite variance. We numerically test and verify this by comparing the methodology here to that of a newly proposed particle filtering methodology. We test this on a range of diffusion models, which include different Ornstein–Uhlenbeck processes and the Fitzhugh–Nagumo model, arising in neuroscience.

Spatiotemporal blocking of the bouncy particle sampler for efficient inference in state-space models

We propose a novel blocked version of the continuous-time bouncy particle sampler of Bouchard-Côté et al. (J Am Stat Assoc 113(522):855-867, 2018) which is applicable to any differentiable probability density. This alternative implementation is motivated by blocked Gibbs sampling for state-space models (Singh et al. in Biometrika 104(4):953-969, 2017) and leads to significant improvement in terms of effective sample size per second, and furthermore, allows for significant parallelization of the resulting algorithm. The new algorithms are particularly efficient for latent state inference in high-dimensional state-space models, where blocking in both space and time is necessary to avoid degeneracy of MCMC. The efficiency of our blocked bouncy particle sampler, in comparison with both the standard implementation of the bouncy particle sampler and the particle Gibbs algorithm of Andrieu et al. (J R Stat Soc Ser B Stat Methodol 72(3):269-342, 2010), is illustrated numerically for both simulated data and a challenging real-world financial dataset.

SUPPLEMENTARY INFORMATION

The online version supplementary material available at 10.1007/s11222-021-10034-6.

Conditional particle filters with diffuse initial distributions

Conditional particle filters (CPFs) are powerful smoothing algorithms for general nonlinear/non-Gaussian 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 time-varying 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.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11222-020-09975-1.