Smoothing With Couplings of Conditional Particle Filters

Auxiliary two-filter particle smoothing for one generalized hidden Markov model

This paper develops two-filter particle smoothing (TFPS) algorithms for the nonlinear fixed-interval smoothing problem of one generalized hidden Markov model (GHMM), where the current observation depends not only on the current state, but also on one-step previous state. Firstly, by Bayesian approach, the two-filter smoothing (TFS) formula for GHMM is established to calculate smoothing densities. In this TFS formula, the backward information prediction density is generally not a density of the state. This results in a difficulty that the normal sequential Monte Carlo (SMC) sampling technique cannot be directly applied to design corresponding TFPS algorithms based on the TFS formula. To solve this difficulty, a generalized TFS formula for GHMM is then proposed by introducing a sequence of artificial densities. By combining this generalized TFS formula, SMC, and the auxiliary variable sampling technique, a basic auxiliary TFPS (ATFPS) algorithm with quadratic computational complexity is proposed, and a simplified ATFPS algorithm with linear computational complexity is further developed. Finally, the effectiveness and superiority of the two proposed ATFPS algorithms for GHMM are verified via simulation examples and real experimental data.

Unbiased filtering of a class of partially observed diffusions

In this article we consider a Monte-Carlo-based method to filter partially observed diffusions observed at regular and discrete times. Given access only to Euler discretizations of the diffusion process, we present a new procedure which can return online estimates of the filtering distribution with no time-discretization bias and finite variance. Our approach is based upon a novel double application of the randomization methods of Rhee and Glynn (Operat. Res.63, 2015) along with the multilevel particle filter (MLPF) approach of Jasra et al. (SIAM J. Numer. Anal.55, 2017). A numerical comparison of our new approach with the MLPF, on a single processor, shows that similar errors are possible for a mild increase in computational cost. However, the new method scales strongly to arbitrarily many processors.

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.

Inferring medication adherence from time-varying health measures

Medication adherence is a problem of widespread concern in clinical care. Poor adherence is a particular problem for patients with chronic diseases requiring long-term medication because poor adherence can result in less successful treatment outcomes and even preventable deaths. Existing methods to collect information about patient adherence are resource-intensive or do not successfully detect low-adherers with high accuracy. Acknowledging that health measures recorded at clinic visits are more reliably recorded than a patient's adherence, we have developed an approach to infer medication adherence rates based on longitudinally recorded health measures that are likely impacted by time-varying adherence behaviors. Our framework permits the inclusion of baseline health characteristics and socio-demographic data. We employ a modular inferential approach. First, we fit a two-component model on a training set of patients who have detailed adherence data obtained from electronic medication monitoring. One model component predicts adherence behaviors only from baseline health and socio-demographic information, and the other predicts longitudinal health measures given the adherence and baseline health measures. Posterior draws of relevant model parameters are simulated from this model using Markov chain Monte Carlo methods. Second, we develop an approach to infer medication adherence from the time-varying health measures using a sequential Monte Carlo algorithm applied to a new set of patients for whom no adherence data are available. We apply and evaluate the method on a cohort of hypertensive patients, using baseline health comorbidities, socio-demographic measures, and blood pressure measured over time to infer patients' adherence to antihypertensive medication.

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.

Central limit theorems for coupled particle filters

In this article we prove new central limit theorems (CLTs) for several coupled particle filters (CPFs). CPFs are used for the sequential estimation of the difference of expectations with respect to filters which are in some sense close. Examples include the estimation of the filtering distribution associated to different parameters (finite difference estimation) and filters associated to partially observed discretized diffusion processes (PODDP) and the implementation of the multilevel Monte Carlo (MLMC) identity. We develop new theory for CPFs, and based upon several results, we propose a new CPF which approximates the maximal coupling (MCPF) of a pair of predictor distributions. In the context of ML estimation associated to PODDP with time-discretization $\Delta_l=2^{-l}$ , $l\in\{0,1,\dots\}$ , we show that the MCPF and the approach of Jasra, Ballesio, et al. (2018) have, under certain assumptions, an asymptotic variance that is bounded above by an expression that is of (almost) the order of $\Delta_l$ ( $\mathcal{O}(\Delta_l)$ ), uniformly in time. The $\mathcal{O}(\Delta_l)$ bound preserves the so-called forward rate of the diffusion in some scenarios, which is not the case for the CPF in Jasra et al. (2017).