Particle Marginal Metropolis-Hastings Research Articles

Estimating temporally variable selection intensity from ancient DNA data with the flexibility of modelling linkage and epistasis.

Innovations in ancient DNA (aDNA) preparation and sequencing technologies have exponentially increased the quality and quantity of aDNA data extracted from ancient biological materials. The additional temporal component from the incoming aDNA data can provide improved power to address fundamental evolutionary questions like characterizing selection processes that shape the phenotypes and genotypes of contemporary populations or species. However, utilizing aDNA to study past selection processes still involves considerable hurdles like how to eliminate the confounding factor of genetic interactions in the inference of selection. To address this issue, we extend the approach of He et al., 2023 to infer temporally variable selection from the aDNA data in the form of genotype likelihoods with the flexibility of modelling linkage and epistasis in this work. Our posterior computation is carried out by a robust adaptive version of the particle marginal Metropolis-Hastings algorithm with a coerced acceptance rate. Our extension inherits the desirable features of He et al., 2023 such as modelling sample uncertainty resulting from the damage and fragmentation of aDNA molecules and reconstructing underlying gamete frequency trajectories of the population. We evaluate its performance through extensive simulations and show its utility with an application to the aDNA data from pigmentation loci in horses.

Comparison of Markov chains via weak Poincaré inequalities with application to pseudo-marginal MCMC

We investigate the use of a certain class of functional inequalities known as weak Poincaré inequalities to bound convergence of Markov chains to equilibrium. We show that this enables the straightforward and transparent derivation of subgeometric convergence bounds for methods such as the Independent Metropolis–Hastings sampler and pseudo-marginal methods for intractable likelihoods, the latter being subgeometric in many practical settings. These results rely on novel quantitative comparison theorems between Markov chains. Associated proofs are simpler than those relying on drift/minorisation conditions and the tools developed allow us to recover and further extend known results as particular cases. We are then able to provide new insights into the practical use of pseudo-marginal algorithms, analyse the effect of averaging in Approximate Bayesian Computation (ABC) and the use of products of independent averages and also to study the case of log-normal weights relevant to particle marginal Metropolis–Hastings (PMMH).

Estimating Distributions of Parameters in Nonlinear State Space Models with Replica Exchange Particle Marginal Metropolis-Hastings Method.

Extracting latent nonlinear dynamics from observed time-series data is important for understanding a dynamic system against the background of the observed data. A state space model is a probabilistic graphical model for time-series data, which describes the probabilistic dependence between latent variables at subsequent times and between latent variables and observations. Since, in many situations, the values of the parameters in the state space model are unknown, estimating the parameters from observations is an important task. The particle marginal Metropolis–Hastings (PMMH) method is a method for estimating the marginal posterior distribution of parameters obtained by marginalization over the distribution of latent variables in the state space model. Although, in principle, we can estimate the marginal posterior distribution of parameters by iterating this method infinitely, the estimated result depends on the initial values for a finite number of times in practice. In this paper, we propose a replica exchange particle marginal Metropolis–Hastings (REPMMH) method as a method to improve this problem by combining the PMMH method with the replica exchange method. By using the proposed method, we simultaneously realize a global search at a high temperature and a local fine search at a low temperature. We evaluate the proposed method using simulated data obtained from the Izhikevich neuron model and Lévy-driven stochastic volatility model, and we show that the proposed REPMMH method improves the problem of the initial value dependence in the PMMH method, and realizes efficient sampling of parameters in the state space models compared with existing methods.

Detecting and Quantifying Natural Selection at Two Linked Loci from Time Series Data of Allele Frequencies with Forward-in-Time Simulations.

Recent advances in DNA sequencing techniques have made it possible to monitor genomes in great detail over time. This improvement provides an opportunity for us to study natural selection based on time serial samples of genomes while accounting for genetic recombination effect and local linkage information. Such time series genomic data allow for more accurate estimation of population genetic parameters and hypothesis testing on the recent action of natural selection. In this work, we develop a novel Bayesian statistical framework for inferring natural selection at a pair of linked loci by capitalising on the temporal aspect of DNA data with the additional flexibility of modeling the sampled chromosomes that contain unknown alleles. Our approach is built on a hidden Markov model where the underlying process is a two-locus Wright-Fisher diffusion with selection, which enables us to explicitly model genetic recombination and local linkage. The posterior probability distribution for selection coefficients is computed by applying the particle marginal Metropolis-Hastings algorithm, which allows us to efficiently calculate the likelihood. We evaluate the performance of our Bayesian inference procedure through extensive simulations, showing that our approach can deliver accurate estimates of selection coefficients, and the addition of genetic recombination and local linkage brings about significant improvement in the inference of natural selection. We also illustrate the utility of our method on real data with an application to ancient DNA data associated with white spotting patterns in horses.

A flexible particle Markov chain Monte Carlo method

Particle Markov Chain Monte Carlo methods are used to carry out inference in nonlinear and non-Gaussian state space models, where the posterior density of the states is approximated using particles. Current approaches usually perform Bayesian inference using either a particle marginal Metropolis–Hastings (PMMH) algorithm or a particle Gibbs (PG) sampler. This paper shows how the two ways of generating variables mentioned above can be combined in a flexible manner to give sampling schemes that converge to a desired target distribution. The advantage of our approach is that the sampling scheme can be tailored to obtain good results for different applications. For example, when some parameters and the states are highly correlated, such parameters can be generated using PMMH, while all other parameters are generated using PG because it is easier to obtain good proposals for the parameters within the PG framework. We derive some convergence properties of our sampling scheme and also investigate its performance empirically by applying it to univariate and multivariate stochastic volatility models and comparing it to other PMCMC methods proposed in the literature.

Particle filters for inference of high-dimensional multivariate stochastic volatility models with cross-leverage effects

Multivariate stochastic volatility models are a popular and well-known class of models in the analysis of financial time series because of their abilities to capture the important stylized facts of financial returns data. We consider the problems of filtering distribution estimation and also marginal likelihood calculation for multivariate stochastic volatility models with cross-leverage effects in the high dimensional case, that is when the number of financial time series that we analyze simultaneously (denoted by $ d $) is large. The standard particle filter has been widely used in the literature to solve these intractable inference problems. It has excellent performance in low to moderate dimensions, but collapses in the high dimensional case. In this article, two new and advanced particle filters proposed in [4], named the space-time particle filter and the marginal space-time particle filter, are explored for these estimation problems. The better performance in both the accuracy and stability for the two advanced particle filters are shown using simulation and empirical studies in comparison with the standard particle filter. In addition, Bayesian static model parameter estimation problem is considered with the advances in particle Markov chain Monte Carlo methods. The particle marginal Metropolis-Hastings algorithm is applied together with the likelihood estimates from the space-time particle filter to infer the static model parameter successfully when that using the likelihood estimates from the standard particle filter fails.

Functional programming for modular Bayesian inference

We present an architectural design of a library for Bayesian modelling and inference in modern functional programming languages. The novel aspect of our approach are modular implementations of existing state-of-the-art inference algorithms. Our design relies on three inherently functional features: higher-order functions, inductive data-types, and support for either type-classes or an expressive module system. We provide a performant Haskell implementation of this architecture, demonstrating that high-level and modular probabilistic programming can be added as a library in sufficiently expressive languages. We review the core abstractions in this architecture: inference representations, inference transformations, and inference representation transformers. We then implement concrete instances of these abstractions, counterparts to particle filters and Metropolis-Hastings samplers, which form the basic building blocks of our library. By composing these building blocks we obtain state-of-the-art inference algorithms: Resample-Move Sequential Monte Carlo, Particle Marginal Metropolis-Hastings, and Sequential Monte Carlo Squared. We evaluate our implementation against existing probabilistic programming systems and find it is already competitively performant, although we conjecture that existing functional programming optimisation techniques could reduce the overhead associated with the abstractions we use. We show that our modular design enables deterministic testing of inherently stochastic Monte Carlo algorithms. Finally, we demonstrate using OCaml that an expressive module system can also implement our design.

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

We establish quantitative bounds for rates of convergence and asymptotic variances for iterated conditional sequential Monte Carlo (i-cSMC) Markov chains and associated particle Gibbs samplers [J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 (2010) 269–342]. Our main findings are that the essential boundedness of potential functions associated with the i-cSMC algorithm provide necessary and sufficient conditions for the uniform ergodicity of the i-cSMC Markov chain, as well as quantitative bounds on its (uniformly geometric) rate of convergence. Furthermore, we show that the i-cSMC Markov chain cannot even be geometrically ergodic if this essential boundedness does not hold in many applications of interest. Our sufficiency and quantitative bounds rely on a novel non-asymptotic analysis of the expectation of a standard normalizing constant estimate with respect to a “doubly conditional” SMC algorithm. In addition, our results for i-cSMC imply that the rate of convergence can be improved arbitrarily by increasing $N$, the number of particles in the algorithm, and that in the presence of mixing assumptions, the rate of convergence can be kept constant by increasing $N$ linearly with the time horizon. We translate the sufficiency of the boundedness condition for i-cSMC into sufficient conditions for the particle Gibbs Markov chain to be geometrically ergodic and quantitative bounds on its geometric rate of convergence, which imply convergence of properties of the particle Gibbs Markov chain to those of its corresponding Gibbs sampler. These results complement recently discovered, and related, conditions for the particle marginal Metropolis–Hastings (PMMH) Markov chain.

Bayesian inference for duplication-mutation with complementarity network models.

We observe an undirected graph G without multiple edges and self-loops, which is to represent a protein–protein interaction (PPI) network. We assume that G evolved under the duplication–mutation with complementarity (DMC) model from a seed graph, G0, and we also observe the binary forest Γ that represents the duplication history of G. A posterior density for the DMC model parameters is established, and we outline a sampling strategy by which one can perform Bayesian inference; that sampling strategy employs a particle marginal Metropolis–Hastings (PMMH) algorithm. We test our methodology on numerical examples to demonstrate a high accuracy and precision in the inference of the DMC model's mutation and homodimerization parameters.

A simulation approach for change-points on phylogenetic trees.

We observe n sequences at each of m sites and assume that they have evolved from an ancestral sequence that forms the root of a binary tree of known topology and branch lengths, but the sequence states at internal nodes are unknown. The topology of the tree and branch lengths are the same for all sites, but the parameters of the evolutionary model can vary over sites. We assume a piecewise constant model for these parameters, with an unknown number of change-points and hence a transdimensional parameter space over which we seek to perform Bayesian inference. We propose two novel ideas to deal with the computational challenges of such inference. Firstly, we approximate the model based on the time machine principle: the top nodes of the binary tree (near the root) are replaced by an approximation of the true distribution; as more nodes are removed from the top of the tree, the cost of computing the likelihood is reduced linearly in n. The approach introduces a bias, which we investigate empirically. Secondly, we develop a particle marginal Metropolis-Hastings (PMMH) algorithm, that employs a sequential Monte Carlo (SMC) sampler and can use the first idea. Our time-machine PMMH algorithm copes well with one of the bottle-necks of standard computational algorithms: the transdimensional nature of the posterior distribution. The algorithm is implemented on simulated and real data examples, and we empirically demonstrate its potential to outperform competing methods based on approximate Bayesian computation (ABC) techniques.

Particle MCMC for Bayesian Microwave Control

We consider the problem of local radioelectric property estimation from global electromagnetic scattering measurements. This challenging ill-posed high dimensional inverse problem can be explored by intensive computations of a parallel Maxwell solver on a petaflopic supercomputer. Then, it is shown how Bayesian inference can be perfomed with a Particle Marginal Metropolis-Hastings (PMMH) approach, which includes a Rao-Blackwellised Sequential Monte Carlo algorithm with interacting Kalman filters. Material properties, including a multiple components "Debye relaxation"/"Lorenzian resonant" material model, are estimated; it is illustrated on synthetic data. Eventually, we propose different ways to deal with higher dimensional problems, from parallelization to the original introduction of efficient sequential data assimilation techniques, widely used in weather forecasting, oceanography, geophysics, etc.

Twisting the Alive Particle Filter

This work focuses on sampling from hidden Markov models (Cappe et al. 2005) whose observations have intractable density functions. We develop a new sequential Monte Carlo (e.g. Doucet, 2011) algorithm and a new particle marginal Metropolis-Hastings (Andrieu et al J R Statist Soc Ser B 72:269-342, 2010) algorithm for these purposes. We build from Jasra et al (2013) and Whiteley and Lee (Ann Statist 42:115-141, 2014) to construct the sequential Monte Carlo (SMC) algorithm, which we call the alive twisted particle filter. Like the alive particle filter (Amrein and Künsch, 2011, Jasra et al, 2013), our new SMC algorithm adopts an approximate Bayesian computation (Tavare et al. Genetics 145:505-518, 1997) estimate of the HMM. Our alive twisted particle filter also uses a twisted proposal as in Whiteley and Lee (Ann Statist 42:115-141, 2014) to obtain a low-variance estimate of the HMM normalising constant. We demonstrate via numerical examples that, in some scenarios, this estimate has a much lower variance than that of the estimate obtained via the alive particle filter. The low variance of this normalising constant estimate encourages the implementation of our SMC algorithm within a particle marginal Metropolis-Hastings (PMMH) scheme, and we call the resulting methodology “alive twisted PMMH”. We numerically demonstrate, on a stochastic volatility model, how our alive twisted PMMH can converge faster than the standard alive PMMH of Jasra et al (2013).

A Particle Marginal Metropolis-Hastings Multi-Target Tracker

We propose a Bayesian multi-target batch processing algorithm capable of tracking an unknown number of targets that move close and/or cross each other in a dense clutter environment. The optimal Bayes multitarget tracking problem is formulated in the random finite set framework and a particle marginal Metropolis-Hastings (PMMH) technique which is a combination of the Metropolis-Hastings (MH) algorithm and sequential Monte Carlo methods is applied to compute the multi-target posterior distribution. The PMMH technique is used to design a high-dimensional proposal distributions for the MH algorithm and allows the proposed batch process multi-target tracker to handle a large number of tracks in a computationally feasible manner. Our simulations show that the proposed tracker reliably estimates the number of tracks and their trajectories in scenarios with a large number of closely spaced tracks in a dense clutter environment albeit, more expensive than online methods.

On the Use of Particle Markov Chain Monte Carlo in Parameter Estimation of Space-Time Interacting Discs

A space-time random set is defined and methods of its parameters estimation are investigated. The evolution in discrete time is described by a state-space model. The observed output is a planar union of interacting discs given by a probability density with respect to a reference Poisson process of discs. The state vector is to be estimated together with auxiliary parameters of transitions caused by a random walk. Three methods of parameters estimation are involved, first of which is the maximum likelihood estimation (MLE) for individual outputs at fixed times. In the space-time model the state vector can be estimated by the particle filter (PF), where MLE serves to the estimation of auxiliary parameters. In the present paper the aim is to compare MLE and PF with particle Markov chain Monte Carlo (PMCMC). From the group of PMCMC methods we use specially the particle marginal Metropolis-Hastings (PMMH) algorithm which updates simultaneously the state vector and the auxiliary parameters. A simulation study is presented in which all estimators are compared by means of the integrated mean square error. New data are then simulated repeatedly from the model with parameters estimated by PMMH and the fit with the original model is quantified by means of the spherical contact distribution function.

Particle Efficient Importance Sampling

The efficient importance sampling (EIS) method is a general principle for the numerical evaluation of high-dimensional integrals that uses the sequential structure of target integrands to build variance minimising importance samplers. Despite a number of successful applications in high dimensions, it is well known that importance sampling strategies are subject to an exponential growth in variance as the dimension of the integration increases. We solve this problem by recognising that the EIS framework has an offline sequential Monte Carlo interpretation. The particle EIS method is based on non-standard resampling weights that take into account the look-ahead construction of the importance sampler. We apply the method for a range of univariate and bivariate stochastic volatility specifications. We also develop a new application of the EIS approach to state space models with Student's t state innovations. Our results show that the particle EIS method strongly outperforms both the standard EIS method and particle filters for likelihood evaluation in high dimensions. Moreover, the ratio between the variances of the particle EIS and particle filter methods remains stable as the time series dimension increases. We illustrate the efficiency of the method for Bayesian inference using the particle marginal Metropolis-Hastings and importance sampling squared algorithms.

On Disturbance State-Space Models and the Particle Marginal Metropolis-Hastings Sampler

We investigate nonlinear state-space models without a closed-form transition density, and propose reformulating such models over their latent noise variables rather than their latent state variables. In doing so the tractable noise density emerges in place of the intractable transition density. For importance sampling methods such as the auxiliary particle filter, this enables importance weights to be computed where they could not be otherwise. As case studies we take two multivariate marine biogeochemical models and perform state and parameter estimation using the particle marginal Metropolis-Hastings sampler. For the particle filter within this sampler, we compare several proposal strategies over noise variables, all based on lookaheads with the unscented Kalman filter. These strategies are compared using conventional means for assessing Metropolis-Hastings efficiency, as well as with a novel metric called the conditional acceptance rate for assessing the consequences of using an estimated, and not exact, likelihood. Results indicate the utility of reformulating the model over noise variables, particularly for fast-mixing process models.