Particle MCMC Research Articles

Particle MCMC in Forecasting Frailty-Correlated Default Models with Expert Opinion

Predicting corporate default risk has long been a crucial topic in the finance field, as bankruptcies impose enormous costs on market participants as well as the economy as a whole. This paper aims to forecast frailty-correlated default models with subjective judgements on a sample of U.S. public non-financial firms spanning January 1980–June 2019. We consider a reduced-form model and adopt a Bayesian approach coupled with the Particle Markov Chain Monte Carlo (Particle MCMC) algorithm to scrutinize this problem. The findings show that the 1-year prediction for frailty-correlated default models with different prior distributions is relatively good, whereas the prediction accuracy ratios for frailty-correlated default models with non-informative and subjective prior distributions over various prediction horizons are not significantly different.

Accelerating inference for stochastic kinetic models

Stochastic kinetic models (SKMs) are increasingly used to account for the inherent stochasticity exhibited by interacting populations of species in areas such as epidemiology, population ecology and systems biology. Species numbers are modelled using a continuous-time stochastic process, and, depending on the application area of interest, this will typically take the form of a Markov jump process or an Itô diffusion process. Widespread use of these models is typically precluded by their computational complexity. In particular, performing exact fully Bayesian inference in either modelling framework is challenging due to the intractability of the observed data likelihood, necessitating the use of computationally intensive techniques such as particle Markov chain Monte Carlo (particle MCMC). It is proposed to increase the computational and statistical efficiency of this approach by leveraging the tractability of an inexpensive surrogate derived directly from either the jump or diffusion process. The surrogate is used in three ways: in the design of a gradient-based parameter proposal, to construct an appropriate bridge and in the first stage of a delayed-acceptance step. The resulting approach, which exactly targets the posterior of interest, offers substantial gains in efficiency over a standard particle MCMC implementation.

Exact Bayesian Inference for Discretely Observed Markov Jump Processes Using Finite Rate Matrices

ABSTRACT We present new methodologies for Bayesian inference on the rate parameters of a discretely observed continuous-time Markov jump process with a countably infinite statespace. The usual method of choice for inference, particle Markov chain Monte Carlo (particle MCMC), struggles when the observation noise is small. We consider the most challenging regime of exact observations and provide two new methodologies for inference in this case: the minimal extended statespace algorithm (MESA) and the nearly minimal extended statespace algorithm (nMESA). By extending the Markov chain Monte Carlo statespace, both MESA and nMESA use the exponentiation of finite rate matrices to perform exact Bayesian inference on the Markov jump process even though its statespace is countably infinite. Numerical experiments show improvements over particle MCMC of between a factor of three and several orders of magnitude. Supplementary materials for this article are available online.

Direct statistical inference for finite Markov jump processes via the matrix exponential

Given noisy, partial observations of a time-homogeneous, finite-statespace Markov chain, conceptually simple, direct statistical inference is available, in theory, via its rate matrix, or infinitesimal generator, {mathsf {Q}}, since exp ({mathsf {Q}}t) is the transition matrix over time t. However, perhaps because of inadequate tools for matrix exponentiation in programming languages commonly used amongst statisticians or a belief that the necessary calculations are prohibitively expensive, statistical inference for continuous-time Markov chains with a large but finite state space is typically conducted via particle MCMC or other relatively complex inference schemes. When, as in many applications {mathsf {Q}} arises from a reaction network, it is usually sparse. We describe variations on known algorithms which allow fast, robust and accurate evaluation of the product of a non-negative vector with the exponential of a large, sparse rate matrix. Our implementation uses relatively recently developed, efficient, linear algebra tools that take advantage of such sparsity. We demonstrate the straightforward statistical application of the key algorithm on a model for the mixing of two alleles in a population and on the Susceptible-Infectious-Removed epidemic model.

Stochastic Epidemic Models inference and diagnosis with Poisson Random Measure Data Augmentation

We present a new Bayesian inference method for compartmental models that takes into account the intrinsic stochasticity of the process. We show how to formulate a SIR-type Markov jump process as the solution of a stochastic differential equation with respect to a Poisson Random Measure (PRM), and how to simulate the process trajectory deterministically from a parameter value and a PRM realization.This forms the basis of our Data Augmented MCMC, which consists of augmenting parameter space with the unobserved PRM value. The resulting simple Metropolis–Hastings sampler acts as an efficient simulation-based inference method, that can easily be transferred from model to model.Compared with a recent Data Augmentation method based on Gibbs sampling of individual infection histories, PRM-augmented MCMC scales much better with epidemic size and is far more flexible. It is also found to be competitive with Particle MCMC for moderate epidemics when using approximate simulations.PRM-augmented MCMC also yields a posteriori estimates of the PRM, that represent process stochasticity, and which can be used to validate the model. A pattern of deviation from the PRM prior distribution will indicate that the model underfits the data and help to understand the cause. We illustrate this by fitting a non-seasonal model to some simulated seasonal case count data.Applied to the Zika epidemic of 2013 in French Polynesia, our approach shows that a simple SEIR model cannot correctly reproduce both the initial sharp increase in the number of cases as well as the final proportion of seropositive.PRM augmentation thus provides a coherent story for Stochastic Epidemic Model inference, where explicitly inferring process stochasticity helps with model validation.

Particle MCMC With Poisson Resampling: Parallelization and Continuous Time Models

We introduce a new version of particle filter in which the number of “children” of a particle at a given time has a Poisson distribution. As a result, the number of particles is random and varies with time. An advantage of this scheme is that descendants of different particles can evolve independently. It makes easy to parallelize computations. Moreover, particle filter with Poisson resampling is readily adapted to the case when a hidden process is a continuous time, piecewise deterministic semi-Markov process. We show that the basic techniques of particle MCMC, namely particle independent Metropolis-Hastings, particle Gibbs sampler and its version with ancestor sampling, work under our Poisson resampling scheme. Our version of particle Gibbs sampler is uniformly ergodic under the same assumptions as its standard counterpart. We present simulation results which indicate that our algorithms can compete with the existing methods. Supplemental materials for this article are available online.

Particle Methods for Stochastic Differential Equation Mixed Effects Models

Parameter inference for stochastic differential equation mixed effects models (SDEMEMs) is challenging. Analytical solutions for these models are rarely available, which means that the likelihood is also intractable. In this case, exact inference (up to the discretisation of the stochastic differential equation) is possible using particle MCMC methods. Although the exact posterior is targeted by these methods, a naive implementation for SDEMEMs can be highly inefficient. Our article develops three extensions to the naive approach which exploit specific aspects of SDEMEMs and other advances such as correlated pseudo-marginal methods. We compare these methods on simulated data and data from a tumour xenography study on mice.

A Bayesian Dynamic Compositional Model for Large Density Combinations in Finance

A Bayesian dynamic compositional model is introduced that can deal with combining a large set of predictive densities. It extends the mixture of experts and the smoothly mixing regression models by allowing for combination weight dependence across models and time. A compositional model with Logistic-normal noise is specified for the latent weight dynamics and the class-preserving property of the logistic-normal is used to reduce the dimension of the latent space and to build a compositional factor model. The projection used in the dimensionality reduction is based on a dynamic clustering process which partitions the large set of predictive densities into a smaller number of subsets. We exploit the state space form of the model to provide an efficient inference procedure based on Particle MCMC. The approach is applied to track the Standard \& Poor 500 index combining 3712 predictive densities, based on 1856 US individual stocks, clustered in relatively small number of model sets. For the period 2007-2009, which included the financial crisis, substantial predictive gains are obtained, in particular, in the tails using Value-at-Risk. Similar predictive gains are obtained for the US Treasury Bill yield using a large set of macroeconomic variables. Evidence obtained on model set incompleteness and dynamic patterns in the financial clusters provide valuable signals for improved modelling and more effective economic and financial decisions.

Efficient Particle MCMC with GMM Likelihood Representation

Particle Markov Chain Monte Carlo (PMCMC) is a widely used method to handle estimation problem in the context of nonlinear structural dynamic models whose likelihood function is analytically intractable. PMCMC can be constructed upon a GMM likelihood representation when one does not want to rely on the structural form of the measurement equation (Gallant et al 2016). It only requires to compute moment conditions available from the structural model. However, particle filter with GMM may suffer from high degeneracy of particle weights which severely affects the accuracy of Monte Carlo approximations and in turn Markov Chain Monte Carlo estimates. This work is concerned with revising particle GMM algorithm as proposed in Gallant et al in order to reduce the depletion problem. Estimation results of stochastic volatility models show that the efficient block sampling strategy as proposed in Doucet et al (2006) can outperform particle GMM and in turn deliver more reliable MCMC estimates. Auxiliary particle filter (Doucet et al, 2011) is also proposed as an alternative strategy to the block sampling approach. However, in the intended experiments it does not seem to be very effective. Thus some of the assumptions needed to estimate structural nonlinear state space models can be weakened and requiring only available moment conditions without affecting dramatically the conclusions.

Ensemble Kalman Methods for High-Dimensional Hierarchical Dynamic Space-Time Models

We propose a new class of filtering and smoothing methods for inference in high-dimensional, nonlinear, non-Gaussian, spatio-temporal state-space models. The main idea is to combine the ensemble Kalman filter and smoother, developed in the geophysics literature, with state-space algorithms from the statistics literature. Our algorithms address a variety of estimation scenarios, including online and off-line state and parameter estimation. We take a Bayesian perspective, for which the goal is to generate samples from the joint posterior distribution of states and parameters. The key benefit of our approach is the use of ensemble Kalman methods for dimension reduction, which allows inference for high-dimensional state vectors. We compare our methods to existing ones, including ensemble Kalman filters, particle filters, and particle MCMC. Using a real data example of cloud motion and data simulated under a number of nonlinear and non-Gaussian scenarios, we show that our approaches outperform these existing methods. Supplementary materials for this article are available online.

Particle swarm optimization–Markov Chain Monte Carlo for accurate visual tracking with adaptive template update

A novel tracking method is proposed, which infers a target state and appearance template simultaneously. With this simultaneous inference, the method accurately estimates the target state and robustly updates the target template. The joint inference is performed by using the proposed particle swarm optimization–Markov chain Monte Carlo (PSO–MCMC) sampling method. PSO–MCMC is a combination of the particle swarm optimization (PSO) and Markov chain Monte Carlo sampling (MCMC), in which the PSO evolutionary algorithm and MCMC aim to find the target state and appearance template, respectively. The PSO can handle multi-modality in the target state and is therefore superior to a standard particle filter. Thus, PSO–MCMC achieves better performance in terms of accuracy when compared to the recently proposed particle MCMC. Experimental results demonstrate that the proposed tracker adaptively updates the target template and outperforms state-of-the-art tracking methods on a benchmark dataset.

Bayesian Forecasting for Time Series of Count Data

Time series of count data is not a widely studied research topic. This paper develops Bayesian forecasting method of counts whose conditional distributions given past observations and explanatory variables follow a Poisson distribution. To determine a forecasted value of an observation-driven integer valued autoregressive model, a series of well thought alghoritm needs to be developed, resulting in the use of Bayesian framework. This effective alghoritm sets is then used for the aforementioned calculation for the modelling of time series count data. To get the demanded results, a particle MCMC algorithm for the autoregressive Poisson regression model is introduced in the equation. Two real-life data sets, monthly demand for medicines type X and type Y (2016–2018) are successfully be analysed. We also illustrate that the Bayesian forecasting is more accurate than the corresponding frequentist’s approach.

Bayesian learning of weakly structural Markov graph laws using sequential Monte Carlo methods

We present a sequential sampling methodology for weakly structural Markov laws, arising naturally in a Bayesian structure learning context for decomposable graphical models. As a key component of our suggested approach, we show that the problem of graph estimation, which in general lacks natural sequential interpretation, can be recast into a sequential setting by proposing a recursive Feynman-Kac model that generates a flow of junction tree distributions over a space of increasing dimensions. We focus on particle McMC methods to provide samples on this space, in particular on particle Gibbs (PG), as it allows for generating McMC chains with global moves on an underlying space of decomposable graphs. To further improve the PG mixing properties, we incorporate a systematic refreshment step implemented through direct sampling from a backward kernel. The theoretical properties of the algorithm are investigated, showing that the proposed refreshment step improves the performance in terms of asymptotic variance of the estimated distribution. The suggested sampling methodology is illustrated through a collection of numerical examples demonstrating high accuracy in Bayesian graph structure learning in both discrete and continuous graphical models.

Efficient Particle MCMC with GMM Likelihood Representation

Particle Markov Chain Monte Carlo (PMCMC) is a widely used method to handle estimation problem in the context of nonlinear structural dynamic models whose likelihood function is analytically intractable. PMCMC can be constructed upon a GMM likelihood representation when one does not want to rely on the structural form of the measurement equation (Gallant et al 2016). It only requires to compute moment conditions available from the structural model. However, particle filter with GMM may suffer from high degeneracy of particle weights which severely affects the accuracy of Monte Carlo approximations and in turn Markov Chain Monte Carlo estimates. This work is concerned with revising particle GMM algorithm as proposed in Gallant et al in order to reduce the depletion problem. Estimation results of stochastic volatility models show that the efficient block sampling strategy as proposed in Doucet et al (2006) can outperform particle GMM and in turn deliver more reliable MCMC estimates. Auxiliary particle filter (Doucet et al, 2011) is also proposed as an alternative strategy to the block sampling approach. However, in the intended experiments it does not seem to be very effective. Thus some of the assumptions needed to estimate structural nonlinear state space models can be weakened and requiring only available moment conditions without affecting dramatically the conclusions.

Estimating Long-Run Risks in Developed and Developing Countries: A Particle MCMC Approach

We estimate a generalized version of the Long-Run Risk model in a panel of developed and developing countries using consumption, dividend growth, and asset returns data by utilizing the particle filter, while allowing for measurement errors in consumption data at quarterly and annual frequencies. We extend the exercise by Schorfheide et al (2018) to non-U.S. countries. Our estimations provide further evidence in support of the Long-Run Risk model and in the existence of a common small persistent component in consumption growth, dividend growth, and cash-flow growth. In certain developing countries with a short data series such as Malaysia, it is important to include asset return data to generate a large posterior estimate on the persistence parameter ρ. Our results thus further challenge the view that consumption growth is an iid process with international evidence. The estimated model can produce asset-pricing moments consistent with data.

Accounting for non-stationarity in epidemiology by embedding time-varying parameters in stochastic models.

The spread of disease through human populations is complex. The characteristics of disease propagation evolve with time, as a result of a multitude of environmental and anthropic factors, this non-stationarity is a key factor in this huge complexity. In the absence of appropriate external data sources, to correctly describe the disease propagation, we explore a flexible approach, based on stochastic models for the disease dynamics, and on diffusion processes for the parameter dynamics. Using such a diffusion process has the advantage of not requiring a specific mathematical function for the parameter dynamics. Coupled with particle MCMC, this approach allows us to reconstruct the time evolution of some key parameters (average transmission rate for instance). Thus, by capturing the time-varying nature of the different mechanisms involved in disease propagation, the epidemic can be described. Firstly we demonstrate the efficiency of this methodology on a toy model, where the parameters and the observation process are known. Applied then to real datasets, our methodology is able, based solely on simple stochastic models, to reconstruct complex epidemics, such as flu or dengue, over long time periods. Hence we demonstrate that time-varying parameters can improve the accuracy of model performances, and we suggest that our methodology can be used as a first step towards a better understanding of a complex epidemic, in situation where data is limited and/or uncertain.

A generalized Schwartz model for energy spot prices — Estimation using a particle MCMC method

We investigate a large set of energy models that account for the stylized properties in energy prices, especially stochastic volatility and spikes. The models under consideration belong to the class of factor models while our full model features a two-factor price process and a two-component stochastic volatility process. The first factor in the price process captures the normal variations; the second accounts for spikes. The two-component volatility allows for a flexible autocorrelation structure. Instead of using various filtering techniques for splitting the factors, as often found in the literature, we estimate the model in one step using the particle MCMC method. We fit the models to both the spot market and the forward market for UK natural gas. We find that the inclusion of stochastic volatility is crucial for the statistical fit of spot prices whereas the spikes are important for explaining forward prices.

Model selection for time series of count data

Selecting between competing statistical models is a challenging problem especially when the competing models are non-nested. An effective algorithm is developed in a Bayesian framework for selecting between a parameter-driven autoregressive Poisson regression model and an observation-driven integer valued autoregressive model when modelling time series count data. In order to achieve this a particle MCMC algorithm for the autoregressive Poisson regression model is introduced. The particle filter underpinning the particle MCMC algorithm plays a key role in estimating the marginal likelihood of the autoregressive Poisson regression model via importance sampling and is also utilised to estimate the DIC. The performance of the model selection algorithms are assessed via a simulation study. Two real-life data sets, monthly US polio cases (1970–1983) and monthly benefit claims from the logging industry to the British Columbia Workers Compensation Board (1985–1994) are successfully analysed.

A Nearly-Black-Box Online Algorithm for Joint Parameter and State Estimation in Temporal Models

Online joint parameter and state estimation is a core problem for temporal models.Most existing methods are either restricted to a particular class of models (e.g., the Storvik filter) or computationally expensive (e.g., particle MCMC). We propose a novel nearly-black-box algorithm, the Assumed Parameter Filter (APF), a hybrid of particle filtering for state variables and assumed density filtering for parameter variables.It has the following advantages:(a) it is online and computationally efficient;(b) it is applicable to both discrete and continuous parameter spaces with arbitrary transition dynamics.On a variety of toy and real models, APF generates more accurate results within a fixed computation budget compared to several standard algorithms from the literature.

Particle MCMC algorithms and architectures for accelerating inference in state-space models

Particle Markov Chain Monte Carlo (pMCMC) is a stochastic algorithm designed to generate samples from a probability distribution, when the density of the distribution does not admit a closed form expression. pMCMC is most commonly used to sample from the Bayesian posterior distribution in State-Space Models (SSMs), a class of probabilistic models used in numerous scientific applications. Nevertheless, this task is prohibitive when dealing with complex SSMs with massive data, due to the high computational cost of pMCMC and its poor performance when the posterior exhibits multi-modality. This paper aims to address both issues by: 1) Proposing a novel pMCMC algorithm (denoted ppMCMC), which uses multiple Markov chains (instead of the one used by pMCMC) to improve sampling efficiency for multi-modal posteriors, 2) Introducing custom, parallel hardware architectures, which are tailored for pMCMC and ppMCMC. The architectures are implemented on Field Programmable Gate Arrays (FPGAs), a type of hardware accelerator with massive parallelization capabilities. The new algorithm and the two FPGA architectures are evaluated using a large-scale case study from genetics. Results indicate that ppMCMC achieves 1.96x higher sampling efficiency than pMCMC when using sequential CPU implementations. The FPGA architecture of pMCMC is 12.1x and 10.1x faster than state-of-the-art, parallel CPU and GPU implementations of pMCMC and up to 53x more energy efficient; the FPGA architecture of ppMCMC increases these speedups to 34.9x and 41.8x respectively and is 173x more power efficient, bringing previously intractable SSM-based data analyses within reach.