Variational Bayesian methods for spatial data analysis

With scientific data available at geocoded locations, investigators are increasingly turning to spatial process models for carrying out statistical inference. However, fitting spatial models often involves expensive matrix decompositions, whose computational complexity increases in cubic order with the number of spatial locations. This situation is aggravated in Bayesian settings where such computations are required once at every iteration of the Markov chain Monte Carlo (MCMC) algorithms. In this paper, we describe the use of Variational Bayesian (VB) methods as an alternative to MCMC to approximate the posterior distributions of complex spatial models. Variational methods, which have been used extensively in Bayesian machine learning for several years, provide a lower bound on the marginal likelihood, which can be computed efficiently. We provide results for the variational updates in several models especially emphasizing their use in multivariate spatial analysis. We demonstrate estimation and model comparisons from VB methods by using simulated data as well as environmental data sets and compare them with inference from MCMC.     

Using a parallelized MCMC algorithm in R to identify appropriate likelihood functions for SWAT

Markov Chain Monte Carlo (MCMC) algorithms allow the analysis of parameter uncertainty. This analysis can inform the choice of appropriate likelihood functions, thereby advancing hydrologic modeling with improved parameter and quantity estimates and more reliable assessment of uncertainty. For long-running models, the Differential Evolution Adaptive Metropolis (DREAM) algorithm offers spectacular reductions in time required for MCMC analysis. This is partly due to multiple parameter sets being evaluated simultaneously. The ability to use this feature is hindered in models that have a large number of input files, such as SWAT. A conceptually simple, robust method for applying DREAM to SWAT in R is provided. The general approach is transferrable to any executable that reads input files. We provide this approach to reduce barriers to the use of MCMC algorithms and to promote the development of appropriate likelihood functions.     

Computational tools for comparing asymmetric GARCH models via Bayes factors

In this paper we use Markov chain Monte Carlo (MCMC) methods in order to estimate and compare GARCH models from a Bayesian perspective. We allow for possibly heavy tailed and asymmetric distributions in the error term. We use a general method proposed in the literature to introduce skewness into a continuous unimodal and symmetric distribution. For each model we compute an approximation to the marginal likelihood, based on the MCMC output. From these approximations we compute Bayes factors and posterior model probabilities.     

Simulation-based Bayesian inference for epidemic models

A powerful and flexible method for fitting dynamic models to missing and censored data is to use the Bayesian paradigm via data-augmented Markov chain Monte Carlo (DA-MCMC). This samples from the joint posterior for the parameters and missing data, but requires high memory overheads for large-scale systems. In addition, designing efficient proposal distributions for the missing data is typically challenging. Pseudo-marginal methods instead integrate across the missing data using a Monte Carlo estimate for the likelihood, generated from multiple independent simulations from the model. These techniques can avoid the high memory requirements of DA-MCMC, and under certain conditions produce the exact marginal posterior distribution for parameters. A novel method is presented for implementing importance sampling for dynamic epidemic models, by conditioning the simulations on sets of validity criteria (based on the model structure) as well as the observed data. The flexibility of these techniques is illustrated using both removal time and final size data from an outbreak of smallpox. It is shown that these approaches can circumvent the need for reversible-jump MCMC, and can allow inference in situations where DA-MCMC is impossible due to computationally infeasible likelihoods.     

Combining particle MCMC with Rao-Blackwellized Monte Carlo data association for parameter estimation in multiple target tracking

We consider state and parameter estimation in multiple target tracking problems with data association uncertainties and unknown number of targets. We show how the problem can be recast into a conditionally linear Gaussian state-space model with unknown parameters and present an algorithm for computationally efficient inference on the resulting model. The proposed algorithm is based on combining the Rao-Blackwellized Monte Carlo data association algorithm with particle Markov chain Monte Carlo algorithms to jointly estimate both parameters and data associations. Both particle marginal Metropolis–Hastings and particle Gibbs variants of particle MCMC are considered. We demonstrate the performance of the method both using simulated data and in a real-data case study of using multiple target tracking to estimate the brown bear population in Finland.     

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.     

Parameterisation and efficient MCMC estimation of non-Gaussian state space models

The impact of parameterisation on the simulation efficiency of Bayesian Markov chain Monte Carlo (MCMC) algorithms for two non-Gaussian state space models is examined. Specifically, focus is given to particular forms of the stochastic conditional duration (SCD) model and the stochastic volatility (SV) model, with four alternative parameterisations of each model considered. A controlled experiment using simulated data reveals that relationships exist between the simulation efficiency of the MCMC sampler, the magnitudes of the population parameters and the particular parameterisation of the state space model. Results of an empirical analysis of two separate transaction data sets for the SCD model, as well as equity and exchange rate data sets for the SV model, are also reported. Both the simulation and empirical results reveal that substantial gains in simulation efficiency can be obtained from simple reparameterisations of both types of non-Gaussian state space models.     

Classification with Bayesian MARS

We present a new method for classification using a Bayesian version of the Multivariate Adaptive Regression Spline (MARS) model of J.H. Friedman (Annals of Statistics, 19, 1–141, 1991). Special attention is paid to the use of Markov chain Monte Carlo (MCMC) simulation to gain inference under the model. In particular we discuss three important developments in MCMC methodology. First, we describe the reversible jump MCMC algorithm of P.J. Green (Biometrika, 82, 711–732, 1995) which allows inference on a varying dimensional, possibly uncountable, model space. This allows us to consider MARS models of differing numbers and positions of splines. Secondly, we discuss marginalisation which is used to reduce the effective dimension of the parameter space under consideration. Thirdly, we describe the use of latent variables to improve the MCMC computation. Our methods are generic and can be applied to any basis function model including, wavelets, artificial neural nets and radial basis functions. We present examples to show that the Bayesian MARS classifier is competitive with other approaches on a number of benchmark data sets.     

Expectation-maximization algorithms for inference in Dirichlet processes mixture

Mixture models are ubiquitous in applied science. In many real-world applications, the number of mixture components needs to be estimated from the data. A popular approach consists of using information criteria to perform model selection. Another approach which has become very popular over the past few years consists of using Dirichlet processes mixture (DPM) models. Both approaches are computationally intensive. The use of information criteria requires computing the maximum likelihood parameter estimates for each candidate model whereas DPM are usually trained using Markov chain Monte Carlo (MCMC) or variational Bayes (VB) methods. We propose here original batch and recursive expectation-maximization algorithms to estimate the parameters of DPM. The performance of our algorithms is demonstrated on several applications including image segmentation and image classification tasks. Our algorithms are computationally much more efficient than MCMC and VB and outperform VB on an example.     

A Markov basis for conditional test of common diagonal effect in quasi-independence model for square contingency tables

In two-way contingency tables we sometimes find that frequencies along the diagonal cells are relatively larger (or smaller) compared to off-diagonal cells, particularly in square tables with the common categories for the rows and the columns. In this case the quasi-independence model with an additional parameter for each of the diagonal cells is usually fitted to the data. A simpler model than the quasi-independence model is to assume a common additional parameter for all the diagonal cells. We consider testing the goodness of fit of the common diagonal effect by the Markov chain Monte Carlo (MCMC) method. We derive an explicit form of a Markov basis for performing the conditional test of the common diagonal effect. Once a Markov basis is given, MCMC procedure can be easily implemented by techniques of algebraic statistics. We illustrate the procedure with some real data sets.     

Estimation of sample selection models with two selection mechanisms

This paper focuses on estimating sample selection models with two incidentally truncated outcomes and two corresponding selection mechanisms. The method of estimation is an extension of the Markov chain Monte Carlo (MCMC) sampling algorithm from Chib (2007) and Chib et al. (2009). Contrary to conventional data augmentation strategies when dealing with missing data, the proposed algorithm augments the posterior with only a small subset of the total missing data caused by sample selection. This results in improved convergence of the MCMC chain and decreased storage costs, while maintaining tractability in the sampling densities. The methods are applied to estimate the effects of residential density on vehicle miles traveled and vehicle holdings in California.     

Efficient direct sampling MCEM algorithm for latent variable models with binary responses

While latent variable models have been successfully applied in many fields and underpin various modeling techniques, their ability to incorporate categorical responses is hindered due to the lack of accurate and efficient estimation methods. Approximation procedures, such as penalized quasi-likelihood, are computationally efficient, but the resulting estimators can be seriously biased for binary responses. Gauss-Hermite quadrature and Markov Chain Monte Carlo (MCMC) integration based methods can yield more accurate estimation, but they are computationally much more intensive. Estimation methods that can achieve both computational efficiency and estimation accuracy are still under development. This paper proposes an efficient direct sampling based Monte Carlo EM algorithm (DSMCEM) for latent variable models with binary responses. Mixed effects and item factor analysis models with binary responses are used to illustrate this algorithm. Results from two simulation studies and a real data example suggest that, as compared with MCMC based EM, DSMCEM can significantly improve computational efficiency as well as produce equally accurate parameter estimates. Other aspects and extensions of the algorithm are discussed.     

Semiparametric regression with shape-constrained penalized splines

In semiparametric regression models, penalized splines can be used to describe complex, non-linear relationships between the mean response and covariates. In some applications it is desirable to restrict the shape of the splines so as to enforce properties such as monotonicity or convexity on regression functions. We describe a method for imposing such shape constraints on penalized splines within a linear mixed model framework. We employ Markov chain Monte Carlo (MCMC) methods for model fitting, using a truncated prior distribution to impose the requisite shape restrictions. We develop a computationally efficient MCMC sampler by using a correspondingly truncated multivariate normal proposal distribution, which is a restricted version of the approximate sampling distribution of the model parameters in an unconstrained version of the model. We also describe a cheap approximation to this methodology that can be applied for shape-constrained scatterplot smoothing. Our methods are illustrated through two applications, the first involving the length of dugongs and the second concerned with growth curves for sitka spruce trees.     

Using capture-recapture data and hybrid Monte Carlo sampling to estimate an animal population affected by an environmental catastrophe

We propose a dynamic model for the evolution of an open animal population that is subject to an environmental catastrophe. The model incorporates a capture-recapture experiment often conducted for studying wildlife population, and enables inferences on the population size and possible effect of the catastrophe. A Bayesian approach is used to model unobserved quantities in the problem as latent variables and Markov chain Monte Carlo (MCMC) is used for posterior computation. Because the particular interrelationship between observed and latent variables negates the feasibility of standard MCMC methods, we propose a hybrid Monte Carlo approach that integrates a Gibbs sampler with the strategies of sequential importance sampling (SIS) and acceptance-rejection (AR) sampling for model estimation. We develop results on how to construct effective proposal densities for the SIS scheme. The approach is illustrated through a simulation study, and is applied to data from a mountain pygmy possum (Burramys Parvus) population that was affected by a bushfire.     

Exchange Monte Carlo Sampling From Bayesian Posterior for Singular Learning Machines

Many singular learning machines such as neural networks, normal mixtures, Bayesian networks, and hidden Markov models belong to singular learning machines and are widely used in practical information systems. In these learning machines, it is well known that Bayesian learning provides better generalization performance than maximum-likelihood estimation. However, it needs huge computational cost to sample from a Bayesian posterior distribution of a singular learning machine by a conventional Markov chain Monte Carlo (MCMC) method, such as the metropolis algorithm, because of singularities. Recently, the exchange Monte Carlo (MC) method, which is well known as an improved algorithm of MCMC method, has been proposed to apply to Bayesian neural network learning in the literature. In this paper, we propose the idea that the exchange MC method has a better effect on Bayesian learning in singular learning machines than that in regular learning machines, and show its effectiveness by comparing the numerical stochastic complexity with the theoretical one.      

Modeling individual email patterns over time with latent variable models

As digital communication devices play an increasingly prominent role in our daily lives, the ability to analyze and understand our communication patterns becomes more important. In this paper, we investigate a latent variable modeling approach for extracting information from individual email histories, focusing in particular on understanding how an individual communicates over time with recipients in their social network. The proposed model consists of latent groups of recipients, each of which is associated with a piecewise-constant Poisson rate over time. Inference of group memberships, temporal changepoints, and rate parameters is carried out via Markov Chain Monte Carlo (MCMC) methods. We illustrate the utility of the model by applying it to both simulated and real-world email data sets.     

Mining boundary effects in areally referenced spatial data using the Bayesian information criterion

Statistical models for areal data are primarily used for smoothing maps revealing spatial trends. Subsequent interest often resides in the formal identification of ‘boundaries’ on the map. Here boundaries refer to ‘difference boundaries’, representing significant differences between adjacent regions. Recently, Lu and Carlin (Geogr Anal 37:265–285, 2005) discussed a Bayesian framework to carry out edge detection employing a spatial hierarchical model that is estimated using Markov chain Monte Carlo (MCMC) methods. Here we offer an alternative that avoids MCMC and is easier to implement. Our approach resembles a model comparison problem where the models correspond to different underlying edge configurations across which we wish to smooth (or not). We incorporate these edge configurations in spatially autoregressive models and demonstrate how the Bayesian Information Criteria (BIC) can be used to detect difference boundaries in the map. We illustrate our methods with a Minnesota Pneumonia and Influenza Hospitalization dataset to elicit boundaries detected from the different models.     

Model-based clustering for longitudinal data

A model-based clustering method is proposed for clustering individuals on the basis of measurements taken over time. Data variability is taken into account through non-linear hierarchical models leading to a mixture of hierarchical models. We study both frequentist and Bayesian estimation procedures. From a classical viewpoint, we discuss maximum likelihood estimation of this family of models through the EM algorithm. From a Bayesian standpoint, we develop appropriate Markov chain Monte Carlo (MCMC) sampling schemes for the exploration of target posterior distribution of parameters. The methods are illustrated with the identification of hormone trajectories that are likely to lead to adverse pregnancy outcomes in a group of pregnant women.     

Fast sampling methods for Bayesian max-margin models

Bayesian max-margin models have shown superiority in various practical applications, such as text categorization, collaborative prediction, social network link prediction and crowdsourcing, and they conjoin the flexibility of Bayesian modeling and predictive strengths of max-margin learning. However, Monte Carlo sampling for these models still remains challenging, especially for applications that involve large-scale datasets. In this paper, we present the stochastic subgradient Hamiltonian Monte Carlo (HMC) methods, which are easy to implement and computationally efficient. We show the approximate detailed balance property of subgradient HMC which reveals a natural and validated generalization of the ordinary HMC. Furthermore, we investigate the variants that use stochastic subsampling and thermostats for better scalability and mixing. Using stochastic subgradient Markov Chain Monte Carlo (MCMC), we efficiently solve the posterior inference task of various Bayesian max-margin models and extensive experimental results demonstrate the effectiveness of our approach.     

Stratified Markov Chain Monte Carlo Light Transport

Markov chain Monte Carlo (MCMC) sampling is a powerful approach to generate samples from an arbitrary distribution. The application to light transport simulation allows us to efficiently handle complex light transport such as highly occluded scenes. Since light transport paths in MCMC methods are sampled according to the path contributions over the sampling domain covering the whole image, bright pixels receive more samples than dark pixels to represent differences in the brightness. This variation in the number of samples per pixel is a fundamental property of MCMC methods. This property often leads to uneven convergence over the image, which is a notorious and fundamental issue of any MCMC method to date. We present a novel stratification method of MCMC light transport methods. Our stratification method, for the first time, breaks the fundamental limitation that the number of samples per pixel is uncontrollable. Our method guarantees that every pixel receives a specified number of samples by running a single Markov chain per pixel. We rely on the fact that different MCMC processes should converge to the same result when the sampling domain and the integrand are the same. We thus subdivide an image into multiple overlapping tiles associated with each pixel, run an independent MCMC process in each of them, and then align all of the tiles such that overlapping regions match. This can be formulated as an optimization problem similar to the reconstruction step for gradient-domain rendering. Further, our method can exploit the coherency of integrands among neighboring pixels via coherent Markov chains and replica exchange. Images rendered with our method exhibit much more predictable convergence compared to existing MCMC methods.