Bayesian Inference for Retrospective Population Genetics Models Using Markov Chain Monte Carlo Methods

Genetics, the science of heredity and variation in living organisms, has a central role in medicine, in breeding crops and livestock, and in studying fundamental topics of biological sciences such as evolution and cell functioning. Currently the field of genetics is under a rapid development because of the recent advances in technologies by which molecular data can be obtained from living organisms. In order that most information from such data can be extracted, the analyses need to be carried out using statistical models that are tailored to take account of the particular genetic processes. In this thesis we formulate and analyze Bayesian models for genetic marker data of contemporary individuals. The major focus is on the modeling of the unobserved recent ancestry of the sampled individuals (say, for tens of generations or so), which is carried out by using explicit probabilistic reconstructions of the pedigree structures accompanied by the gene flows at the marker loci. For such a recent history, the recombination process is the major genetic force that shapes the genomes of the individuals, and it is included in the model by assuming that the recombination fractions between the adjacent markers are known. The posterior distribution of the unobserved history of the individuals is studied conditionally on the observed marker data by using a Markov chain Monte Carlo algorithm (MCMC). The example analyses consider estimation of the population structure, relatedness structure (both at the level of whole genomes as well as at each marker separately), and haplotype configurations. For situations where the pedigree structure is partially known, an algorithm to create an initial state for the MCMC algorithm is given. Furthermore, the thesis includes an extension of the model for the recent genetic history to situations where also a quantitative phenotype has been measured from the contemporary individuals. In that case the goal is to identify positions on the genome that affect the observed phenotypic values. This task is carried out within the Bayesian framework, where the number and the relative effects of the quantitative trait loci are treated as random variables whose posterior distribution is studied conditionally on the observed genetic and phenotypic data. In addition, the thesis contains an extension of a widely-used haplotyping method, the PHASE algorithm, to settings where genetic material from several individuals has been pooled together, and the allele frequencies of each pool are determined in a single genotyping.

Risk-sensitive control of continuous time Markov chains

We study risk-sensitive control of continuous time Markov chains taking values in discrete state space. We study both finite and infinite horizon problems. In the finite horizon problem we characterize the value function via Hamilton Jacobi Bellman equation and obtain an optimal Markov control. We do the same for infinite horizon discounted cost case. In the infinite horizon average cost case we establish the existence of an optimal stationary control under certain Lyapunov condition. We also develop a policy iteration algorithm for finding an optimal control.

Nonparametric Bayesian Density Modeling with Gaussian Processes

We present the Gaussian process density sampler (GPDS), an exchangeable generative model for use in nonparametric Bayesian density estimation. Samples drawn from the GPDS are consistent with exact, independent samples from a distribution defined by a density that is a transformation of a function drawn from a Gaussian process prior. Our formulation allows us to infer an unknown density from data using Markov chain Monte Carlo, which gives samples from the posterior distribution over density functions and from the predictive distribution on data space. We describe two such MCMC methods. Both methods also allow inference of the hyperparameters of the Gaussian process.

Tractable nonparametric bayesian inference in poisson processes with Gaussian process intensities

The inhomogeneous Poisson process is a point process that has varying intensity across its domain (usually time or space). For nonparametric Bayesian modeling, the Gaussian process is a useful way to place a prior distribution on this intensity. The combination of a Poisson process and GP is known as a Gaussian Cox process, or doubly-stochastic Poisson process. Likelihood-based inference in these models requires an intractable integral over an infinite-dimensional random function. In this paper we present the first approach to Gaussian Cox processes in which it is possible to perform inference without introducing approximations or finitedimensional proxy distributions. We call our method the Sigmoidal Gaussian Cox Process, which uses a generative model for Poisson data to enable tractable inference via Markov chain Monte Carlo. We compare our methods to competing methods on synthetic data and apply it to several real-world data sets. Copyright 2009.

Tractable nonparametric Bayesian inference in Poisson processes with Gaussian process intensities

The inhomogeneous Poisson process is a point process that has varying intensity across its domain (usually time or space). For nonparametric Bayesian modeling, the Gaussian process is a useful way to place a prior distribution on this intensity. The combination of a Poisson process and GP is known as a Gaussian Cox process, or doubly-stochastic Poisson process. Likelihood-based inference in these models requires an intractable integral over an infinite-dimensional random function. In this paper we present the first approach to Gaussian Cox processes in which it is possible to perform inference without introducing approximations or finite-dimensional proxy distributions. We call our method the Sigmoidal Gaussian Cox Process, which uses a generative model for Poisson data to enable tractable inference via Markov chain Monte Carlo. We compare our methods to competing methods on synthetic data and apply it to several real-world data sets.

Markov Switching Risk Premium and the term structure of interest rates. Empirical evidence from US post-war interest rates

This paper considers the basic present value model of interest rates under rational expectations with two additional features. First, following McCallum (1994), the model assumes a policy reaction function where changes in the short-term interest rate are determined by the long-short spread. Second, the short-term interest rate and the risk premium processes are characterized by a Markov regime-switching model. Using US post-war interest rate data, this paper finds evidence that a two-regime switching model fits the data better than the basic model. The estimation results also show the presence of two alternative states displaying quite different features.

Segmentação de nome e endereço por meio de modelos escondidos de Markov e sua aplicação em processos de vinculação de registros

A segmentação dos nomes nas suas partes constitutivas é uma etapa fundamental no processo de integração de bases de dados por meio das técnicas de vinculação de registros. Esta separação dos nomes pode ser realizada de diferentes maneiras. Este estudo teve como objetivo avaliar a utilização do Modelo Escondido de Markov (HMM) na segmentação nomes e endereços de pessoas e a eficiência desta segmentação no processo de vinculação de registros. Foram utilizadas as bases do Sistema de Informações sobre Mortalidade (SIM) e do Subsistema de Informação de Procedimentos de Alta Complexidade (APAC) do estado do Rio de Janeiro no período entre 1999 a 2004. Uma metodologia foi proposta para a segmentação de nome e endereço sendo composta por oito fases, utilizando rotinas implementadas em PL/SQL e a biblioteca JAHMM, implementação na linguagem Java de algoritmos de HMM. Uma amostra aleatória de 100 registros de cada base foi utilizada para verificar a correção do processo de segmentação por meio do modelo HMM.Para verificar o efeito da segmentação do nome por meio do HMM, três processos de vinculação foram aplicados sobre uma amostra das duas bases citadas acima, cada um deles utilizando diferentes estratégias de segmentação, a saber: 1) divisão dos nomes pela primeira parte, última parte e iniciais do nome do meio; 2) divisão do nome em cinco partes; (3) segmentação segundo o HMM. A aplicação do modelo HMM como mecanismo de segmentação obteve boa concordância quando comparado com o observador humano. As diferentes estratégias de segmentação geraram resultados bastante similares na vinculação de registros, tendo a estratégia 1 obtido um desempenho pouco melhor que as demais. Este estudo sugere que a segmentação de nomes brasileiros por meio do modelo escondido de Markov não é mais eficaz do que métodos tradicionais de segmentação.

Tree-Structured Stick Breaking Processes for Hierarchical Data

Many data are naturally modeled by an unobserved hierarchical structure. In this paper we propose a flexible nonparametric prior over unknown data hierarchies. The approach uses nested stick-breaking processes to allow for trees of unbounded width and depth, where data can live at any node and are infinitely exchangeable. One can view our model as providing infinite mixtures where the components have a dependency structure corresponding to an evolutionary diffusion down a tree. By using a stick-breaking approach, we can apply Markov chain Monte Carlo methods based on slice sampling to perform Bayesian inference and simulate from the posterior distribution on trees. We apply our method to hierarchical clustering of images and topic modeling of text data.

Copula Processes

We define a copula process which describes the dependencies between arbitrarily many random variables independently of their marginal distributions. As an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), to predict the latent standard deviations of a sequence of random variables. To make predictions we use Bayesian inference, with the Laplace approximation, and with Markov chain Monte Carlo as an alternative. We find both methods comparable. We also find our model can outperform GARCH on simulated and financial data. And unlike GARCH, GCPV can easily handle missing data, incorporate covariates other than time, and model a rich class of covariance structures.

Enhanced poisson sum representation for alpha-stable processes

In this paper we present Poisson sum series representations for α-stable (αS) random variables and a-stable processes, in particular concentrating on continuous-time autoregressive (CAR) models driven by α-stable Lévy processes. Our representations aim to provide a conditionally Gaussian framework, which will allow parameter estimation using Rao-Blackwellised versions of state of the art Bayesian computational methods such as particle filters and Markov chain Monte Carlo (MCMC). To overcome the issues due to truncation of the series, novel residual approximations are developed. Simulations demonstrate the potential of these Poisson sum representations for inference in otherwise intractable α-stable models. © 2011 IEEE.

Gaussian processes for fast policy optimisation of POMDP-based dialogue managers

Modelling dialogue as a Partially Observable Markov Decision Process (POMDP) enables a dialogue policy robust to speech understanding errors to be learnt. However, a major challenge in POMDP policy learning is to maintain tractability, so the use of approximation is inevitable. We propose applying Gaussian Processes in Reinforcement learning of optimal POMDP dialogue policies, in order (1) to make the learning process faster and (2) to obtain an estimate of the uncertainty of the approximation. We first demonstrate the idea on a simple voice mail dialogue task and then apply this method to a real-world tourist information dialogue task. © 2010 Association for Computational Linguistics.

Bayesian Structured Prediction Using Gaussian Processes

We introduce a conceptually novel structured prediction model, GPstruct, which is kernelized, non-parametric and Bayesian, by design. We motivate the model with respect to existing approaches, among others, conditional random fields (CRFs), maximum margin Markov networks (M3N), and structured support vector machines (SVMstruct), which embody only a subset of its properties. We present an inference procedure based on Markov Chain Monte Carlo. The framework can be instantiated for a wide range of structured objects such as linear chains, trees, grids, and other general graphs. As a proof of concept, the model is benchmarked on several natural language processing tasks and a video gesture segmentation task involving a linear chain structure. We show prediction accuracies for GPstruct which are comparable to or exceeding those of CRFs and SVMstruct.

Gaussian processes for POMDP-based dialogue manager optimization

A partially observable Markov decision process (POMDP) has been proposed as a dialog model that enables automatic optimization of the dialog policy and provides robustness to speech understanding errors. Various approximations allow such a model to be used for building real-world dialog systems. However, they require a large number of dialogs to train the dialog policy and hence they typically rely on the availability of a user simulator. They also require significant designer effort to hand-craft the policy representation. We investigate the use of Gaussian processes (GPs) in policy modeling to overcome these problems. We show that GP policy optimization can be implemented for a real world POMDP dialog manager, and in particular: 1) we examine different formulations of a GP policy to minimize variability in the learning process; 2) we find that the use of GP increases the learning rate by an order of magnitude thereby allowing learning by direct interaction with human users; and 3) we demonstrate that designer effort can be substantially reduced by basing the policy directly on the full belief space thereby avoiding ad hoc feature space modeling. Overall, the GP approach represents an important step forward towards fully automatic dialog policy optimization in real world systems. © 2013 IEEE.

Classification using log Gaussian Cox processes

McCullagh and Yang (2006) suggest a family of classification algorithms based on Cox processes. We further investigate the log Gaussian variant which has a number of appealing properties. Conditioned on the covariates, the distribution over labels is given by a type of conditional Markov random field. In the supervised case, computation of the predictive probability of a single test point scales linearly with the number of training points and the multiclass generalization is straightforward. We show new links between the supervised method and classical nonparametric methods. We give a detailed analysis of the pairwise graph representable Markov random field, which we use to extend the model to semi-supervised learning problems, and propose an inference method based on graph min-cuts. We give the first experimental analysis on supervised and semi-supervised datasets and show good empirical performance.

Equivalence and Reduction of Hidden Markov Models

This report studies when and why two Hidden Markov Models (HMMs) may represent the same stochastic process. HMMs are characterized in terms of equivalence classes whose elements represent identical stochastic processes. This characterization yields polynomial time algorithms to detect equivalent HMMs. We also find fast algorithms to reduce HMMs to essentially unique and minimal canonical representations. The reduction to a canonical form leads to the definition of 'Generalized Markov Models' which are essentially HMMs without the positivity constraint on their parameters. We discuss how this generalization can yield more parsimonious representations of stochastic processes at the cost of the probabilistic interpretation of the model parameters.