Sinan Yıldırım

An Online Expectation-Maximization Algorithm for Changepoint Models

Changepoint models are widely used to model the heterogeneity of sequential data. We present a novel sequential Monte Carlo (SMC) online expectation–maximization (EM) algorithm for estimating the static parameters of such models. The SMC online EM algorithm has a cost per time which is linear in the number of particles and could be particularly important when the data is representable as a long sequence of observations, since it drastically reduces the computational requirements for implementation. We present an asymptotic analysis for the stability of the SMC estimates used in the online EM algorithm and demonstrate the performance of this scheme by using both simulated and real data originating from DNA analysis. The supplementary materials for the article are available online.

Parameter Estimation in Hidden Markov Models With Intractable Likelihoods Using Sequential Monte Carlo

We propose sequential Monte Carlo-based algorithms for maximum likelihood estimation of the static parameters in hidden Markov models with an intractable likelihood using ideas from approximate Bayesian computation. The static parameter estimation algorithms are gradient-based and cover both offline and online estimation. We demonstrate their performance by estimating the parameters of three intractable models, namely the α-stable distribution, g-and-k distribution, and the stochastic volatility model with α-stable returns, using both real and synthetic data.

A Bayesian Deconvolution Approach for Receiver Function Analysis

In this paper, we propose a Bayesian methodology for receiver function analysis, a key tool in determining the deep structure of the Earths crust. We exploit the assumption of sparsity for receiver functions to develop a Bayesian deconvolution method as an alternative to the widely used iterative deconvolution. We model samples of a sparse signal as i.i.d. Student-t random variables. Gibbs sampling and variational Bayes techniques are investigated for our specific posterior inference problem. We used those techniques within the expectation-maximization (EM) algorithm to estimate our unknown model parameters. The superiority of the Bayesian deconvolution is demonstrated by the experiments on both simulated and real earthquake data.

Calibrating the Gaussian multi-target tracking model

We present novel batch and online (sequential) versions of the expectation–maximisation (EM) algorithm for inferring the static parameters of a multiple target tracking (MTT) model. Online EM is of particular interest as it is a more practical method for long data sets since in batch EM, or a full Bayesian approach, a complete browse of the data is required between successive parameter updates. Online EM is also suited to MTT applications that demand real-time processing of the data. Performance is assessed in numerical examples using simulated data for various scenarios. For batch estimation our method significantly outperforms an existing gradient based maximum likelihood technique, which we show to be significantly biased.

Bayesian Tracking and Parameter Learning for Non-Linear Multiple Target Tracking Models

This paper proposes a new Bayesian tracking and parameter learning algorithm for non-linear and non-Gaussian multiple target tracking (MTT) models. A Markov chain Monte Carlo (MCMC) algorithm is designed to sample from the posterior distribution of the target states, birth and death times, and association of observations to targets, which constitutes the solution to the tracking problem, as well as the model parameters. The numerical section presents performance comparisons with several competing techniques and demonstrates significant performance improvements in all cases.

MCMC Shape Sampling for Image Segmentation with Nonparametric Shape Priors

Segmenting images of low quality or with missing data is a challenging problem. Integrating statistical prior information about the shapes to be segmented can improve the segmentation results significantly. Most shape-based segmentation algorithms optimize an energy functional and find a point estimate for the object to be segmented. This does not provide a measure of the degree of confidence in that result, neither does it provide a picture of other probable solutions based on the data and the priors. With a statistical view, addressing these issues would involve the problem of characterizing the posterior densities of the shapes of the objects to be segmented. For such characterization, we propose a Markov chain Monte Carlo (MCMC) sampling-based image segmentation algorithm that uses statistical shape priors. In addition to better characterization of the statistical structure of the problem, such an approach would also have the potential to address issues with getting stuck at local optima, suffered by existing shape-based segmentation methods. Our approach is able to characterize the posterior probability density in the space of shapes through its samples, and to return multiple solutions, potentially from different modes of a multimodal probability density, which would be encountered, e.g., in segmenting objects from multiple shape classes. We present promising results on a variety of data sets. We also provide an extension for segmenting shapes of objects with parts that can go through independent shape variations. This extension involves the use of local shape priors on object parts and provides robustness to limitations in shape training data size.

A hybrid method for deconvolution of Bernoulli-Gaussian processes

We investigate a hybrid method which improves the quality of state inference and parameter estimation in blind deconvolution of a sparse source modeled by a Bernoulli-Gaussian process. In this problem, when both the signal and the filter are jointly estimated, the true posterior is typically highly multimodal. Therefore, when not properly initialized, standard stochastic inference methods, (MCEM, SEM or SAEM), tend to get stuck and suffer from poor convergence. In our approach, we first relax the Bernoulli-Gaussian prior model by a Student-t model. Our simulations suggest that deterministic inference in the relaxed model is not only efficient, but also provides a very good initialization for the Bernoulli-Gaussian model. We provide simulation studies that compare the results obtained with and without our initialization method for several combinations of state inference and parameter estimation methods used for the Bernoulli-Gaussian model.

An online expectation-maximisation algorithm for nonnegative matrix factorisation models

In this paper we formulate the nonnegative matrix factorisation (NMF) problem as a maximum likelihood estimation problem for hidden Markov models and propose online expectation-maximisation (EM) algorithms to estimate the NMF and the other unknown static parameters. We also propose a sequential Monte Carlo approximation of our online EM algorithm. We show the performance of the proposed method with two numerical examples.

A Markov chain Monte Carlo algorithm for Bayesian policy search

ABSTRACT Policy search algorithms have facilitated application of Reinforcement Learning (RL) to dynamic systems, such as control of robots. Many policy search algorithms are based on the policy gradient, and thus may suffer from slow convergence or local optima complications. In this paper, we take a Bayesian approach to policy search under RL paradigm, for the problem of controlling a discrete time Markov decision process with continuous state and action spaces and with a multiplicative reward structure. For this purpose, we assume a prior over policy parameters and aim for the ‘posterior’ distribution where the ‘likelihood’ is the expected reward. We propound a Markov chain Monte Carlo algorithm as a method of generating samples for policy parameters from this posterior. The proposed algorithm is compared with certain well-known policy gradient-based RL methods and exhibits more appropriate performance in terms of time response and convergence rate, when applied to a nonlinear model of a Cart-Pole benchmark.

Maximum likelihood parameter estimation in time series models using sequential Monte Carlo

Time series models are used to characterise uncertainty in many real-world dynamical phenomena. A time series model typically contains a static variable, called parameter, which parametrizes the joint law of the random variables involved in the definition of the model. When a time series model is to be fitted to some sequentially observed data, it is essential to decide on the value of the parameter that describes the data best, a procedure generally called parameter estimation. This thesis comprises novel contributions to the methodology on parameter estimation in time series models. Our primary interest is online estimation, although batch estimation is also considered. The developed methods are based on batch and online versions of expectation-maximisation (EM) and gradient ascent, two widely popular algorithms for maximum likelihood estimation (MLE). In the last two decades, the range of statistical models where parameter estimation can be performed has been significantly extended with the development of Monte Carlo methods. We provide contribution to the field in a similar manner, namely by combining EM and gradient ascent algorithms with sequential Monte Carlo (SMC) techniques. The time series models we investigate are widely used in statistical and engineering applications. The original work of this thesis is organised in Chapters 4 to 7. Chapter 4 contains an online EM algorithm using SMC for MLE in changepoint models, which are widely used to model heterogeneity in sequential data. In Chapter 5, we present batch and online EM algorithms using SMC for MLE in linear Gaussian multiple target tracking models. Chapter 6 contains a novel methodology for implementing MLE in a hidden Markov model having intractable probability densities for its observations. Finally, in Chapter 7 we formulate the nonnegative matrix factorisation problem as MLE in a specific hidden Markov model and propose online EM algorithms using SMC to perform MLE.