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Sumeetpal S. Singh

Publications and source records attributed to Sumeetpal S. Singh.

36 records · Page 2Linked to original sources
arXiv2018

On the loss of Fisher information in some multi-object tracking observation models

The concept of Fisher information can be useful even in cases where the probability distributions of interest are not absolutely continuous with respect to the natural reference measure on the underlying space. Practical examples where this extension is useful are provided in the context of multi-object tracking statistical models. Upon defining the Fisher information without introducing a reference measure, we provide remarkably concise proofs of the loss of Fisher information in some widely used multi-object tracking observation models.

math.ST↗
arXiv2017

Approximate Smoothing and Parameter Estimation in High-Dimensional State-Space Models

We present approximate algorithms for performing smoothing in a class of high-dimensional state-space models via sequential Monte Carlo methods ("particle filters"). In high dimensions, a prohibitively large number of Monte Carlo samples ("particles") -- growing exponentially in the dimension of the state space -- is usually required to obtain a useful smoother. Using blocking strategies as in Rebeschini and Van Handel (2015) (and earlier pioneering work on blocking), we exploit the spatial ergodicity properties of the model to circumvent this curse of dimensionality. We thus obtain approximate smoothers that can be computed recursively in time and in parallel in space. First, we show that the bias of our blocked smoother is bounded uniformly in the time horizon and in the model dimension. We then approximate the blocked smoother with particles and derive the asymptotic variance of idealised versions of our blocked particle smoother to show that variance is no longer adversely effected by the dimension of the model. Finally, we employ our method to successfully perform maximum-likelihood estimation via stochastic gradient-ascent and stochastic expectation--maximisation algorithms in a 100-dimensional state-space model.

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arXiv2017

Identification of multi-object dynamical systems: consistency and Fisher information

Learning the model parameters of a multi-object dynamical system from partial and perturbed observations is a challenging task. Despite recent numerical advancements in learning these parameters, theoretical guarantees are extremely scarce. In this article, we study the identifiability of these parameters and the consistency of the corresponding maximum likelihood estimate (MLE) under assumptions on the different components of the underlying multi-object system. In order to understand the impact of the various sources of observation noise on the ability to learn the model parameters, we study the asymptotic variance of the MLE through the associated Fisher information matrix. For example, we show that specific aspects of the multi-target tracking (MTT) problem such as detection failures and unknown data association lead to a loss of information which is quantified in special cases of interest.

math.ST↗
arXiv2016

Tracking multiple moving objects in images using Markov Chain Monte Carlo

A new Bayesian state and parameter learning algorithm for multiple target tracking (MTT) models with image observations is proposed. Specifically, a Markov chain Monte Carlo algorithm is designed to sample from the posterior distribution of the unknown number of targets, their birth and death times, states and model parameters, which constitutes the complete solution to the tracking problem. The conventional approach is to pre-process the images to extract point observations and then perform tracking. We model the image generation process directly to avoid potential loss of information when extracting point observations. Numerical examples show that our algorithm has improved tracking performance over commonly used techniques, for both synthetic examples and real florescent microscopy data, especially in the case of dim targets with overlapping illuminated regions.

stat.AP↗
arXiv2015

Blocking Strategies and Stability of Particle Gibbs Samplers

Sampling from the conditional (or posterior) probability distribution of the latent states of a Hidden Markov Model, given the realization of the observed process, is a non-trivial problem in the context of Markov Chain Monte Carlo. To do this Andrieu et al. (2010) constructed a Markov kernel which leaves this conditional distribution invariant using a Particle Filter. From a practitioner's point of view, this Markov kernel attempts to mimic the act of sampling all the latent state variables as one block from the posterior distribution but for models where exact simulation is not possible. There are some recent theoretical results that establish the uniform ergodicity of this Markov kernel and that the mixing rate does not diminish provided the number of particles grows at least linearly with the number of latent states in the posterior. This gives rise to a cost, per application of the kernel, that is quadratic in the number of latent states which could be prohibitive for long observation sequences. We seek to answer an obvious but important question: is there a different implementation with a cost per-iteration that grows linearly with the number of latent states, but which is still stable in the sense that its mixing rate does not deteriorate? We address this problem using blocking strategies, which are easily parallelizable, and prove stability of the resulting sampler.

math.ST↗
arXiv2015

On Particle Methods for Parameter Estimation in State-Space Models

Nonlinear non-Gaussian state-space models are ubiquitous in statistics, econometrics, information engineering and signal processing. Particle methods, also known as Sequential Monte Carlo (SMC) methods, provide reliable numerical approximations to the associated state inference problems. However, in most applications, the state-space model of interest also depends on unknown static parameters that need to be estimated from the data. In this context, standard particle methods fail and it is necessary to rely on more sophisticated algorithms. The aim of this paper is to present a comprehensive review of particle methods that have been proposed to perform static parameter estimation in state-space models. We discuss the advantages and limitations of these methods and illustrate their performance on simple models.

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arXiv2015

On particle Gibbs sampling

The particle Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm to sample from the full posterior distribution of a state-space model. It does so by executing Gibbs sampling steps on an extended target distribution defined on the space of the auxiliary variables generated by an interacting particle system. This paper makes the following contributions to the theoretical study of this algorithm. Firstly, we present a coupling construction between two particle Gibbs updates from different starting points and we show that the coupling probability may be made arbitrarily close to one by increasing the number of particles. We obtain as a direct corollary that the particle Gibbs kernel is uniformly ergodic. Secondly, we show how the inclusion of an additional Gibbs sampling step that reselects the ancestors of the particle Gibbs' extended target distribution, which is a popular approach in practice to improve mixing, does indeed yield a theoretically more efficient algorithm as measured by the asymptotic variance. Thirdly, we extend particle Gibbs to work with lower variance resampling schemes. A detailed numerical study is provided to demonstrate the efficiency of particle Gibbs and the proposed variants.

stat.CO↗
arXiv2015

Particle ancestor sampling for near-degenerate or intractable state transition models

We consider Bayesian inference in sequential latent variable models in general, and in nonlinear state space models in particular (i.e., state smoothing). We work with sequential Monte Carlo (SMC) algorithms, which provide a powerful inference framework for addressing this problem. However, for certain challenging and common model classes the state-of-the-art algorithms still struggle. The work is motivated in particular by two such model classes: (i) models where the state transition kernel is (nearly) degenerate, i.e. (nearly) concentrated on a low-dimensional manifold, and (ii) models where point-wise evaluation of the state transition density is intractable. Both types of models arise in many applications of interest, including tracking, epidemiology, and econometrics. The difficulties with these types of models is that they essentially rule out forward-backward-based methods, which are known to be of great practical importance, not least to construct computationally efficient particle Markov chain Monte Carlo (PMCMC) algorithms. To alleviate this, we propose a "particle rejuvenation" technique to enable the use of the forward-backward strategy for (nearly) degenerate models and, by extension, for intractable models. We derive the proposed method specifically within the context of PMCMC, but we emphasise that it is applicable to any forward-backward-based Monte Carlo method.

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arXiv2014

Bayesian tracking and parameter learning for non-linear multiple target tracking models

We propose a new Bayesian tracking and parameter learning algorithm for non-linear non-Gaussian multiple target tracking (MTT) models. We design a Markov chain Monte Carlo (MCMC) algorithm 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. In the numerical section, we present performance comparisons with several competing techniques and demonstrate significant performance improvements in all cases.

stat.AP↗
arXiv2014

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.

cs.LG↗
arXiv2012

Estimating the Static Parameters in Linear Gaussian Multiple Target Tracking Models

We present both offline and online maximum likelihood estimation (MLE) techniques for inferring the static parameters of a multiple target tracking (MTT) model with linear Gaussian dynamics. We present the batch and online versions of the expectation-maximisation (EM) algorithm for short and long data sets respectively, and we show how Monte Carlo approximations of these methods can be implemented. Performance is assessed in numerical examples using simulated data for various scenarios and a comparison with a Bayesian estimation procedure is also provided.

stat.AP↗
arXiv2012

Bayesian learning of noisy Markov decision processes

We consider the inverse reinforcement learning problem, that is, the problem of learning from, and then predicting or mimicking a controller based on state/action data. We propose a statistical model for such data, derived from the structure of a Markov decision process. Adopting a Bayesian approach to inference, we show how latent variables of the model can be estimated, and how predictions about actions can be made, in a unified framework. A new Markov chain Monte Carlo (MCMC) sampler is devised for simulation from the posterior distribution. This step includes a parameter expansion step, which is shown to be essential for good convergence properties of the MCMC sampler. As an illustration, the method is applied to learning a human controller.

stat.ML↗
arXiv2012

Approximate Bayesian Computation for Smoothing

We consider a method for approximate inference in hidden Markov models (HMMs). The method circumvents the need to evaluate conditional densities of observations given the hidden states. It may be considered an instance of Approximate Bayesian Computation (ABC) and it involves the introduction of auxiliary variables valued in the same space as the observations. The quality of the approximation may be controlled to arbitrary precision through a parameter ε>0 . We provide theoretical results which quantify, in terms of ε, the ABC error in approximation of expectations of additive functionals with respect to the smoothing distributions. Under regularity assumptions, this error is O(nε), where n is the number of time steps over which smoothing is performed. For numerical implementation we adopt the forward-only sequential Monte Carlo (SMC) scheme of [16] and quantify the combined error from the ABC and SMC approximations. This forms some of the first quantitative results for ABC methods which jointly treat the ABC and simulation errors, with a finite number of data and simulated samples. When the HMM has unknown static parameters, we consider particle Markov chain Monte Carlo [2] (PMCMC) methods for batch statistical inference.

stat.CO↗
arXiv2012

Distributed Maximum Likelihood for Simultaneous Self-localization and Tracking in Sensor Networks

We show that the sensor self-localization problem can be cast as a static parameter estimation problem for Hidden Markov Models and we implement fully decentralized versions of the Recursive Maximum Likelihood and on-line Expectation-Maximization algorithms to localize the sensor network simultaneously with target tracking. For linear Gaussian models, our algorithms can be implemented exactly using a distributed version of the Kalman filter and a novel message passing algorithm. The latter allows each node to compute the local derivatives of the likelihood or the sufficient statistics needed for Expectation-Maximization. In the non-linear case, a solution based on local linearization in the spirit of the Extended Kalman Filter is proposed. In numerical examples we demonstrate that the developed algorithms are able to learn the localization parameters.

math.OC↗
arXiv2011

Asymptotic Behaviour of Approximate Bayesian Estimators

Although approximate Bayesian computation (ABC) has become a popular technique for performing parameter estimation when the likelihood functions are analytically intractable there has not as yet been a complete investigation of the theoretical properties of the resulting estimators. In this paper we give a theoretical analysis of the asymptotic properties of ABC based parameter estimators for hidden Markov models and show that ABC based estimators satisfy asymptotically biased versions of the standard results in the statistical literature.

math.ST↗
arXiv2011

Parameter Estimation for Hidden Markov Models with Intractable Likelihoods

Approximate Bayesian computation (ABC) is a popular technique for approximating likelihoods and is often used in parameter estimation when the likelihood functions are analytically intractable. Although the use of ABC is widespread in many fields, there has been little investigation of the theoretical properties of the resulting estimators. In this paper we give a theoretical analysis of the asymptotic properties of ABC based maximum likelihood parameter estimation for hidden Markov models. In particular, we derive results analogous to those of consistency and asymptotic normality for standard maximum likelihood estimation. We also discuss how Sequential Monte Carlo methods provide a natural method for implementing likelihood based ABC procedures.

math.ST↗
arXiv2009

A Backward Particle Interpretation of Feynman-Kac Formulae

We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can be used to compute normalized additive functionals "on-the-fly" as well as their limiting occupation measures with a given precision degree that does not depend on the final time horizon. We provide uniform convergence results w.r.t. the time horizon parameter as well as functional central limit theorems and exponential concentration estimates. We also illustrate these results in the context of computational physics and imaginary time Schroedinger type partial differential equations, with a special interest in the numerical approximation of the invariant measure associated to $h$-processes.

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