Phase Transitions in Nonlinear Filtering

It has been established under very general conditions that the ergodic properties of Markov processes are inherited by their conditional distributions given partial information. While the existing theory provides a rather complete picture of classical filtering models, many infinite-dimensional problems are outside its scope. Far from being a technical issue, the infinite-dimensional setting gives rise to surprising phenomena and new questions in filtering theory. The aim of this paper is to discuss some elementary examples, conjectures, and general theory that arise in this setting, and to highlight connections with problems in statistical mechanics and ergodic theory. In particular, we exhibit a simple example of a uniformly ergodic model in which ergodicity of the filter undergoes a phase transition, and we develop some qualitative understanding as to when such phenomena can and cannot occur. We also discuss closely related problems in the setting of conditional Markov random fields.Comment: 51 page

Infinite Hidden Conditional Random Fields for the Recognition of Human Behaviour

While detecting and interpreting temporal patterns of nonverbal behavioral cues in a given context is a natural and often unconscious process for humans, it remains a rather difficult task for computer systems. In this thesis we are primarily motivated by the problem of recognizing expressions of high--level behavior, and specifically agreement and disagreement. We thoroughly dissect the problem by surveying the nonverbal behavioral cues that could be present during displays of agreement and disagreement; we discuss a number of methods that could be used or adapted to detect these suggested cues; we list some publicly available databases these tools could be trained on for the analysis of spontaneous, audiovisual instances of agreement and disagreement, we examine the few existing attempts at agreement and disagreement classification, and we discuss the challenges in automatically detecting agreement and disagreement. We present experiments that show that an existing discriminative graphical model, the Hidden Conditional Random Field (HCRF) is the best performing on this task. The HCRF is a discriminative latent variable model which has been previously shown to successfully learn the hidden structure of a given classification problem (provided an appropriate validation of the number of hidden states). We show here that HCRFs are also able to capture what makes each of these social attitudes unique. We present an efficient technique to analyze the concepts learned by the HCRF model and show that these coincide with the findings from social psychology regarding which cues are most prevalent in agreement and disagreement. Our experiments are performed on a spontaneous expressions dataset curated from real televised debates. The HCRF model outperforms conventional approaches such as Hidden Markov Models and Support Vector Machines. Subsequently, we examine existing graphical models that use Bayesian nonparametrics to have a countably infinite number of hidden states and adapt their complexity to the data at hand. We identify a gap in the literature that is the lack of a discriminative such graphical model and we present our suggestion for the first such model: an HCRF with an infinite number of hidden states, the Infinite Hidden Conditional Random Field (IHCRF). In summary, the IHCRF is an undirected discriminative graphical model for sequence classification and uses a countably infinite number of hidden states. We present two variants of this model. The first is a fully nonparametric model that relies on Hierarchical Dirichlet Processes and a Markov Chain Monte Carlo inference approach. The second is a semi--parametric model that uses Dirichlet Process Mixtures and relies on a mean--field variational inference approach. We show that both models are able to converge to a correct number of represented hidden states, and perform as well as the best finite HCRFs ---chosen via cross--validation--- for the difficult tasks of recognizing instances of agreement, disagreement, and pain in audiovisual sequences.Open Acces

Conditional ergodicity in infinite dimension

The goal of this paper is to develop a general method to establish conditional ergodicity of infinite-dimensional Markov chains. Given a Markov chain in a product space, we aim to understand the ergodic properties of its conditional distributions given one of the components. Such questions play a fundamental role in the ergodic theory of nonlinear filters. In the setting of Harris chains, conditional ergodicity has been established under general nondegeneracy assumptions. Unfortunately, Markov chains in infinite-dimensional state spaces are rarely amenable to the classical theory of Harris chains due to the singularity of their transition probabilities, while topological and functional methods that have been developed in the ergodic theory of infinite-dimensional Markov chains are not well suited to the investigation of conditional distributions. We must therefore develop new measure-theoretic tools in the ergodic theory of Markov chains that enable the investigation of conditional ergodicity for infinite dimensional or weak-* ergodic processes. To this end, we first develop local counterparts of zero-two laws that arise in the theory of Harris chains. These results give rise to ergodic theorems for Markov chains that admit asymptotic couplings or that are locally mixing in the sense of H. F\"{o}llmer, and to a non-Markovian ergodic theorem for stationary absolutely regular sequences. We proceed to show that local ergodicity is inherited by conditioning on a nondegenerate observation process. This is used to prove stability and unique ergodicity of the nonlinear filter. Finally, we show that our abstract results can be applied to infinite-dimensional Markov processes that arise in several settings, including dissipative stochastic partial differential equations, stochastic spin systems and stochastic differential delay equations.Comment: Published in at http://dx.doi.org/10.1214/13-AOP879 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Tutorial on Bayesian Nonparametric Models

A key problem in statistical modeling is model selection, how to choose a model at an appropriate level of complexity. This problem appears in many settings, most prominently in choosing the number ofclusters in mixture models or the number of factors in factor analysis. In this tutorial we describe Bayesian nonparametric methods, a class of methods that side-steps this issue by allowing the data to determine the complexity of the model. This tutorial is a high-level introduction to Bayesian nonparametric methods and contains several examples of their application.Comment: 28 pages, 8 figure

The Entropy of a Binary Hidden Markov Process

The entropy of a binary symmetric Hidden Markov Process is calculated as an expansion in the noise parameter epsilon. We map the problem onto a one-dimensional Ising model in a large field of random signs and calculate the expansion coefficients up to second order in epsilon. Using a conjecture we extend the calculation to 11th order and discuss the convergence of the resulting series

Many Roads to Synchrony: Natural Time Scales and Their Algorithms

We consider two important time scales---the Markov and cryptic orders---that monitor how an observer synchronizes to a finitary stochastic process. We show how to compute these orders exactly and that they are most efficiently calculated from the epsilon-machine, a process's minimal unifilar model. Surprisingly, though the Markov order is a basic concept from stochastic process theory, it is not a probabilistic property of a process. Rather, it is a topological property and, moreover, it is not computable from any finite-state model other than the epsilon-machine. Via an exhaustive survey, we close by demonstrating that infinite Markov and infinite cryptic orders are a dominant feature in the space of finite-memory processes. We draw out the roles played in statistical mechanical spin systems by these two complementary length scales.Comment: 17 pages, 16 figures: http://cse.ucdavis.edu/~cmg/compmech/pubs/kro.htm. Santa Fe Institute Working Paper 10-11-02