Bayesian Nonparametric Hidden Semi-Markov Models

There is much interest in the Hierarchical Dirichlet Process Hidden Markov Model (HDP-HMM) as a natural Bayesian nonparametric extension of the ubiquitous Hidden Markov Model for learning from sequential and time-series data. However, in many settings the HDP-HMM's strict Markovian constraints are undesirable, particularly if we wish to learn or encode non-geometric state durations. We can extend the HDP-HMM to capture such structure by drawing upon explicit-duration semi-Markovianity, which has been developed mainly in the parametric frequentist setting, to allow construction of highly interpretable models that admit natural prior information on state durations. In this paper we introduce the explicit-duration Hierarchical Dirichlet Process Hidden semi-Markov Model (HDP-HSMM) and develop sampling algorithms for efficient posterior inference. The methods we introduce also provide new methods for sampling inference in the finite Bayesian HSMM. Our modular Gibbs sampling methods can be embedded in samplers for larger hierarchical Bayesian models, adding semi-Markov chain modeling as another tool in the Bayesian inference toolbox. We demonstrate the utility of the HDP-HSMM and our inference methods on both synthetic and real experiments

Diffusion of Context and Credit Information in Markovian Models

This paper studies the problem of ergodicity of transition probability matrices in Markovian models, such as hidden Markov models (HMMs), and how it makes very difficult the task of learning to represent long-term context for sequential data. This phenomenon hurts the forward propagation of long-term context information, as well as learning a hidden state representation to represent long-term context, which depends on propagating credit information backwards in time. Using results from Markov chain theory, we show that this problem of diffusion of context and credit is reduced when the transition probabilities approach 0 or 1, i.e., the transition probability matrices are sparse and the model essentially deterministic. The results found in this paper apply to learning approaches based on continuous optimization, such as gradient descent and the Baum-Welch algorithm.Comment: See http://www.jair.org/ for any accompanying file

Retrospective Higher-Order Markov Processes for User Trails

Users form information trails as they browse the web, checkin with a geolocation, rate items, or consume media. A common problem is to predict what a user might do next for the purposes of guidance, recommendation, or prefetching. First-order and higher-order Markov chains have been widely used methods to study such sequences of data. First-order Markov chains are easy to estimate, but lack accuracy when history matters. Higher-order Markov chains, in contrast, have too many parameters and suffer from overfitting the training data. Fitting these parameters with regularization and smoothing only offers mild improvements. In this paper we propose the retrospective higher-order Markov process (RHOMP) as a low-parameter model for such sequences. This model is a special case of a higher-order Markov chain where the transitions depend retrospectively on a single history state instead of an arbitrary combination of history states. There are two immediate computational advantages: the number of parameters is linear in the order of the Markov chain and the model can be fit to large state spaces. Furthermore, by providing a specific structure to the higher-order chain, RHOMPs improve the model accuracy by efficiently utilizing history states without risks of overfitting the data. We demonstrate how to estimate a RHOMP from data and we demonstrate the effectiveness of our method on various real application datasets spanning geolocation data, review sequences, and business locations. The RHOMP model uniformly outperforms higher-order Markov chains, Kneser-Ney regularization, and tensor factorizations in terms of prediction accuracy

An Overview of Schema Theory

The purpose of this paper is to give an introduction to the field of Schema Theory written by a mathematician and for mathematicians. In particular, we endeavor to to highlight areas of the field which might be of interest to a mathematician, to point out some related open problems, and to suggest some large-scale projects. Schema theory seeks to give a theoretical justification for the efficacy of the field of genetic algorithms, so readers who have studied genetic algorithms stand to gain the most from this paper. However, nothing beyond basic probability theory is assumed of the reader, and for this reason we write in a fairly informal style. Because the mathematics behind the theorems in schema theory is relatively elementary, we focus more on the motivation and philosophy. Many of these results have been proven elsewhere, so this paper is designed to serve a primarily expository role. We attempt to cast known results in a new light, which makes the suggested future directions natural. This involves devoting a substantial amount of time to the history of the field. We hope that this exposition will entice some mathematicians to do research in this area, that it will serve as a road map for researchers new to the field, and that it will help explain how schema theory developed. Furthermore, we hope that the results collected in this document will serve as a useful reference. Finally, as far as the author knows, the questions raised in the final section are new.Comment: 27 pages. Originally written in 2009 and hosted on my website, I've decided to put it on the arXiv as a more permanent home. The paper is primarily expository, so I don't really know where to submit it, but perhaps one day I will find an appropriate journa