Localizing the Latent Structure Canonical Uncertainty: Entropy Profiles for Hidden Markov Models

This report addresses state inference for hidden Markov models. These models rely on unobserved states, which often have a meaningful interpretation. This makes it necessary to develop diagnostic tools for quantification of state uncertainty. The entropy of the state sequence that explains an observed sequence for a given hidden Markov chain model can be considered as the canonical measure of state sequence uncertainty. This canonical measure of state sequence uncertainty is not reflected by the classic multivariate state profiles computed by the smoothing algorithm, which summarizes the possible state sequences. Here, we introduce a new type of profiles which have the following properties: (i) these profiles of conditional entropies are a decomposition of the canonical measure of state sequence uncertainty along the sequence and makes it possible to localize this uncertainty, (ii) these profiles are univariate and thus remain easily interpretable on tree structures. We show how to extend the smoothing algorithms for hidden Markov chain and tree models to compute these entropy profiles efficiently.Comment: Submitted to Journal of Machine Learning Research; No RR-7896 (2012

Statistical Physics Analysis of Maximum a Posteriori Estimation for Multi-channel Hidden Markov Models

The performance of Maximum a posteriori (MAP) estimation is studied analytically for binary symmetric multi-channel Hidden Markov processes. We reduce the estimation problem to a 1D Ising spin model and define order parameters that correspond to different characteristics of the MAP-estimated sequence. The solution to the MAP estimation problem has different operational regimes separated by first order phase transitions. The transition points for $L$-channel system with identical noise levels, are uniquely determined by $L$ being odd or even, irrespective of the actual number of channels. We demonstrate that for lower noise intensities, the number of solutions is uniquely determined for odd $L$, whereas for even $L$ there are exponentially many solutions. We also develop a semi analytical approach to calculate the estimation error without resorting to brute force simulations. Finally, we examine the tradeoff between a system with single low-noise channel and one with multiple noisy channels.Comment: The paper has been submitted to Journal of Statistical Physics with submission number JOSS-S-12-0039

Bayesian regularization of hidden Markov models with an application to bioinformatics

This paper discusses a Bayesian approach to regularizing hidden Markov models and demonstrates an application of this scheme to Bioinformatics

A generalized risk approach to path inference based on hidden Markov models

Motivated by the unceasing interest in hidden Markov models (HMMs), this paper re-examines hidden path inference in these models, using primarily a risk-based framework. While the most common maximum a posteriori (MAP), or Viterbi, path estimator and the minimum error, or Posterior Decoder (PD), have long been around, other path estimators, or decoders, have been either only hinted at or applied more recently and in dedicated applications generally unfamiliar to the statistical learning community. Over a decade ago, however, a family of algorithmically defined decoders aiming to hybridize the two standard ones was proposed (Brushe et al., 1998). The present paper gives a careful analysis of this hybridization approach, identifies several problems and issues with it and other previously proposed approaches, and proposes practical resolutions of those. Furthermore, simple modifications of the classical criteria for hidden path recognition are shown to lead to a new class of decoders. Dynamic programming algorithms to compute these decoders in the usual forward-backward manner are presented. A particularly interesting subclass of such estimators can be also viewed as hybrids of the MAP and PD estimators. Similar to previously proposed MAP-PD hybrids, the new class is parameterized by a small number of tunable parameters. Unlike their algorithmic predecessors, the new risk-based decoders are more clearly interpretable, and, most importantly, work "out of the box" in practice, which is demonstrated on some real bioinformatics tasks and data. Some further generalizations and applications are discussed in conclusion.Comment: Section 5: corrected denominators of the scaled beta variables (pp. 27-30), => corrections in claims 1, 3, Prop. 12, bottom of Table 1. Decoder (49), Corol. 14 are generalized to handle 0 probabilities. Notation is more closely aligned with (Bishop, 2006). Details are inserted in eqn-s (43); the positivity assumption in Prop. 11 is explicit. Fixed typing errors in equation (41), Example

Probabilistic Methodology and Techniques for Artefact Conception and Development

The purpose of this paper is to make a state of the art on probabilistic methodology and techniques for artefact conception and development. It is the 8th deliverable of the BIBA (Bayesian Inspired Brain and Artefacts) project. We first present the incompletness problem as the central difficulty that both living creatures and artefacts have to face: how can they perceive, infer, decide and act efficiently with incomplete and uncertain knowledge?. We then introduce a generic probabilistic formalism called Bayesian Programming. This formalism is then used to review the main probabilistic methodology and techniques. This review is organized in 3 parts: first the probabilistic models from Bayesian networks to Kalman filters and from sensor fusion to CAD systems, second the inference techniques and finally the learning and model acquisition and comparison methodologies. We conclude with the perspectives of the BIBA project as they rise from this state of the art