Distribution of Mutual Information

The mutual information of two random variables i and j with joint probabilities t_ij is commonly used in learning Bayesian nets as well as in many other fields. The chances t_ij are usually estimated by the empirical sampling frequency n_ij/n leading to a point estimate I(n_ij/n) for the mutual information. To answer questions like "is I(n_ij/n) consistent with zero?" or "what is the probability that the true mutual information is much larger than the point estimate?" one has to go beyond the point estimate. In the Bayesian framework one can answer these questions by utilizing a (second order) prior distribution p(t) comprising prior information about t. From the prior p(t) one can compute the posterior p(t|n), from which the distribution p(I|n) of the mutual information can be calculated. We derive reliable and quickly computable approximations for p(I|n). We concentrate on the mean, variance, skewness, and kurtosis, and non-informative priors. For the mean we also give an exact expression. Numerical issues and the range of validity are discussed.Comment: 8 page

Model Selection for Gaussian Mixture Models

This paper is concerned with an important issue in finite mixture modelling, the selection of the number of mixing components. We propose a new penalized likelihood method for model selection of finite multivariate Gaussian mixture models. The proposed method is shown to be statistically consistent in determining of the number of components. A modified EM algorithm is developed to simultaneously select the number of components and to estimate the mixing weights, i.e. the mixing probabilities, and unknown parameters of Gaussian distributions. Simulations and a real data analysis are presented to illustrate the performance of the proposed method

Intentional Motion On-line Learning and Prediction

International audiencePredicting motion of humans, animals and other objects which move according to internal plans is a challenging problem. Most existing approaches operate in two stages: a) learning typical motion patterns by observing an environment and b) predicting future motion on the basis of the learned patterns. In existing techniques, learning is performed off-line, hence, it is impossible to refine the existing knowledge on the basis of the new observations obtained during the prediction phase. We propose an approach which uses Hidden Markov Models to represent motion patterns. It is different from similar approaches because it is able to learn and predict in a concurrent fashion thanks to a novel approximate learning approach, based on the Growing Neural Gas algorithm, which estimates both HMM parameters and structure. The found structure has the property of being a planar graph, thus enabling exact inference in linear time with respect to the number of states in the model. Our experiments demonstrate that the technique works in real-time, and is able to produce sound long-term predictions of people motion

Learning from Partial Labels with Minimum Entropy

This paper introduces the minimum entropy regularizer for learning from partial labels. This learning problem encompasses the semi-supervised setting, where a decision rule is to be learned from labeled and unlabeled examples. The minimum entropy regularizer applies to diagnosis models, i.e. models of the posterior probabilities of classes. It is shown to include other approaches to the semi-supervised problem as particular or limiting cases. A series of experiments illustrates that the proposed criterion provides solutions taking advantage of unlabeled examples when the latter convey information. Even when the data are sampled from the distribution class spanned by a generative model, the proposed approach improves over the estimated generative model when the number of features is of the order of sample size. The performances are definitely in favor of minimum entropy when the generative model is slightly misspecified. Finally, the robustness of the learning scheme is demonstrated: in situations where unlabeled examples do not convey information, minimum entropy returns a solution discarding unlabeled examples and performs as well as supervised learning. Cet article introduit le régularisateur à entropie minimum pour l'apprentissage d'étiquettes partielles. Ce problème d'apprentissage incorpore le cadre non supervisé, où une règle de décision doit être apprise à partir d'exemples étiquetés et non étiquetés. Le régularisateur à entropie minimum s'applique aux modèles de diagnostics, c'est-à-dire aux modèles des probabilités postérieures de classes. Nous montrons comment inclure d'autres approches comme un cas particulier ou limité du problème semi-supervisé. Une série d'expériences montrent que le critère proposé fournit des solutions utilisant les exemples non étiquetés lorsque ces dernières sont instructives. Même lorsque les données sont échantillonnées à partir de la classe de distribution balayée par un modèle génératif, l'approche mentionnée améliore le modèle génératif estimé lorsque le nombre de caractéristiques est de l'ordre de la taille de l'échantillon. Les performances avantagent certainement l'entropie minimum lorsque le modèle génératif est légèrement mal spécifié. Finalement, la robustesse de ce cadre d'apprentissage est démontré : lors de situations où les exemples non étiquetés n'apportent aucune information, l'entropie minimum retourne une solution rejetant les exemples non étiquetés et est aussi performante que l'apprentissage supervisé.discriminant learning, semi-supervised learning, minimum entropy, apprentissage discriminant, apprentissage semi-supervisé, entropie minimum

Learning Model Structure from Data : an Application to On-Line Handwriting

We present a learning strategy for Hidden Markov Models that may be used to cluster handwriting sequences or to learn a character model by identifying its main writing styles. Our approach aims at learning both the structure and parameters of a Hidden Markov Model (HMM) from the data. A byproduct of this learning strategy is the ability to cluster signals and identify allograph. We provide experimental results on artificial data that demonstrate the possibility to learn from data HMM parameters and topology. For a given topology, our approach outperforms in some cases that we identify standard Maximum Likelihood learning scheme. We also apply our unsupervised learning scheme on on-line handwritten signals for allograph clustering as well as for learning HMM models for handwritten digit recognition

Model Predictive Control and Fault Detection and Diagnostics of a Building Heating, Ventilation, and Air Conditioning System

The paper presents Model Predictive Control (MPC) and Fault Detection and Diagnostics (FDD) technologies, their on-line implementation, and results from several demonstrations conducted for a large-size HVAC system. The two technologies are executed at the supervisory level in a hierarchical control architecture as extensions of a baseline Building Management System (BMS). The MPC algorithm generates optimal set points for the HVAC actuator loops which minimize energy consumption while meeting equipment operational constraints and occupant comfort constraints. The MPC algorithm is implemented using a new tool, the Berkeley Library for Optimization Modeling (BLOM), which generates automatically an efficient optimization formulation directly from a simulation model. The FDD algorithm detects and classifies in real-time potential faults of the HVAC actuators based on data from multiple sensors. The performance and limitations of FDD and MPC algorithms are illustrated and discussed based on measurement data recorded from multiple tests

On Separation Between Learning and Control in Partially Observed Markov Decision Processes

Cyber-physical systems (CPS) encounter a large volume of data which is added to the system gradually in real time and not altogether in advance. As the volume of data increases, the domain of the control strategies also increases, and thus it becomes challenging to search for an optimal strategy. Even if an optimal control strategy is found, implementing such strategies with increasing domains is burdensome. To derive an optimal control strategy in CPS, we typically assume an ideal model of the system. Such model-based control approaches cannot effectively facilitate optimal solutions with performance guarantees due to the discrepancy between the model and the actual CPS. Alternatively, traditional supervised learning approaches cannot always facilitate robust solutions using data derived offline. Similarly, applying reinforcement learning approaches directly to the actual CPS might impose significant implications on safety and robust operation of the system. The goal of this chapter is to provide a theoretical framework that aims at separating the control and learning tasks which allows us to combine offline model-based control with online learning approaches, and thus circumvent the challenges in deriving optimal control strategies for CPS.Comment: 18 pages, 5 figures. arXiv admin note: text overlap with arXiv:2101.1099

Structure Learning in Conditional Probability Models via an Entropic Prior and Parameter Extinction

We introduce an entropic prior for multinomial parameter estimation problems and solve for its maximum..