Approximate probability propagation with mixtures of truncated exponentials

AbstractMixtures of truncated exponentials (MTEs) are a powerful alternative to discretisation when working with hybrid Bayesian networks. One of the features of the MTE model is that standard propagation algorithms can be used. However, the complexity of the process is too high and therefore approximate methods, which tradeoff complexity for accuracy, become necessary. In this paper we propose an approximate propagation algorithm for MTE networks which is based on the Penniless propagation method already known for discrete variables. We also consider how to use Markov Chain Monte Carlo to carry out the probability propagation. The performance of the proposed methods is analysed in a series of experiments with random networks

Approximate Probability Propagation with Mixtures of Truncated Exponentials*

Mixtures of truncated exponentials (MTEs) are a powerful alternative to discretisation when working with hybrid Bayesian networks. One of the features of the MTE model is that standard propagation algorithms can be used. However, the complexity of the process is too high and therefore approximate methods, which tradeoff complexity for accuracy, become necessary. In this paper we propose an approximate propagation algorithm for MTE networks which is based on the Penniless propagation method already known for discrete variables. We also consider how to use Markov Chain Monte Carlo to carry out the probability propagation. The performance of the proposed methods is analysed in a series of experiments with random networks

Supervised Classification Using Probabilistic Decision Graphs

A new model for supervised classification based on probabilistic decision graphs is introduced. A probabilistic decision graph (PDG) is a graphical model that efficiently captures certain context specific independencies that are not easily represented by other graphical models traditionally used for classification, such as the Naïve Bayes (NB) or Classification Trees (CT). This means that the PDG model can capture some distributions using fewer parameters than classical models. Two approaches for constructing a PDG for classification are proposed. The first is to directly construct the model from a dataset of labelled data, while the second is to transform a previously obtained Bayesian classifier into a PDG model that can then be refined. These two approaches are compared with a wide range of classical approaches to the supervised classification problem on a number of both real world databases and artificially generated data

Answering queries in hybrid Bayesian networks using importance sampling

In this paper we propose an algorithm for answering queries in hybrid Bayesian networks where the underlying probability distribution is of class MTE (mixture of truncated exponentials). The algorithm is based on importance sampling simulation. We show how, like existing importance sampling algorithms for discrete networks, it is able to provide answers to multiple queries simultaneously using a single sample. The behaviour of the new algorithm is experimentally tested and compared with previous methods existing in the literature

Learning hybrid Bayesian networks using mixtures of truncated exponentials

In this paper we introduce an algorithm for learning hybrid Bayesian networks from data. The result of the algorithm is a network where the conditional distribution for each variable is a mixture of truncated exponentials (MTE), so that no restrictions on the network topology are imposed. The structure of the network is obtained by searching over the space of candidate networks using optimisation methods. The conditional densities are estimated by means of Gaussian kernel densities that afterwards are approximated by MTEs, so that the resulting network is appropriate for using standard algorithms for probabilistic reasoning. The behaviour of the proposed algorithm is tested using a set of real-world and arti cially generated databases

Parameter Estimation and Model Selection for Mixtures of Truncated Exponentials

AbstractBayesian networks with mixtures of truncated exponentials (MTEs) support efficient inference algorithms and provide a flexible way of modeling hybrid domains (domains containing both discrete and continuous variables). On the other hand, estimating an MTE from data has turned out to be a difficult task, and most prevalent learning methods treat parameter estimation as a regression problem. The drawback of this approach is that by not directly attempting to find the parameter estimates that maximize the likelihood, there is no principled way of performing subsequent model selection using those parameter estimates. In this paper we describe an estimation method that directly aims at learning the parameters of an MTE potential following a maximum likelihood approach. Empirical results demonstrate that the proposed method yields significantly better likelihood results than existing regression-based methods. We also show how model selection, which in the case of univariate MTEs amounts to partitioning the domain and selecting the number of exponential terms, can be performed using the BIC score

Unsupervised naive Bayes for data clustering with mixtures of truncated exponentials

In this paper we propose a naive Bayes model for unsupervised data clustering, where the class variable is hidden. The feature variables can be discrete or continuous, as the conditional distributions are represented as mixtures of truncated exponentials (MTEs). The number of classes is determined using the data augmentation algorithm. The proposed model is compared with the conditional Gaussian model for some real world and synthetic databases

A Review of Inference Algorithms for Hybrid Bayesian Networks

Hybrid Bayesian networks have received an increasing attention during the last years. The difference with respect to standard Bayesian networks is that they can host discrete and continuous variables simultaneously, which extends the applicability of the Bayesian network framework in general. However, this extra feature also comes at a cost: inference in these types of models is computationally more challenging and the underlying models and updating procedures may not even support closed-form solutions. In this paper we provide an overview of the main trends and principled approaches for performing inference in hybrid Bayesian networks. The methods covered in the paper are organized and discussed according to their methodological basis. We consider how the methods have been extended and adapted to also include (hybrid) dynamic Bayesian networks, and we end with an overview of established software systems supporting inference in these types of models

Estimating mixtures of truncated exponentials from data

The MTE (mixture of truncated exponentials) model allows to deal with Bayesian networks containing discrete and continuous variables simultaneously. One of the features of this model is that standard propagation algorithms can be applied. In this paper, we study the problem of estimating these models from data. We propose an iterative algorithm based on least squares approximation. The performance of the algorithm is tested both with artificial and actual data