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

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

LEARNING BAYESIAN NETWORKS FOR REGRESSION FROM INCOMPLETE DATABASES*

In this paper we address the problem of inducing Bayesian network models for regression from incomplete databases. We use mixtures of truncated exponentials (MTEs) to represent the joint distribution in the induced networks. We consider two particular Bayesian network structures, the so-called na¨ıve Bayes and TAN, which have been successfully used as regression models when learning from complete data. We propose an iterative procedure for inducing the models, based on a variation of the data augmentation method in which the missing values of the explanatory variables are filled by simulating from their posterior distributions, while the missing values of the response variable are generated using the conditional expectation of the response given the explanatory variables. We also consider the refinement of the regression models by using variable selection and bias reduction. We illustrate through a set of experiments with various databases the performance of the proposed algorithms

Modelling Censored Losses Using Splicing: a Global Fit Strategy With Mixed Erlang and Extreme Value Distributions

In risk analysis, a global fit that appropriately captures the body and the tail of the distribution of losses is essential. Modelling the whole range of the losses using a standard distribution is usually very hard and often impossible due to the specific characteristics of the body and the tail of the loss distribution. A possible solution is to combine two distributions in a splicing model: a light-tailed distribution for the body which covers light and moderate losses, and a heavy-tailed distribution for the tail to capture large losses. We propose a splicing model with a mixed Erlang (ME) distribution for the body and a Pareto distribution for the tail. This combines the flexibility of the ME distribution with the ability of the Pareto distribution to model extreme values. We extend our splicing approach for censored and/or truncated data. Relevant examples of such data can be found in financial risk analysis. We illustrate the flexibility of this splicing model using practical examples from risk measurement

Learning Mixtures of Truncated Basis Functions from Data

In this paper we investigate methods for learning hybrid Bayesian networks from data. First we utilize a kernel density estimate of the data in order to translate the data into a mixture of truncated basis functions (MoTBF) representation using a convex optimization technique. When utilizing a kernel density representation of the data, the estimation method relies on the specification of a kernel bandwidth. We show that in most cases the method is robust wrt. the choice of bandwidth, but for certain data sets the bandwidth has a strong impact on the result. Based on this observation, we propose an alternative learning method that relies on the cumulative distribution function of the data. Empirical results demonstrate the usefulness of the approaches: Even though the methods produce estimators that are slightly poorer than the state of the art (in terms of log-likelihood), they are significantly faster, and therefore indicate that the MoTBF framework can be used for inference and learning in reasonably sized domains. Furthermore, we show how a particular subclass of MoTBF potentials (learnable by the proposed methods) can be exploited to significantly reduce complexity during inference

Conditional density approximations with mixtures of polynomials

Mixtures of polynomials (MoPs) are a non-parametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one- and multi-dimensional (marginal) MoPs from data have recently been proposed. In this paper we introduce two methods for learning MoP approximations of conditional densities from data. Both approaches are based on learning MoP approximations of the joint density and the marginal density of the conditioning variables, but they differ as to how the MoP approximation of the quotient of the two densities is found. We illustrate and study the methods using data sampled from known parametric distributions, and we demonstrate their applicability by learning models based on real neuroscience data. Finally, we compare the performance of the proposed methods with an approach for learning mixtures of truncated basis functions (MoTBFs). The empirical results show that the proposed methods generally yield models that are comparable to or significantly better than those found using the MoTBF-based method

Hybrid Bayesian Networks Using Mixtures of Truncated Basis Functions

This paper introduces MoTBFs, an R package for manipulating mixtures of truncated basis functions. This class of functions allows the representation of joint probability distributions involving discrete and continuous variables simultaneously, and includes mixtures of truncated exponentials and mixtures of polynomials as special cases. The package implements functions for learning the parameters of univariate, multivariate, and conditional distributions, and provides support for parameter learning in Bayesian networks with both discrete and continuous variables. Probabilistic inference using forward sampling is also implemented. Part of the functionality of the MoTBFs package relies on the bnlearn package, which includes functions for learning the structure of a Bayesian network from a data set. Leveraging this functionality, the MoTBFs package supports learning of MoTBF-based Bayesian networks over hybrid domains. We give a brief introduction to the methodological context and algorithms implemented in the package. An extensive illustrative example is used to describe the package, its functionality, and its usage