Thomas Dyhre Nielsen

INFERENCE IN HYBRID BAYESIAN NETWORKS

Abstract Since the 1980s, Bayesian networks (BNs) have become increasingly popular for building statistical models of complex systems. This is particularly true for boolean systems, where BNs often prove to be a more efficient modelling framework than traditional reliability techniques (like fault trees and reliability block diagrams). However, limitations in the BNs’ calculation engine have prevented BNs from becoming equally popular for domains containing mixtures of both discrete and continuous variables (the so-called hybrid domains). In this paper we focus on these difficulties, and summarize some of the last decades research on inference in hybrid Bayesian networks. The discussions are linked to an example model for estimating human reliability.

Classification using Hierarchical Naïve Bayes models

Classification problems have a long history in the machine learning literature. One of the simplest, and yet most consistently well-performing set of classifiers is the Naïve Bayes models. However, an inherent problem with these classifiers is the assumption that all attributes used to describe an instance are conditionally independent given the class of that instance. When this assumption is violated (which is often the case in practice) it can reduce classification accuracy due to “information double-counting” and interaction omission.In this paper we focus on a relatively new set of models, termed Hierarchical Naïve Bayes models. Hierarchical Naïve Bayes models extend the modeling flexibility of Naïve Bayes models by introducing latent variables to relax some of the independence statements in these models. We propose a simple algorithm for learning Hierarchical Naïve Bayes models in the context of classification. Experimental results show that the learned models can significantly improve classification accuracy as compared to other frameworks.

Mixtures of truncated basis functions

In this paper we propose a framework, called mixtures of truncated basis functions (MoTBFs), for representing general hybrid Bayesian networks. The proposed framework generalizes both the mixture of truncated exponentials (MTEs) framework and the Mixture of Polynomials (MoPs) framework. Similar to MTEs and MoPs, MoTBFs are defined so that the potentials are closed under combination and marginalization, which ensures that inference in MoTBF networks can be performed efficiently using the Shafer-Shenoy architecture. Based on a generalized Fourier series approximation, we devise a method for efficiently approximating an arbitrary density function using the MoTBF framework. The translation method is more flexible than existing MTE or MoP-based methods, and it supports an online/anytime tradeoff between the accuracy and the complexity of the approximation. Experimental results show that the approximations obtained are either comparable or significantly better than the approximations obtained using existing methods.

Latent variable discovery in classification models

The naive Bayes model makes the often unrealistic assumption that the feature variables are mutually independent given the class variable. We interpret a violation of this assumption as an indication of the presence of latent variables, and we show how latent variables can be detected. Latent variable discovery is interesting, especially for medical applications, because it can lead to a better understanding of application domains. It can also improve classification accuracy and boost user confidence in classification models.

Learning a decision maker's utility function from (possibly) inconsistent behavior

When modeling a decision problem using the influence diagram framework, the quantitative part rests on two principal components: probabilities for representing the decision makers uncertainty about the domain and utilities for representing preferences. Over the last decade, several methods have been developed for learning the probabilities from a database. However, methods for learning the utilities have only received limited attention in the computer science community.A promising approach for learning a decision makers utility function is to take outset in the decision makers observed behavioral patterns, and then find a utility function which (together with a domain model) can explain this behavior. That is, it is assumed that decision makers preferences are reflected in the behavior. Standard learning algorithms also assume that the decision maker is behavioral consistent, i.e., given a model of the decision problem, there exists a utility function which can account for all the observed behavior. Unfortunately, this assumption is rarely valid in real-world decision problems, and in these situations existing learning methods may only identify a trivial utility function. In this paper we relax this consistency assumption, and propose two algorithms for learning a decision makers utility function from possibly inconsistent behavior; inconsistent behavior is interpreted as random deviations from an underlying (true) utility function. The main difference between the two algorithms is that the first facilitates a form of batch learning whereas the second focuses on adaptation and is particularly well-suited for scenarios where the DMs preferences change over time. Empirical results demonstrate the tractability of the algorithms, and they also show that the algorithms converge toward the true utility function for even very small sets of observations.

Learning Probabilistic Automata for Model Checking

Obtaining accurate system models for verification is a hard and time consuming process, which is seen by industry as a hindrance to adopt otherwise powerful model driven development techniques and tools. In this paper we pursue an alternative approach where an accurate high-level model can be automatically constructed from observations of a given black-box embedded system. We adapt algorithms for learning finite probabilistic automata from observed system behaviors. We prove that in the limit of large sample sizes the learned model will be an accurate representation of the data-generating system. In particular, in the large sample limit, the learned model and the original system will define the same probabilities for linear temporal logic (LTL) properties. Thus, we can perform PLTL model-checking on the learned model to infer properties of the system. We perform experiments learning models from system observations at different levels of abstraction. The experimental results show the learned models provide very good approximations for relevant properties of the original system.

Adapting Bayes network structures to non-stationary domains

When an incremental structural learning method gradually modifies a Bayesian network (BN) structure to fit a sequential stream of observations, we call the process structural adaptation. Structural adaptation is useful when the learner is set to work in an unknown environment, where a BN is gradually being constructed as observations of the environment are made. Existing algorithms for incremental learning assume that the samples in the database have been drawn from a single underlying distribution. In this paper we relax this assumption, so that the underlying distribution can change during the sampling of the database. The proposed method can thus be used in unknown environments, where it is not even known whether the dynamics of the environment are stable. We state formal correctness results for our method, and demonstrate its feasibility experimentally.

Parameter estimation and model selection for mixtures of truncated exponentials

Bayesian 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.

A latent model for collaborative filtering

Recommender systems based on collaborative filtering have received a great deal of interest over the last two decades. In particular, recently proposed methods based on dimensionality reduction techniques and using a symmetrical representation of users and items have shown promising results. Following this line of research, we propose a probabilistic collaborative filtering model that explicitly represents all items and users simultaneously in the model. Experimental results show that the proposed system obtains significantly better results than other collaborative filtering systems (evaluated on the MovieLens data set). Furthermore, the explicit representation of all users and items allows the model to e.g. make group-based recommendations balancing the preferences of the individual users.

Sequential influence diagrams: A unified asymmetry framework

We describe a new graphical language for specifying asymmetric decision problems. The language is based on a filtered merge of several existing languages including sequential valuation networks, asymmetric influence diagrams, and unconstrained influence diagrams. Asymmetry is encoded using a structure resembling a clustered decision tree, whereas the representation of the uncertainty model is based on the (unconstrained) influence diagram framework. We illustrate the proposed language by modeling several highly asymmetric decision problems, and we describe an efficient solution procedure.