Alessandro Antonucci

Epistemic irrelevance in credal nets: The case of imprecise Markov trees

We focus on credal nets, which are graphical models that generalise Bayesian nets to imprecise probability. We replace the notion of strong independence commonly used in credal nets with the weaker notion of epistemic irrelevance, which is arguably more suited for a behavioural theory of probability. Focusing on directed trees, we show how to combine the given local uncertainty models in the nodes of the graph into a global model, and we use this to construct and justify an exact message-passing algorithm that computes updated beliefs for a variable in the tree. The algorithm, which is linear in the number of nodes, is formulated entirely in terms of coherent lower previsions, and is shown to satisfy a number of rationality requirements. We supply examples of the algorithms operation, and report an application to on-line character recognition that illustrates the advantages of our approach for prediction. We comment on the perspectives, opened by the availability, for the first time, of a truly efficient algorithm based on epistemic irrelevance.

Decision-theoretic specification of credal networks: A unified language for uncertain modeling with sets of Bayesian networks

Credal networks are models that extend Bayesian nets to deal with imprecision in probability, and can actually be regarded as sets of Bayesian nets. Credal nets appear to be powerful means to represent and deal with many important and challenging problems in uncertain reasoning. We give examples to show that some of these problems can only be modeled by credal nets called non-separately specified. These, however, are still missing a graphical representation language and updating algorithms. The situation is quite the opposite with separately specified credal nets, which have been the subject of much study and algorithmic development. This paper gives two major contributions. First, it delivers a new graphical language to formulate any type of credal network, both separately and non-separately specified. Second, it shows that any non-separately specified net represented with the new language can be easily transformed into an equivalent separately specified net, defined over a larger domain. This result opens up a number of new outlooks and concrete outcomes: first of all, it immediately enables the existing algorithms for separately specified credal nets to be applied to non-separately specified ones. We explore this possibility for the 2U algorithm: an algorithm for exact updating of singly connected credal nets, which is extended by our results to a class of non-separately specified models. We also consider the problem of inference on Bayesian networks, when the reason that prevents some of the variables from being observed is unknown. The problem is first reformulated in the new graphical language, and then mapped into an equivalent problem on a separately specified net. This provides a first algorithmic approach to this kind of inference, which is also proved to be NP-hard by similar transformations based on our formalism.

Bayesian networks with imprecise probabilities: theory and application to classification

Bayesian networks are powerful probabilistic graphical models for modelling uncertainty. Among others, classification represents an important application: some of the most used classifiers are based on Bayesian networks. Bayesian networks are precise models: exact numeric values should be provided for quantification. This requirement is sometimes too narrow. Sets instead of single distributions can provide a more realistic description in these cases. Bayesian networks can be generalized to cope with sets of distributions. This leads to a novel class of imprecise probabilistic graphical models, called credal networks. In particular, classifiers based on Bayesian networks are generalized to so-called credal classifiers. Unlike Bayesian classifiers, which always detect a single class as the one maximizing the posterior class probability, a credal classifier may eventually be unable to discriminate a single class. In other words, if the available information is not sufficient, credal classifiers allow for indecision between two or more classes, this providing a less informative but more robust conclusion than Bayesian classifiers.

Robust classification of multivariate time series by imprecise hidden Markov models

A novel technique to classify time series with imprecise hidden Markov models is presented. The learning of these models is achieved by coupling the EM algorithm with the imprecise Dirichlet model. In the stationarity limit, each model corresponds to an imprecise mixture of Gaussian densities, this reducing the problem to the classification of static, imprecise-probabilistic, information. Two classifiers, one based on the expected value of the mixture, the other on the Bhattacharyya distance between pairs of mixtures, are developed. The computation of the bounds of these descriptors with respect to the imprecise quantification of the parameters is reduced to, respectively, linear and quadratic optimization tasks, and hence efficiently solved. Classification is performed by extending the k-nearest neighbors approach to interval-valued data. The classifiers are credal, meaning that multiple class labels can be returned in the output. Experiments on benchmark datasets for computer vision show that these methods achieve the required robustness whilst outperforming other precise and imprecise methods. Two credal classifiers for multivariate time series based on imprecise HMMs.Classification is achieved by extending the k-NN approach to interval data.Other credal approaches outperformed, compete also with dynamic time warping.

Equivalence Between Bayesian and Credal Nets on an Updating Problem

We establish an intimate connection between Bayesian and credal nets. Bayesian nets are precise graphical models, credal nets extend Bayesian nets to imprecise probability. We focus on traditional belief updating with credal nets, and on the kind of belief updating that arises with Bayesian nets when the reason for the missingness of some of the unobserved variables in the net is unknown. We show that the two updating problems are formally the same.

Credal networks for military identification problems

Credal networks are imprecise probabilistic graphical models generalizing Bayesian networks to convex sets of probability mass functions. This makes credal networks particularly suited to model expert knowledge under very general conditions, including states of qualitative and incomplete knowledge. In this paper, we present a credal network for risk evaluation in case of intrusion of civil aircrafts into a restricted flight area. The different factors relevant for this evaluation, together with an independence structure over them, are initially identified. These factors are observed by sensors, whose reliabilities can be affected by variable external factors, and even by the behaviour of the intruder. A model of these observation processes, and the necessary fusion scheme for the information returned by the sensors measuring the same factor, are both completely embedded into the structure of the credal network. A pool of experts, facilitated in their task by specific techniques to convert qualitative judgements into imprecise probabilistic assessments, has made possible the quantification of the network. We show the capabilities of the proposed model by means of some preliminary tests referred to simulated scenarios. Overall, we can regard this application as a useful tool to support military experts in their decision, but also as a quite general imprecise-probability paradigm for information fusion.

Trading off Speed and Accuracy in Multilabel Classification

In previous work, we devised an approach for multilabel classification based on an ensemble of Bayesian networks. It was characterized by an efficient structural learning and by high accuracy. Its shortcoming was the high computational complexity of the MAP inference, necessary to identify the most probable joint configuration of all classes. In this work, we switch from the ensemble approach to the single model approach. This allows important computational savings. The reduction of inference times is exponential in the difference between the treewidth of the single model and the number of classes. We adopt moreover a more sophisticated approach for the structural learning of the class subgraph. The proposed single models outperforms alternative approaches for multilabel classification such as binary relevance and ensemble of classifier chains.

Credal ensembles of classifiers

It is studied how to aggregate the probabilistic predictions generated by different SPODE (Super-Parent-One-Dependence Estimators) classifiers. It is shown that aggregating such predictions via compression-based weights achieves a slight but consistent improvement of performance over previously existing aggregation methods, including Bayesian Model Averaging and simple average (the approach adopted by the AODE algorithm). Then, attention is given to the problem of choosing the prior probability distribution over the models; this is an important issue in any Bayesian ensemble of models. To robustly deal with the choice of the prior, the single prior over the models is substituted by a set of priors over the models (credal set), thus obtaining a credal ensemble of Bayesian classifiers. The credal ensemble recognizes the prior-dependent instances, namely the instances whose most probable class varies when different prior over the models are considered. When faced with prior-dependent instances, the credal ensemble remains reliable by returning a set of classes rather than a single class. Two credal ensembles of SPODEs are developed; the first generalizes the Bayesian Model Averaging and the second the compression-based aggregation. Extensive experiments show that the novel ensembles compare favorably to traditional methods for aggregating SPODEs and also to previous credal classifiers.

Credal sets approximation by lower probabilities: application to credal networks

Credal sets are closed convex sets of probability mass functions. The lower probabilities specified by a credal set for each element of the power set can be used as constraints defining a second credal set. This simple procedure produces an outer approximation, with a bounded number of extreme points, for general credal sets. The approximation is optimal in the sense that no other lower probabilities can specify smaller supersets of the original credal set. Notably, in order to be computed, the approximation does not need the extreme points of the credal set, but only its lower probabilities. This makes the approximation particularly suited for credal networks, which are a generalization of Bayesian networks based on credal sets. Although most of the algorithms for credal networks updating only return lower posterior probabilities, the suggested approximation can be used to evaluate (as an outer approximation of) the posterior credal set. This makes it possible to adopt more sophisticated decision making criteria, without having to replace existing algorithms. The quality of the approximation is investigated by numerical tests.

Approximating credal network inferences by linear programming

An algorithm for approximate credal network updating is presented. The problem in its general formulation is a multilinear optimization task, which can be linearized by an appropriate rule for fixing all the local models apart from those of a single variable. This simple idea can be iterated and quickly leads to very accurate inferences. The approach can also be specialized to classification with credal networks based on the maximality criterion. A complexity analysis for both the problem and the algorithm is reported together with numerical experiments, which confirm the good performance of the method. While the inner approximation produced by the algorithm gives rise to a classifier which might return a subset of the optimal class set, preliminary empirical results suggest that the accuracy of the optimal class set is seldom affected by the approximate probabilities.