In this paper we formulate the problem of inference under incomplete information in very general terms. This includes modelling the process responsible for the incompleteness, which we call the incompleteness process. We allow the process ’ behaviour to be partly unknown. Then we use Walley’s theory of coherent lower previsions, a generalisation of the Bayesian theory to imprecision, to derive the rule to update beliefs under incompleteness that logically follows from our assumptions, and that we call conservative inference rule. This rule has some remarkable properties: it is an abstract rule to update beliefs that can be applied in any situation or domain; it gives us the opportunity to be neither too optimistic nor too pessimistic about the incompleteness process, which is a necessary condition to draw reliable while strong enough conclusions; and it is a coherent rule, in the sense that it cannot lead to inconsistencies. We give examples to show how the new rule can be applied in expert systems, in parametric statistical inference, and in pattern classification, and discuss more generally the view of incompleteness processes defended here as well as some of its consequences. 1.

We propose a local (or lazy) version of the naive credal classifier. The latter is an extension of naive Bayes to imprecise probability developed to issue reliable classifications despite small amounts of data, which may then be carrying highly uncertain information about a domain. Reliability is maintained because credal classifiers can issue set-valued classifications on instances that are particularly difficult to classify. We show by extensive experiments that the local classifier outperforms the original one, both in terms of accuracy of classification and because it leads to stronger conclusions (i.e., set-valued classifications made by fewer classes). By comparing the local credal classifier with a local version of naive Bayes, we also show that the former reliably deals with instances which are difficult to classify, unlike the local naive Bayes which leads to fragile classifications.

We replace strong independence in credal networks with the weaker notion of epistemic irrelevance. Focusing on directed trees, we show how to combine local credal sets 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 essentially linear in the number of nodes, is formulated entirely in terms of coherent lower previsions. We supply examples of the algorithm’s operation, and report an application to on-line character recognition that illustrates the advantages of our model for prediction.

JNCC2 implements the naive credal classifier 2 (NCC2). This is an extension of naive Bayes to imprecise probabilities that aims at delivering robust classifications also when dealing with small or incomplete data sets. Robustness is achieved by delivering set-valued classifications (that is, returning multiple classes) on the instances for which (i) the learning set is not informative enough to smooth the effect of choice of the prior density or (ii) the uncertainty arising from missing data prevents the reliable indication of a single class. JNCC2 is released under the GNU GPL license.

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 algorithm’s 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.

Abstract. We deal with the arbitrariness in the choice of the prior over the models in Bayesian model averaging (BMA), by modelling prior knowledge by a set of priors (i.e., a prior credal set). We consider Dash and Cooper’s BMA applied to naive Bayesian networks, replacing the single prior over the naive models by a credal set; this models a condition close to prior ignorance about the models, which leads to credal model averaging (CMA). CMA returns an indeterminate classification, i.e., multiple classes, on the instances for which the learning set is not informative enough to smooth the effect of the choice of the prior. We give an algorithm to compute exact credal model averaging for naive networks. Extensive experiments show that indeterminate classifications preserve the reliability of CMA on the instances which are classified in a prior-dependent way by BMA.

Bayesian network 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.

In this paper we present TANC, i.e., a tree-augmented naive credal classifier based on imprecise probabilities; it models prior near-ignorance via the Extreme Imprecise Dirichlet Model (EDM) [1] and deals conservatively with missing data in the training set, without assuming them to be missing-at-random. The EDM is an approximation of the global Imprecise Dirichlet Model (IDM), which considerably simplifies the computation of upper and lower probabilities; yet, having been only recently introduced, the quality of the provided approximation needs still to be verified. As first contribution, we extensively compare the output of the naive credal classifier (one of the few cases in which the global IDM can be exactly implemented) when learned with the EDM and the global IDM; the output of the classifier appears to be identical in the vast majority of cases, thus supporting the adoption of the EDM in real classification problems. Then, by experiments we show that TANC is more reliable than the precise TAN (learned with uniform prior), and also that it provides better performance compared to a previous [13] TAN model based on imprecise probabilities. TANC treats missing data by considering all possible completions of the training set, but avoiding an exponential increase of the computational times; eventually, we present some preliminary results with missing data.

Naive credal classifier 2 (NCC2) extends naive Bayes in order to deliver more robust classifications. NCC2 is based on a set of prior densities rather than on a single prior; as a consequence, when faced with instances whose classification is prior-dependent (and therefore might not be reliable), it returns a set of classes (we call this an indeterminate classification) instead of a single class. Moreover, NCC2 introduces a very general and flexible treatment of missing data, which, under certain circumstances, can also lead to indeterminate classifications. In this case, indeterminacy can be regarded as a way to preserve reliability despite the information hidden by missing values. We call hard-to-classify the instances classified indeterminately by NCC2. Extensive empirical evaluations show that naive Bayes ’ accuracy drops considerably on the hard-toclassify instances identified by NCC2, and that on the other hand, NCC2 has high set-accuracy (the proportion of times that the actual class is contained in the set of returned classes) when it is indeterminate.

(NCC2). JNCC2 is open source; it is hence freely available together with manual, sources and javadoc documentation. NCC2 is an extension of Naive Bayes Classifier (NBC) to imprecise probabilities, designed so as to return robust classifications even on small and/or incomplete data sets. A peculiar feature of NCC2 is that it returns indeterminate classifications (i.e., more than one class) when faced with doubtful instances. The empirical results of Corani and Zaffalon (2008) have shown that NCC2 returns indeterminate judgments on instances whose classification is truly doubtful; in fact, NBC achieves a much higher classification accuracy on the instances precisely classified by NCC2, than on those indeterminately classified by NCC2. 1