Learning from partially supervised data using mixture models and belief functions

Abstract

This paper addresses classification problems in which the class membership of training data is only partially known. Each learning sample is assumed to consist in a feature vector xi ∈ X and an imprecise and/or uncertain "soft" label mi defined as a Dempster-Shafer basic belief assignment over the set of classes. This framework thus generalizes many kinds of learning problems including supervised, unsupervised and semi-supervised learning. Here, it is assumed that the feature vectors are generated from a mixture model. Using the Generalized Bayesian Theorem, an extension of Bayes' theorem in the belief function framework, we derive a criterion generalizing the likelihood function. A variant of the EM algorithm dedicated to the optimization of this criterion is proposed, allowing us to compute estimates of model parameters. Experimental results demonstrate the ability of this approach to exploit partial information about class labels.

Clustering and classification of fuzzy data using the fuzzy EM algorithm

“…In this last case, we followed [55,63] and gave the observed label ω k a degree of possibility δ ik = 1, and the other plausible labels ω k ′ , k ′ = k a degree of possibility pl ik ′ = η.…”

“…Besides, the problem of noisy labels has been addressed in a probabilistic framework [49,50,51]. More recently, the general framework of belief functions has been successfully employed to estimate GMMs from imprecise and uncertain labels [52,53,54,55]. Finally, it should be stressed out that the literature dedicated to fuzzy GMM estimation, or more generally imprecise or uncertain clustering, actually refers to associating precise instances with several classes in a imprecise way.…”

“…This procedure was recently proposed by Denoeux as an extension of the EM algorithm for imprecise data represented by fuzzy numbers [62] and by belief functions [63] 2 . Thus, unlike in [52,53,55], we consider here both attribute and class uncertainty in our estimation process. Note that [15] considered uncertain attribute values represented by intervals; besides, the approach described in [54] made it possible to take into account the fact that some attribute information were missing.…”

Naive possibilistic classifiers for imprecise or uncertain numerical data

“…The choice a weighted sum for N j corresponds to the standard definition of the scalar cardinality of a fuzzy set [47], which leads to the other weighted sums in the same spirit. Besides, it is noticeable that counterpart of these three expressions can be found in the work of Côme et al [15] where the more general setting of belief functions (that encompasses possibility theory) is used for describing uncertain classes. We note that the proposed model, supporting uncertainty in the class labels, also includes the certain case where π k (c j ) is 1 for the the true label and 0 otherwise.…”

Partially-Hidden Markov Models

“…Compared to [3], we take into account the temporal dependency into account, helping in time-series modelling. The proposed approach is based on the Evidential Expectation-Maximisation (E2M) algorithm [5].…”