A min-max regret approach to maximum likelihood inference under incomplete data
Résumé
Various methods have been proposed to express and solve maximum likelihood problems with incomplete data. In some of these approaches, the idea is that incompleteness makes the likelihood function imprecise. Two proposals can be found to cope with this situation: the maximax approach that maximizes the greatest likelihood value induced by precise data sets compatible with the incomplete observations, and the maximin approach that maximizes the least such likelihood value. These approaches prove to display extreme behaviors in some contexts, the maximax approach having a tendency to disambiguate the data, while the maximin approach favors uniform distributions. In this paper, we propose an alternative approach consisting in minimizing a relative regret criterion with respect to maximal likelihood solutions obtained for all precise data sets compatible with the coarse data. In contrast with the maximax and the maximin methods, the min-max-regret method relies on comparing relative likelihoods and obtains results that achieve a trade-off between results of the two other methods. The methods are compared on toy examples and also on simulated random data as well as a supervised classification problem.