Maximum-likelihood principle for possibility distributions viewed as families of probabilities

An acknowledged interpretation of possibility distributions in quantitative possibility theory is in terms of families of probabilities that are upper and lower bounded by the associated possibility and necessity measures. This paper proposes a likelihood function for possibility distributions that agrees with the above-mentioned view of possibility theory in the continuous and in the discrete cases. Especially, we show that, given a set of data following a probability distribution, the optimal possibility distribution with respect to our likelihood function is the distribution obtained as the result of the probability-possibility transformation that obeys the maximal specificity principle. It is also shown that when the optimal distribution is not available, a direct application of this possibilistic likelihood provides more faithful results than approximating the probability distribution and then applying the probability possibility transformation. We detail the particular case of triangular and trapezoidal possibility distributions and we show that any unimodal unknown probability distribution can be faithfully upper approximated by a triangular distribution obtained by optimizing the possibilistic likelihood.