Update of Prior Probabilities by Minimal Divergence

The present paper investigates the update of an empirical probability distribution with the results of a new set of observations. The update reproduces the new observations and interpolates using prior information. The optimal update is obtained by minimizing either the Hellinger distance or the quadratic Bregman divergence. The results obtained by the two methods differ. Updates with information about conditional probabilities are considered as well.

A collaborative filtering algorithm based on item labels and Hellinger distance for sparse data

The Neighbourhood-based collaborative filtering (CF) algorithm has been widely used in recommender systems. To enhance the adaptability to the sparse data, a CF with new similarity measure and prediction method is proposed. The new similarity measure is designed based on the Hellinger distance of item labels, which overcomes the problem of depending on common-rated items (co-rated items). In the proposed prediction method, we present a new strategy to solve the problem that the neighbour users do not rate the target item, that is, the most similar item rated by the neighbour user is used to replace the target item. The proposed prediction method can significantly improve the utilisation of neighbours and obviously increase the accuracy of prediction. The experimental results on two benchmark datasets both confirm that the proposed algorithm can effectively alleviate the sparse data problem and improve the recommendation results.

Rare Event Analysis for Minimum Hellinger Distance Estimators via Large Deviation Theory

Hellinger distance has been widely used to derive objective functions that are alternatives to maximum likelihood methods. While the asymptotic distributions of these estimators have been well investigated, the probabilities of rare events induced by them are largely unknown. In this article, we analyze these rare event probabilities using large deviation theory under a potential model misspecification, in both one and higher dimensions. We show that these probabilities decay exponentially, characterizing their decay via a “rate function” which is expressed as a convex conjugate of a limiting cumulant generating function. In the analysis of the lower bound, in particular, certain geometric considerations arise that facilitate an explicit representation, also in the case when the limiting generating function is nondifferentiable. Our analysis involves the modulus of continuity properties of the affinity, which may be of independent interest.