Interval-valued probability density estimation based on quasi-continuous histograms: Proof of the conjecture

The sensitivity of histogram computation to the choice of a reference interval and number of bins can be attenuated by replacing the crisp partition on which the histogram is built by a fuzzy partition. This involves replacing the crisp counting process by a distributed (weighted) voting process. The counterpart to this low sensitivity is some confusion in the count values: a value of 10 in the accumulator associated with a bin can mean 10 observations in the bin or 40 observations near the bin. This confusion can bias the statistical decision process based on such a histogram. In a recent paper, we proposed a method that links the probability measure associated with any subset of the reference interval with the accumulator values of a fuzzy partition-based histogram. The method consists of transferring counts associated with each bin proportionally to its interaction with the considered subset. Two methods have been proposed which are called precise and imprecise pignistic transfer. Imprecise pignistic transfer accounts for the interactivity of two consecutive cells in order to propagate, in the estimated probability measure, counting confusion due to fuzzy granulation. Imprecise pignistic transfer has been conjectured to include precise pignistic transfer. The present article proposes a proof of this conjecture.