Abstract
In this paper, we introduce a novel objective prior distribution levering on the connections between information, divergence and scoring rules. In particular, we do so from the starting point of convex functions representing information in density functions. This provides a natural route to proper local scoring rules using Bregman divergence. Specifically, we determine the prior which solves setting the score function to be a constant. Although in itself this provides motivation for an objective prior, the prior also minimizes a corresponding information criterion.
Highlights
We have derived a class of objective prior distributions that have the appealing properties of being proper and heavy-tailed
These have been obtained by exploiting a straightforward approach to the construction of score functions
The Hyvärinen score arises with α(u) = u2 ; whereas we have used α(u) = u−2 and used it to construct objective prior distributions using methodology introduced in [7]
Summary
There are two well-known relations that connect information, proper local scoring rules and divergences. The term on the left-hand-side of (1) is the Kullback–Leibler divergence [8] between p and q, the first term on the right-hand-side is the Shannon information associated with density p, and the second term is the expectation of the log-score function. [10,11] characterize all local and proper scoring rules of order m = 2 With this respect, as an additional interesting result, in the Appendix A we present the characterization using measures of information and the Bregman divergence [12].
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