Robust probability updating

Abstract

This paper discusses an alternative to conditioning that may be used when the probability distribution is not fully specified. It does not require any assumptions (such as CAR: coarsening at random) on the unknown distribution. The well-known Monty Hall problem is the simplest scenario where neither naive conditioning nor the CAR assumption suffice to determine an updated probability distribution. This paper thus addresses a generalization of that problem to arbitrary distributions on finite outcome spaces, arbitrary sets of 'messages', and (almost) arbitrary loss functions, and provides existence and characterization theorems for robust probability updating strategies. We find that for logarithmic loss, optimality is characterized by an elegant condition, which we call RCAR (reverse coarsening at random). Under certain conditions, the same condition also characterizes optimality for a much larger class of loss functions, and we obtain an objective and general answer to how one should update probabilities in the light of new information.

Efficient Algorithms for Minimax Decisions Under Tree-Structured Incompleteness

“…A decision problem with tree-structured incompleteness is given by a finite set X , a family Y of two-element subsets of X such that the undirected graph (X , Y) forms a tree, a distribution p over X having p x > 0 everywhere, and a loss function L : X × A → [0, ∞], where A is the set of actions available to the decision maker. We assume that the loss function L satisfies the conditions in Theorem 18 of [16]. 2 A coarsening mechanism for this problem is an (unknown) joint distribution P on X × Y that satisfies P (x, y) = 0 whenever x / ∈ y, and…”

“…It was found in [16] that if the action space is rich enough, this game has a Nash equilibrium, so that neither player benefits from knowing the other's strategy before picking their own. We concentrate on finding a maximin P , because once it is known, minimax optimal strategies A are typically easily determined.…”

“…Note that the second equation holds with equality for y with P (y) > 0. The vector q is called the RCAR vector; it exists and is unique by [16,Lemma 11].…”

“…This type of decision problem with incomplete information was introduced in [16], where minimax optimal strategies are characterized for arbitrary Y, but where the question of how to compute them is not addressed. This computational problem was considered previously in [17], where it was demonstrated that the problem is hard for general Y, but a direct formula could be given for finding a minimax optimal strategy in the special case that Y forms a partition matroid.…”

“…Namely, that the generalized entropy HL[6] is finite and continuous; further, HL or an affine transformation of it is invariant under permutation of x1 and x3 whenever {x1, x2}, {x2, x3} ∈ Y. See[16] for definitions.…”