We consider how an agent should update her uncertainty when it is represented by a set P of probability distributions and the agent observes that a random variable X takes on value x, given that the agent makes decisions using the minimax criterion,
perhaps the best-studied and most commonly-used criterion in the literature. We adopt a game-theoretic framework, where the agent plays against a bookie, who chooses some distribution from P. We consider two reasonable games that differ in what the bookie knows when he makes his choice. Anomalies that have been observed before, like time inconsistency, can be understood as arising because different games are being played, against bookies with different information. We characterize the important special cases in which the optimal decision rules according to the minimax criterion amount to either conditioning or simply ignoring the information. Finally, we consider the relationship between conditioning and calibration when uncertainty is described by sets of probabilities.
It is commonly-accepted wisdom that more information is better, and that information should never be ignored. Here we argue, using both a Bayesian and a non-Bayesian analysis, that in some situations you are better off ignoring information if your un
certainty is represented by a set of probability measures. These include situations in which the information is relevant for the prediction task at hand. In the non-Bayesian analysis, we show how ignoring information avoids dilation, the phenomenon that additional pieces of information sometimes lead to an increase in uncertainty. In the Bayesian analysis, we show that for small sample sizes and certain prediction tasks, the Bayesian posterior based on a noninformative prior yields worse predictions than simply ignoring the given information.
As examples such as the Monty Hall puzzle show, applying conditioning to update a probability distribution on a ``naive space, which does not take into account the protocol used, can often lead to counterintuitive results. Here we examine why. A crit
erion known as CAR (coarsening at random) in the statistical literature characterizes when ``naive conditioning in a naive space works. We show that the CAR condition holds rather infrequently. We then consider more generalized notions of update such as Jeffrey conditioning and minimizing relative entropy (MRE). We give a generalization of the CAR condition that characterizes when Jeffrey conditioning leads to appropriate answers, but show that there are no such conditions for MRE. This generalizes and interconnects previous results obtained in the literature on CAR and MRE.
We give an interpretation of the Maximum Entropy (MaxEnt) Principle in game-theoretic terms. Based on this interpretation, we make a formal distinction between different ways of {em applying/} Maximum Entropy distributions. MaxEnt has frequently been
criticized on the grounds that it leads to highly representation dependent results. Our distinction allows us to avoid this problem in many cases.
Consider the set of all sequences of $n$ outcomes, each taking one of $m$ values, that satisfy a number of linear constraints. If $m$ is fixed while $n$ increases, most sequences that satisfy the constraints result in frequency vectors whose entropy
approaches that of the maximum entropy vector satisfying the constraints. This well-known entropy concentration phenomenon underlies the maximum entropy method. Existing proofs of the concentration phenomenon are based on limits or asymptotics and unrealistically assume that constraints hold precisely, supporting maximum entropy inference more in principle than in practice. We present, for the first time, non-asymptotic, explicit lower bounds on $n$ for a number of variants of the concentration result to hold to any prescribed accuracies, with the constraints holding up to any specified tolerance, taking into account the fact that allocations of discrete units can satisfy constraints only approximately. Again unlike earlier results, we measure concentration not by deviation from the maximum entropy value, but by the $ell_1$ and $ell_2$ distances from the maximum entropy-achieving frequency vector. One of our results holds independently of the alphabet size $m$ and is based on a novel proof technique using the multi-dimensional Berry-Esseen theorem. We illustrate and compare our results using various detailed examples.
We introduce algorithmic information theory, also known as the theory of Kolmogorov complexity. We explain the main concepts of this quantitative approach to defining `information. We discuss the extent to which Kolmogorovs and Shannons information t
heory have a common purpose, and where they are fundamentally different. We indicate how recent developments within the theory allow one to formally distinguish between `structural (meaningful) and `random information as measured by the Kolmogorov structure function, which leads to a mathematical formalization of Occams razor in inductive inference. We end by discussing some of the philosophical implications of the theory.
