Optimal Polynomial-Time Estimators: A Bayesian Notion of Approximation Algorithm

The concept of an “approximation algorithm” is usually only applied to optimization problems, since in optimization problems the performance of the algorithm on any given input is a continuous parameter. We introduce a new concept of approximation applicable to decision problems and functions, inspired by Bayesian probability. From the perspective of a Bayesian reasoner with limited computational resources, the answer to a problem that cannot be solved exactly is uncertain and therefore should be described by a random variable. It thus should make sense to talk about the expected value of this random variable, an idea we formalize in the language of average-case complexity theory by introducing the concept of “optimal polynomialtime estimators.” We prove some existence theorems and completeness results, and show that optimal polynomial-time estimators exhibit many parallels with “classical” probability theory.