We consider the task of forecasting an infinite sequence of future observations based on some number of past observations, where the probability measure generating the observations is "suspected" to satisfy one or more of a set of incomplete models, i.e., convex sets in the space of probability measures. This setting is in some sense intermediate between the realizable setting where the probability measure comes from some known set of probability measures (which can be addressed using, e.g., Bayesian inference) and the unrealizable setting where the probability measure is completely arbitrary. We demonstrate a method of forecasting which guarantees that, whenever the true probability measure satisfies an incomplete model in a given countable set, the forecast converges to the same incomplete model in the (appropriately normalized) Kantorovich-Rubinstein metric. This is analogous to merging of opinions for Bayesian inference, except that convergence in the Kantorovich-Rubinstein metric is weaker than convergence in total variation.

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.