Josiah Lopez-Wild

Appeals to typicality are common in physics, but it is often unclear what it means for a physical state to be typical relative to a probability measure, and correspondingly unclear what a law that appeals to typicality asserts. Here we consider how one might characterize physical typicality using ideas from the theory of algorithmic randomness. As a concrete example, we show how taking a physical state to be typical relative to a computable measure when it is Martin-Löf random allows one to formulate the distribution postulate in Bohmian mechanics as a statistical constraining law of the theory. Using a toy model, we show how this constraint guarantees the standard Born statistics for computable experimental protocols. Algorithmic Bohmian mechanics (aBM) thus illustrates how algorithmic randomness may be used to provide precise content to a statistical law.

Bayesian epistemology is broadly concerned with providing norms for rational belief and learning using the mathematics of probability theory. Many authors have worried that the theory is too idealized to accurately describe real agents. In this paper I argue that an emerging program, computable Bayesian epistemology, can describe more realistic agents while retaining sufficient generality. I situate this program by placing it among the ongoing debate about ideal versus bounded rationality. I then present the basics of computable analysis and demonstrate its usefulness by proving a simple result: there are no computable finitely additive probability measures.

Expected utility theory seeks to define rational choice behavior. Given a collection of acts available to some decision maker, expected utility theorists commonly identify the “rational” act as the act which _maximizes expected utility_ (where the expectation is taken with respect to some probability measure). The mathematical core of expected utility theory is a representation theorem. These theorems link expected utility maximization to a qualitative description of an agent’s choice behavior, captured in a preference relation. We say that an agent’s preferences are _represented_ by a utility function and/or probability measure which is derived from the representation theorem. Representation theorems only prove that such a function exists, but do not show how to find it. Thus one may ask: given an agent’s preference relation, when can we compute a representing utility function? In this paper I prove a computable version of the von Neumann-Morgenstern representation theorem, answering this question in the affirmative.