Probability in two deterministic universes

TL;DR

This paper explores how subjective and objective probabilities can be understood within deterministic many-worlds theories, generalizing existing theorems and analyzing frequency measures across worlds.

Contribution

It generalizes the Deutsch-Wallace theorem to various many-world theories and links subjective probabilities to dynamics, while also showing how relative frequencies approximate objective probabilities.

Findings

Subjective probabilities depend on the theory's dynamics, using different norms.

Relative frequencies in most worlds match the objective probabilities.

The measure of worlds determines the objective probabilities in many-worlds theories.

Abstract

How can probabilities make sense in a deterministic many-worlds theory? We address two facets of this problem: why should rational agents assign subjective probabilities to branching events, and why should branching events happen with relative frequencies matching their objective probabilities. To address the first question, we generalise the Deutsch-Wallace theorem to a wide class of many-world theories, and show that the subjective probabilities are given by a norm that depends on the dynamics of the theory: the 2-norm in the usual Many-Worlds interpretation of quantum mechanics, and the 1-norm in a classical many-worlds theory known as Kent's universe. To address the second question, we show that if one takes the objective probability of an event to be the proportion of worlds in which this event is realised, then in most worlds the relative frequencies will approximate well the objective probabilities. This suggests that the task of determining the objective probabilities in a many-worlds theory reduces to the task of determining how to assign a measure to the worlds.