Probabilistic inference in very large universes

TL;DR

This paper introduces a Bayesian framework for evaluating cosmological theories involving vast universes, emphasizing first-person probabilities and the Principle of Self-Locating Indifference to incorporate observer-specific information.

Contribution

It proposes a novel Bayesian method for assessing large-universe theories using first-person probabilities and the PSLI, addressing observer selection effects and subjective states.

Findings

First-person probabilities depend on observer-instants and subjective states.

Weighting universes by the number of observer-instants improves prediction accuracy.

The approach clarifies issues like Boltzmann brains and old evidence in cosmology.

Abstract

[Abridged] Some cosmological theories propose that the observable universe is a small part of a much larger universe in which parameters describing the low-energy laws of physics vary from region to region. How can we reasonably assess a theory that describes such a mostly unobservable universe? We propose a Bayesian method based on theory-generated probability distributions for our observations. We focus on basic principles, leaving aside concerns about practicality. (We also leave aside the measure problem, to discuss other issues.) We argue that cosmological theories can be tested by standard Bayesian updating, but we need to use theoretical predictions for "first-person" probabilities -- i.e., probabilities for our observations, accounting for all relevant selection effects. These selection effects can depend on the observer, and on time, so in principle first-person probabilities are defined for each observer-instant -- an observer at an instant of time. First-person probabilities should take into account everything the observer believes about herself and her surroundings -- i.e., her "subjective state". We advocate a "Principle of Self-Locating Indifference" (PSLI), asserting that any real observer should make predictions as if she were chosen randomly from the theoretically predicted observer-instants that share her subjective state. We believe the PSLI is intuitively very reasonable, but also argue that it maximizes the expected fraction of observers who will make correct predictions. Cosmological theories will in general predict a set of possible universes, each with a probability. To calculate first-person probabilities, we argue that each possible universe should be weighted by the number of observer-instants in the specified subjective state that it contains. We also discuss Boltzmann brains, the humans/Jovians parable of Hartle and Srednicki, and the use of "old evidence".