Probabilistic inference in very large universes

[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”.

💡 Research Summary

The paper tackles the methodological challenge of testing cosmological theories that posit our observable universe as a tiny domain within a vastly larger, possibly infinite, multiverse where low‑energy physical parameters vary from region to region. The authors adopt a standard Bayesian framework for theory assessment but emphasize that the likelihoods required for Bayesian updating must be “first‑person” probabilities—probabilities for the observations that we, as specific observers, will actually experience.

To bridge the gap between the theory‑generated “third‑person” probabilities (the distribution over whole‑universe states) and the needed first‑person probabilities, the authors introduce the notion of an observer‑instant I = (O, t), where O is a physical information‑gathering system and t is a particular moment. Each observer‑instant possesses a “subjective state” S(I), defined as the total set of beliefs the observer holds at that moment about herself and her surroundings (memories, self‑conception, etc.). The subjective state may be false or incomplete, but it is the information the observer can actually use for prediction.

The central proposal is the Principle of Self‑Locating Indifference (PSLI). PSLI states that a real observer should make predictions as if she were randomly and uniformly selected from the set of all observer‑instants that share her subjective state, as predicted by the theory. In other words, given a theory T, the probability that we will observe outcome E is obtained by summing over all possible universes U_i that T predicts, weighting each universe by both its prior probability p_i and by the number N_i(S) of observer‑instants in that universe that have the same subjective state S. Mathematically,

 P(E | S, T) = ∑_i p_i ·