Algorithmic identification of probabilities is hard

TL;DR

This paper investigates the difficulty of inductively identifying the exact probability parameter of a random binary sequence, proving that uniform successful identification is impossible for all such sequences.

Contribution

It establishes a broad negative result showing that no uniform method can correctly identify the probability parameter for all sequences random for computable Bernoulli measures.

Findings

Uniform identification of the probability parameter is impossible.

The negative result extends beyond Bernoulli measures.

The study highlights fundamental limitations in inductive inference for random sequences.

Abstract

Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better approximations of $p$ as we read the sequence. We study in this paper a similar question, but from the viewpoint of inductive inference. We suppose now that $p$ is a computable real, but one asks for more: as we are reading more and more bits of our random sequence, we have to eventually guess the exact parameter $p$ (in the form of a Turing code). Can one do such a thing uniformly on all sequences that are random for computable Bernoulli measures, or even on a `large enough' fraction of them? In this paper, we give a negative answer to this question. In fact, we prove a very general negative result which extends far beyond the class of Bernoulli measures.