# Algorithmic Identification of Probabilities Is Hard

## 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, and 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 its Turing code). Can one do such a thing uniformly for all sequences that are random for computable Bernoulli measures, or even for 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.

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