# Identification of Probabilities

arXiv: https://arxiv.org/abs/1708.01611 · HTML: https://mlsanity.com/paper/1708.01611/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Aug 2017.

## Built on by (most cited first, 2 of 2)

- [Still no evidence for an effect of the proportion of non-native speakers on language complexity -- A response to Kauhanen, Einhaus & Walkden (2023)](https://mlsanity.com/paper/2305.00217/) (2023)
- [Less than one percent of words would be affected by gender-inclusive language in German press texts](https://mlsanity.com/paper/2402.03870/) (2024)

## Abstract (arXiv)

Within psychology, neuroscience and artificial intelligence, there has been increasing interest in the proposal that the brain builds probabilistic models of sensory and linguistic input: that is, to infer a probabilistic model from a sample. The practical problems of such inference are substantial: the brain has limited data and restricted computational resources. But there is a more fundamental question: is the problem of inferring a probabilistic model from a sample possible even in principle? We explore this question and find some surprisingly positive and general results. First, for a broad class of probability distributions characterised by computability restrictions, we specify a learning algorithm that will almost surely identify a probability distribution in the limit given a finite i.i.d. sample of sufficient but unknown length. This is similarly shown to hold for sequences generated by a broad class of Markov chains, subject to computability assumptions. The technical tool is the strong law of large numbers. Second, for a large class of dependent sequences, we specify an algorithm which identifies in the limit a computable measure for which the sequence is typical, in the sense of Martin-Lof (there may be more than one such measure). The technical tool is the theory of Kolmogorov complexity. We analyse the associated predictions in both cases. We also briefly consider special cases, including language learning, and wider theoretical implications for psychology.