# Constructive mathematics and statistics

by LiKao   Last Updated September 11, 2019 10:19 AM

Usual non-constructive mathematics leads to some paradoxes (e.g. the Banach-Tarski paradox), which are directly related to the axiom of choice. In non-constructive mathematics, the axiom of choice (as well as proofs by contradiction) are not accepted.

For statistics, the axiom of choice implies that for any $$c$$ there must be a model such that the BIC (or also the AIC) is below $$c$$ (see this answer, we can arbitrarily increase the likelihood without increasing $$N$$, the number of parameters), which kind of defeats the purpose of the BIC as a model selection criterion. This method would not work in constructive math, because the space-filling curves or bijections don't exists in constructive mathematics.

Is there a general treatment of statistics in constructive mathematics? AFAIK a lot of parts of statistics are non-constructive (e.g. Gaussian distribution, Beta distribution, etc). It would be interesting to see which parts of statistics are different when the constructive approach is taken.

Tags :

## Related Questions

Updated November 06, 2018 19:19 PM

Updated August 28, 2018 16:19 PM

Updated July 12, 2019 03:19 AM

Updated November 01, 2018 04:19 AM