Statistical Proof - Bayes Theorem

Bayes Theorem

Bayesian statistics are based on a different philosophical approach for proof of inference. The mathematical formula for Bayes's theorem is:

The formula is read as the probability of the parameter (or hypothesis =h, as used in the notation on axioms) “given” the data (or empirical observation), where the horizontal bar refers to "given". The right hand side of the formula calculates the prior probability of a statistical model (Pr ) with the likelihood (Pr ) to produce a posterior probability distribution of the parameter (Pr ). The posterior probability is the likelihood that the parameter is correct given the observed data or samples statistics. Hypotheses can be compared using Bayesian inference by means of the Bayes factor, which is the ratio of the posterior odds to the prior odds. It provides a measure of the data and if it has increased or decreased the likelihood of one hypotheses relative to another.

The statistical proof is the Bayesian demonstration that one hypothesis has a higher (weak, strong, positive) likelihood. There is considerable debate if the Bayesian method aligns with Karl Poppers method of proof of falsification, where some have suggested that "...there is no such thing as "accepting" hypotheses at all. All that one does in science is assign degrees of belief..." According to Popper, hypotheses that have withstood testing and have yet to be falsified are not verified but corroborated. Some researches have suggested that Popper's quest to define corroboration on the premise of probability put his philosophy in line with the Bayesian approach. In this context, the likelihood of one hypothesis relative to another may be an index of corroboration, not confirmation, and thus statistically proven through rigorous objective standing.