What is Bayesian statistics?

Abstract

Bayesian statistics is included in few elementary statistics courses, and many mathematicians have heard of it, perhaps through collateral readings from popular literature or [1], selected as an Editor's Choice in the New York Times Book Review. ‘Bayesian statistics’ provides for a way to incorporate prior beliefs, experience, or information into the analysis of data. Bayesian thinking is natural, and that is an advantage. For example, on a summer morning, if we see dark rain clouds up in the sky, we leave home for work with an umbrella because prior experience tells us that doing so is beneficial. In general, the idea is simple; schematically, it looks like this:(prior belief) + (data: new information) ⇒ (posterior belief).Thus, we begin with a prior belief that we allow to be modified or informed by new data to produce a posterior belief, which then becomes our new prior, and this process is never-ending. We are always willing to update our beliefs according to new information.