Introduction

My purpose here is to give an example of how Bayes's theorem and the Bayesian approach to probabilities model and explain the way we interpret data.

Bayes's theorem tells you how to get from the prior probability density of an unknown parameter to the posterior probability density. These probability densities are just an individual's subjective assessment of how likely various possible values of the parameter are. The prior is the probability density in the person's mind before he or she looked at a particular data set; the posterior is the probability density after looking at the data. Since it tells you how to get from the prior to the posterior -- that is, how a person's ``state of mind'' changed as a result of looking at the data -- Bayes's theorem can be thought of as the way to model what a person has learned from a data set.

The specific example I'm going to focus on is the value of the cosmological density parameter $\Omega$. After giving a bit of background, I will show an admittedly cartoonish representation of my prior probability density for this parameter and what posterior probability density this leads to, given some recent observations.