AP Statistics Curriculum 2007 Bayesian Prelim
From Socr
Bayes Theorem
Bayes theorem, or "Bayes Rule" can be stated succinctly by the equality
In words, "the probability of event A occurring given that event B occurred is equal to the probability of event B occurring given that event A occurred times the probability of event A occurring divided by the probability that event B occurs."
Bayes Theorem can also be written in terms of densities or likelihood functions over continuous random variables. So, if X and Y are random variables, and is a density or likelihood, we can say
What is commonly called Bayesian Statistics is a very special application of Bayes Theorem.
We will examine a number of examples in this Chapter, but to illustrate generally, imagine that x is a fixed collection of data that has been realized from under some known density, , that takes a parameter, μ, whose value is not certainly known.
Using Bayes Theorem we may write
In this formulation, we solve for , the "posterior" density of the population parameter, μ.
For this we utilize the likelihood function of our data given our parameter, , and, importantly, a density f(μ), that describes our "prior" belief in μ.
Since is fixed, is a fixed number -- a "normalizing constant" so to ensure that the posterior density integrates to one.