# 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, *f*(*X*), 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.