# AP Statistics Curriculum 2007 Bayesian Prelim

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What is commonly called '''Bayesian Statistics''' is a very special application of Bayes Theorem. | 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, <math>f( | + | 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, <math>f(X)</math>, that takes a parameter, <math>\mu</math>, whose value is not certainly known. |

Using Bayes Theorem we may write | Using Bayes Theorem we may write |

## Revision as of 20:21, 23 July 2009

**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.