# AP Statistics Curriculum 2007 Bayesian Prelim

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For this we utilize the likelihood function of our data given our parameter, <math>f(\mathbf{x}|\mu) </math>, and, importantly, a density <math>f(\mu)</math>, that describes our "prior" belief in <math>\mu</math>. | For this we utilize the likelihood function of our data given our parameter, <math>f(\mathbf{x}|\mu) </math>, and, importantly, a density <math>f(\mu)</math>, that describes our "prior" belief in <math>\mu</math>. | ||

- | + | Since <math>\mathbf{x}</math> is fixed, <math>f(\mathbf{x})</math>, is a fixed number -- a "normalizing constant" so to assure that the posterior density integrates to one. | |

- | + | <math>f(\mathbf{x}) = \int_{\mu} f(\mu \cap \mathbf{x}) d\mu = \int_{\mu} f( \mathbf{x} | \mu ) d\mu </math> | |

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## Revision as of 20:08, 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 over continuous random variables. So, if is some density, and *X* and *Y* are random variables, then 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 assure that the posterior density integrates to one.