# AP Statistics Curriculum 2007 Bayesian Prelim

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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." | 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. | + | Bayes Theorem can also be written in terms of densities or likelihood functions over continuous random variables. Let's call <math>f(\star)</math> the density (or in some cases, the likelihood) defined by the random process <math>\star</math>. If <math>X</math> and <math>Y</math> are random variables, we can say |

<math>f(Y|X) = \frac{f(X|Y) \cdot f(Y)} { f(X) }</math> | <math>f(Y|X) = \frac{f(X|Y) \cdot f(Y)} { f(X) }</math> |

## Revision as of 20:46, 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. Let's call the density (or in some cases, the likelihood) defined by the random process . If *X* and *Y* are random variables, 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.