# AP Statistics Curriculum 2007 Bayesian Prelim

### From Socr

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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. So, if <math>X</math> and <math>Y</math> are random variables, and <math>f(\cdot)</math> is a density or likelihood, | + | Bayes Theorem can also be written in terms of densities or likelihood functions over continuous random variables. So, if <math>X</math> and <math>Y</math> are random variables, and <math>f(\cdot)</math> is a density or likelihood, 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> | ||

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

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

## Revision as of 20:20, 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, , 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.