# AP Statistics Curriculum 2007 Bayesian Prelim

### From Socr

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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(\cdot)</math> that takes a parameter, <math>\mu</math> whose value is not certainly known. | + | 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(\cdot)</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 | ||

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<math>f(\mu|\mathbf{x}) = \frac{f(\mathbf{x}|\mu) \cdot f(\mu)} { f(\mathbf{x}) }</math> | <math>f(\mu|\mathbf{x}) = \frac{f(\mathbf{x}|\mu) \cdot f(\mu)} { f(\mathbf{x}) }</math> | ||

- | In this formulation, we solve for <math>f(\mu|\mathbf{x})</math>, the "posterior" density of the population parameter <math>\mu</math>. | + | In this formulation, we solve for <math>f(\mu|\mathbf{x})</math>, the "posterior" density of the population parameter, <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>. | 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> | + | 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(\mu \cap \mathbf{x}) d\mu = \int_{\mu} f( \mathbf{x} | \mu ) f(\mu) d\mu </math> | <math>f(\mathbf{x}) = \int_{\mu} f(\mu \cap \mathbf{x}) d\mu = \int_{\mu} f( \mathbf{x} | \mu ) f(\mu) d\mu </math> |

## Revision as of 20:13, 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 ensure that the posterior density integrates to one.