# AP Statistics Curriculum 2007 Bayesian Prelim

## Probability and Statistics Ebook - Bayes Theorem

### Introduction

Bayes Theorem, or "Bayes Rule" can be stated succinctly by the equality

$P(A|B) = \frac{P(B|A) \cdot P(A)} {P(B)}$

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 $$f(\star)$$ the density (or in some cases, the likelihood) defined by the random process $$\star$$. If $$X$$ and $$Y$$ are random variables, we can say

$$f(Y|X) = \frac{f(X|Y) \cdot f(Y)} { f(X) }$$

### Example

Suppose a laboratory blood test is used as evidence for a disease. Assume P(positive Test| Disease) = 0.95, P(positive Test| no Disease)=0.01 and P(Disease) = 0.005. Find P(Disease|positive Test)=?

Denote D = {the test person has the disease}, $$D^c$$ = {the test person does not have the disease} and T = {the test result is positive}. Then

$$P(D | T) = {P(T | D) P(D) \over P(T)} = {P(T | D) P(D) \over P(T|D)P(D) + P(T|D^c)P(D^c)}=$$ $$={0.95\times 0.005 \over {0.95\times 0.005 +0.01\times 0.995}}=0.3231293.$$

### Bayesian Statstics

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 some known density, $$f(X)$$, that takes a parameter, $$\mu$$, whose value is not certainly known.

Using Bayes Theorem we may write

$f(\mu|\mathbf{x}) = \frac{f(\mathbf{x}|\mu) \cdot f(\mu)} { f(\mathbf{x}) }$

In this formulation, we solve for $$f(\mu|\mathbf{x})$$, the "posterior" density of the population parameter, $$\mu$$.

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

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

$$f(\mathbf{x}) = \int_{\mu} f( \mathbf{x} \cap \mu) d\mu = \int_{\mu} f( \mathbf{x} | \mu ) f(\mu) d\mu$$