AP Statistics Curriculum 2007 Bayesian Prelim

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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.

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

See also

References




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