AP Statistics Curriculum 2007 Bayesian Prelim

Bayes Theorem

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. So, if \(X\) and \(Y\) are random variables, and \(f(\cdot)\) is a density or likelihood, then we can say

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

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, \(f(\cdot)\), 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.

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