AP Statistics Curriculum 2007 Bayesian Prelim

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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, 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(X), that takes a parameter, μ, 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, μ.

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

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

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