# AP Statistics Curriculum 2007 Bayesian Prelim

(Difference between revisions)
 Revision as of 20:09, 23 July 2009 (view source)DaveZes (Talk | contribs)← Older edit Revision as of 20:13, 23 July 2009 (view source)DaveZes (Talk | contribs) Newer edit → Line 13: Line 13: 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, $f(\cdot)$ that takes a parameter, $\mu$ 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, $f(\cdot)$, that takes a parameter, $\mu$, whose value is not certainly known. Using Bayes Theorem we may write Using Bayes Theorem we may write Line 19: Line 19: $f(\mu|\mathbf{x}) = \frac{f(\mathbf{x}|\mu) \cdot f(\mu)} { f(\mathbf{x}) }$ $f(\mu|\mathbf{x}) = \frac{f(\mathbf{x}|\mu) \cdot f(\mu)} { f(\mathbf{x}) }$ - In this formulation, we solve for $f(\mu|\mathbf{x})$, the "posterior" density of the population parameter $\mu$. + 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 $\mu$. 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 assure that the posterior density integrates to one. + 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. $f(\mathbf{x}) = \int_{\mu} f(\mu \cap \mathbf{x}) d\mu = \int_{\mu} f( \mathbf{x} | \mu ) f(\mu) d\mu$ $f(\mathbf{x}) = \int_{\mu} f(\mu \cap \mathbf{x}) d\mu = \int_{\mu} f( \mathbf{x} | \mu ) f(\mu) d\mu$

## Revision as of 20:13, 23 July 2009

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 over continuous random variables. So, if $f(\cdot)$ is some density, and X and Y are random variables, 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, 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.

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