Difference between revisions of "AP Statistics Curriculum 2007 Bayesian Prelim"
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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, <math>f(\cdot)</math> that takes a parameter, <math>\mu</math> 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, <math>f(\cdot)</math>, that takes a parameter, <math>\mu</math>, whose value is not certainly known. |
Using Bayes Theorem we may write | Using Bayes Theorem we may write | ||
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<math>f(\mu|\mathbf{x}) = \frac{f(\mathbf{x}|\mu) \cdot f(\mu)} { f(\mathbf{x}) }</math> | <math>f(\mu|\mathbf{x}) = \frac{f(\mathbf{x}|\mu) \cdot f(\mu)} { f(\mathbf{x}) }</math> | ||
| − | In this formulation, we solve for <math>f(\mu|\mathbf{x})</math>, the "posterior" density of the population parameter <math>\mu</math>. | + | In this formulation, we solve for <math>f(\mu|\mathbf{x})</math>, the "posterior" density of the population parameter, <math>\mu</math>. |
For this we utilize the likelihood function of our data given our parameter, <math>f(\mathbf{x}|\mu) </math>, and, importantly, a density <math>f(\mu)</math>, that describes our "prior" belief in <math>\mu</math>. | For this we utilize the likelihood function of our data given our parameter, <math>f(\mathbf{x}|\mu) </math>, and, importantly, a density <math>f(\mu)</math>, that describes our "prior" belief in <math>\mu</math>. | ||
| − | Since <math>\mathbf{x}</math> is fixed, <math>f(\mathbf{x})</math> | + | Since <math>\mathbf{x}</math> is fixed, <math>f(\mathbf{x})</math> is a fixed number -- a "normalizing constant" so to ensure that the posterior density integrates to one. |
<math>f(\mathbf{x}) = \int_{\mu} f(\mu \cap \mathbf{x}) d\mu = \int_{\mu} f( \mathbf{x} | \mu ) f(\mu) d\mu </math> | <math>f(\mathbf{x}) = \int_{\mu} f(\mu \cap \mathbf{x}) d\mu = \int_{\mu} f( \mathbf{x} | \mu ) f(\mu) d\mu </math> | ||
Revision as of 15:19, 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, \(\mu\), 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.
\(f(\mathbf{x}) = \int_{\mu} f(\mu \cap \mathbf{x}) d\mu = \int_{\mu} f( \mathbf{x} | \mu ) f(\mu) d\mu \)
