AP Statistics Curriculum 2007 Bayesian Prelim

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'''Bayes Theorem'''
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==[[EBook | Probability and Statistics Ebook]] - Bayes Theorem==
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Bayes theorem can be stated succinctly by the equality
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===Introduction===
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Bayes Theorem, or "Bayes Rule" can be stated succinctly by the equality
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P(A|B) = P(B|A)*P(A)/P(B)
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: <math>P(A|B) = \frac{P(B|A) \cdot P(A)} {P(B)}</math>
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."
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."
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Bayes Theorem can also be written in terms of densities over continuous random variables. So, if <math>f(.)</math> is some density, and <math>X</math> and <math>Y</math> are random variables, then we can say
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Bayes Theorem can also be written in terms of densities or likelihood functions over continuous random variables. Let's call <math>f(\star)</math> the density (or in some cases, the likelihood) defined by the random process <math>\star</math>.  If <math>X</math> and <math>Y</math> are random variables, we can say
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<math>f(Y|X) = f(X|Y) \cdot f(Y) / f(X) here</math>
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<math>f(Y|X) = \frac{f(X|Y) \cdot f(Y)} { f(X) }</math>
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===Example===
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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)=?
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Denote D = {the test person has the disease}, <math>D^c</math> = {the test person does not have the disease} and  T = {the test result is positive}. Then
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<center><math>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)}=</math>
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<math>={0.95\times 0.005 \over {0.95\times 0.005 +0.01\times 0.995}}=0.3231293.</math></center>
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is associated with probability statements that relate conditional and marginal properties of two random events. These statements are often written in the form "the probability of A, given B" and denoted P(A|B) = P(B|A)*P(A)/P(B) where P(B) not equal to 0.  
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===Bayesian Statstics===
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What is commonly called '''Bayesian Statistics''' is a very special application of Bayes Theorem.
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P(A) is often known as the Prior Probability (or as the Marginal Probability)
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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, <math>f(X)</math>, that takes a parameter, <math>\mu</math>, whose value is not certainly known.
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P(A|B) is known as the Posterior Probability (Conditional Probability)
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Using Bayes Theorem we may write
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P(B|A) is the conditional probability of B given A (also known as the likelihood function)
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: <math>f(\mu|\mathbf{x}) = \frac{f(\mathbf{x}|\mu) \cdot f(\mu)} { f(\mathbf{x}) }</math>
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P(B) is the prior on B and acts as the normalizing constant. In the Bayesian framework, the posterior probability is equal to the prior belief on A times the likelihood function given by P(B|A).
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In this formulation, we solve for <math>f(\mu|\mathbf{x})</math>, the "posterior" density of the population parameter, <math>\mu</math>.
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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>.
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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.
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<math>f(\mathbf{x}) = \int_{\mu} f( \mathbf{x} \cap \mu) d\mu =  \int_{\mu} f( \mathbf{x} | \mu ) f(\mu) d\mu </math>
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==See also==
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* [[EBook#Chapter_III:_Probability |Probability Chapter]]
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==References==
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<hr>
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* SOCR Home page: http://www.socr.ucla.edu
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{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Bayesian_Prelim}}

Current revision as of 18:24, 20 December 2010

Contents

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}, Dc = {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, 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

See also

References




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