DNA Identification:Bayesian Belief Update
Cybergenetics © 2003-2010Cybergenetics © 2003-2010
TrueAlleleTrueAllele®® Lectures LecturesFall, 2010Fall, 2010
Mark W Perlin, PhD, MD, PhDMark W Perlin, PhD, MD, PhDCybergenetics, Pittsburgh, PACybergenetics, Pittsburgh, PA
Bayes Theorem
Bayes Theorem
Our belief in a hypothesisafter seeing datais proportional to how well that hypothesis explains the datatimes our initial belief.
All hypotheses must be considered. Need computers to do this properly.
Find the probability of causes by examining effects.
Rev Bayes, 1763
Computers, 1992
Pr X=xdata{ }∝Pr dataX=x{ }⋅Pr X=x{ }
Bayesian Update
Posterior probability Likelihood function Prior probability
Pr X=xdata{ }= Pr dataX=x{ }⋅Pr X=x{ }Pr dataX=′x{ }⋅Pr X=′x{ }
′x∈X∑
Pr X=xdata{ }∝Pr dataX=x{ }⋅Pr X=x{ }
Bayesian Update
Posterior probability Likelihood function Prior probability
Consider all possibilities
Pr X=x( )∝xa−1⋅(1−x)b−1PrkX=x( )∝xk⋅(1−x)n−kPrX=xk( )∝PrkX=x( )⋅PrX=x( )
∝xk+a( )−1⋅(1−x)b+n−k( )−1
Parameter Update
Beta distribution
Beta distribution
Binomial distribution
Bayes original example
Disease Prevalence
With a positive medical test result, what is the chance of having this rare disease?
Free of Disease
Got the Disease
Pr(Free)99.9%
(= 100% - 0.1%)
Pr(Got)0.1%1 in a thousand
Medical Test: Likelihood
DataFree of Disease
Got the Disease
Positive Test
False positivePr(Pos | Free)5%
5 in a hundred
True positivePr(Pos | Got)99%
(= 100% - 1%)
Negative Test
True negativePr(Neg | Free)
95%(= 100% -
5%)
False negativePr(Neg | Got)1%
1 in a hundred
Probability of DiseaseWith a positive test (Pos),
what is the probability of having the disease (Got)?
Posterior Odds=PrGotPos( )PrFreePos( )=
PrPosGot( )⋅PrGot( )PrPosFree( )⋅PrFree( )
=PrPosGot( )PrPosFree( )⋅
PrGot( )PrFree( )
=99%5%⋅0.1%99.9%Likelihood Ratio ⋅ Prior Odds=20⋅11000=150=2%
Odds of DiseaseWith a positive test (Pos),
what are the odds of having the disease (Got vs. Free)?
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