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Unformatted text preview: Chapter 3 A Recipe for Inference 3.0.3 Pure Inference The projection operation is very useful for Bayesian inference when expressed in the following form: n i =0 Pr( A | B i ) Pr( B i ) = Pr( A ) This enables us to rewrite Bayes Law in the form: Pr( H | O ) = Pr( O | H ) Pr( H ) Pr( O ) = Pr( O | H ) Pr( H ) h Pr( O | h ) Pr( h ) where the hidden variable h is summed over all possible disjoint values, and H represents one specific possible value of h , whose posterior probability Pr( H | O ) we wish to calculate. This can be considered a pure form of inference in that it replaces the somewhat mysterious term for the probability of the observations Pr( O ) , with an explicit calculation that is entirely based on inference models. In other words, the entire calculation is supplied purely by models (specifically, their likelihood functions and priors). The denominator in this expression can be considered a normalization in the sense that it is simply a summation of the term in the numerator over all possible values of the hidden variable h . Whereas its often not obvious how to calculate the probability of the observations Pr( O ) directly, we now have a simple expression that depends only on the likelihood model Pr( O | h ) and the prior Pr( h ) , the same factors that we need for the numerator. 3.1 Inference Examples Lets see how we can use these ideas to solve problems. 3.1.1 Estimating Disease Risk Lets say a recessive disease gene has been mapped to the X chromosome. Since women have two copies of the X chromosome (they have two sex chromosomes, XX ) and men have only one copy (they have one X chromosome and one Y chromosome, XY ), men are much more likely to get disease symptoms. A disease is defined as recessive if a single copy of the normal gene is sufficient to prevent disease, even if one copy of the genetic variant that causes disease is also present. For a man, a single bad copy of the gene (which we will symbolize as x ) will give him disease. Such a man will be xY , whereas a woman with one copy of the disease gene ( xX ) will not develop disease symptoms. Such a woman is referred to as a disease carrier. Only women with two bad copies of the gene ( xx ) will get the disease. Consider a woman who is a disease carrier ( xX ); she will have no symptoms (which we will symbolize as- ), but her sons are at high risk for the disease, because they only inherit the X chromosome from their mother (they inherit a male Y chromosome from their father; only daughters inherit an X chromosome from the father). Since a son is 31 32 CHAPTER 3. A RECIPE FOR INFERENCE equally likely to inherit either copy of the X chromosome from the mother, he has a 50% probability of inheriting the disease allele ( x ) and developing disease symptoms, which we will symbolize as + ....
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