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# MoG,II2011 - Algorithm and its derivation MOG,II WHEN PDF...

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3/1/2011 1 MOG,II Algorithm and its derivation WHEN PDF SAMPLE IS UNAVAILABLE ´ Now, we consider the case when the values of PDF f ( x ) is not available ´ But assume: we can sample a random variable X~ f ( x ) ´ It is indeed more convenient! ´ A prototype of MCMC ´ A good example for HMM ´ The EM algorithm works (to some degree)

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3/1/2011 2 A BABY EXAMPLE ´ Problem: Model the PDF of the height of students (of two obvious groups: male and female) ´ Given: measurement results x 1 , x 2 , x 3 , …, x N ´ Assumption: Each height follows a Gaussian g ( x; μ 1 , σ 1 ) or g ( x; μ 2 , σ 2 ) depending on the sex ´ Easy case: we know the sex, say Male: x 1 , x 4 ,..., x N-1 Female: x 2 , x 3 ,…, x N Then it’s easy to estimate the means and variances of each Gaussian ´ Hard case: we do not know the sex of the person Note that we can not sample the PDF itself directly: what is the value of f ( x )? Samples of X~ f ( x ) EXAMPLE ´ Assume p(male)= p and p(female)= q ´ X~ f(x)=pg(x; μ 1 , σ 1 )+qg(x; μ 2 , σ 2 ) ´ What is p(male|x i )?
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