I1 although this proof yields the form of the

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Unformatted text preview: xponential family if every member of F has the form: � � (16) p(yi |θ) = f (yi )g (θ) exp φ(θ)T u(yi ) , for some f (·), g (·), φ(·), and u(·). Essentially all of the distributions that we typically work with (normal, ex­ ponential, Poisson, beta, gamma, binomial, Bernoulli,...) are exponential families. The next theorem tells us when we can expect to have a conjugate prior. Theorem 1 (Exponential families and conjugate priors). If the likeli­ hood model is an exponential family, then there exists a conjugate prior. Proof. Consider the likelihood of our iid data y = {y1 , . . . , ym }: p(y |θ) = = m m p(yi |θ) = i=1 im m m m � � f (yi )g (θ) exp φ(θ)T u(yi ) i=1 � f (yi ) g (θ)m exp φ(θ)T i=1 m m � u(yi ) . i=1 Take the prior distribution to be: p(θ) = � � g (θ)η exp φ(θ)T ν , g (θ' )η exp (φ(θ' )T ν ) dθ' 13 (17) where η and ν are prior hyperparameters. Then the posterior will be p(θ|y ) ∝ p(y |θ)p(θ) � im � m m m = f (yi ) g (θ)m exp φ(θ)T u(yi ) i=1 i=1 � ∝ g (θ)η+m exp φ(θ)T � ν+ m m � � g (θ)η exp φ(θ)T ν g (θ' )η...
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This note was uploaded on 03/24/2014 for the course MIT 15.097 taught by Professor Cynthiarudin during the Spring '12 term at MIT.

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