Proof we will rely on jensens inequality which states

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Unformatted text preview: , u(yi ) = yi , φ(θ) = −θ, and g (θ) = θ. Then, f (yi )g (θ) exp (φ(θ)u(yi )) = θe−θyi = p(yi |θ). Thus the exponential distribution is an exponential family. 4 Posterior asymptotics Up to this point, we have defined a likelihood model that is parameterized by θ, assigned a prior distribution to θ, and then computed the posterior p(θ|y ). There are two natural questions that arise. First, what if we choose the ‘wrong’ likelihood model? That is, what if the data were actually gener­ ated by some distribution q (y ) such that q (y ) = p(y |θ) for any θ, but we use p(y |θ) as our likelihood model? Second, what if we assign the ‘wrong’ prior? We can answer both of these questions asymptotically as m → ∞. First we 15 must develop a little machinery from information theory. A useful way to measure the dissimilarity between two probability distribu­ tions is the Kullback-Leibler (KL) divergence, defined for two distributions p(y ) and q (y ) as: � q (y ) q (y ) = q (y ) log dy. D(q (·)||p(·)) := 1y∼q...
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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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