MIT15_097S12_lec15

2i 2 2 2 2 let f yi 21 2 exp yi 2 2 uyi yi

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Unformatted text preview: exp (φ(θ' )T ν ) dθ' �� u(yi ) (dropping non-θ terms), i=1 which is in the same family as the prior (17), with the posterior hyperparam­ eters being η + m and ν + m u(yi ). i=1 Although this proof yields the form of the conjugate prior, we may not always be able to compute the partition function. So we can’t always gain something in practice from it. It turns out that in general, the converse to Theorem 1 is true, and exponen­ tial families are the only distributions with (non-trivial) conjugate priors. Coin Flip Example Part 5. Returning again to the coin flip example, let us first verify that the Bernoulli distribution is an exponential family: p(yi |θ) = θyi (1 − θ)1−yi = exp (yi log θ + (1 − yi ) log(1 − θ)) = exp (yi log θ − yi log(1 − θ) + log(1 − θ)) θ = (1 − θ) exp yi log , 1−θ and we see that the Bernoulli distribution is an exponential family according θ to (16) with f (yi ) = 1, g (θ) = 1 − θ, u(yi ) = yi , and φ(θ) = log 1−θ . Th...
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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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