Mixtures of conjugate priors p θ k x k 1 w k p k θ

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Mixtures of Conjugate Priors p ( θ ) = K X k =1 w k p k ( θ ) I w k is mixture weight 0 < w k < 1 and k w k = 1 I p k ( θ ) is a conjugate prior for θ p k ( θ ) = c k c ( θ ) n 0 k exp( θ n 0 k t 0 k ) where c k is the normalizing constant. Bayes Theorem: p ( θ ) = X w k c k c ( θ ) n 0 k exp( θ n 0 k t 0 k ) p ( Y | θ ) = c Y c ( θ ) n exp( θ n ¯ t ( Y )) p ( θ | Y ) p ( θ ) p ( Y | θ )
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Result p ( θ | Y ) c Y c ( θ ) n exp( θ n ¯ t ( Y )) X w k c k c ( θ ) n 0 k exp( θ n 0 k t 0 k ) Subject to integration to 1; update
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