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Unformatted text preview: apply Jensen’s inequality:
�
− 1y∼q(y)[f (X )] ≥ f (1y∼q(�)[X ])
y
q (y ) log p(y )
dy ≥ − log
q (y ) q (y ) p(y )
dy
q (y ) �
≥ − log p(y )dy ≥ − log 1 = 0 so,
�
q (y ) log q (y )
dy ≥ 0,
p(y ) with equality under the same conditions required for equality in Jensen’s
inequality: if and only if X is constant, that is, q (y ) = p(y ) ∀y .
16 We will use the KL divergence to ﬁnd the distribution from the likelihood
family that is ‘closest’ to the true generating distribution:
θ∗ = arg min D(q (·)p(·θ)).
θ∈Θ (18) For convenience, we will suppose that the arg min in (18) is unique. The
results can be easily extended to the case where the arg min is not unique. The
main results of this section are two theorems, the ﬁrst for discrete parameter
spaces and the second for continuous parameter spaces. The intuition for the
theorem is that as long as there is some probability in the prior that θ = θ∗ ,
then as m → ∞ the whole posterior will be close to θ∗ .
Theorem 3 (Posterior consistency in ﬁnite parameter space). Let F be...
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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.
 Spring '12
 CynthiaRudin

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