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Unformatted text preview: → ∞).Thus as m → ∞, (19) obeys
m p(θy )
p(θ) m
p(yi θ)
log
= log
+
log
→ −∞,
∗ y )
∗)
p(θ
p(θ
p(yi θ∗ )
i=1
and
log p(θy )
p(θy )
→ −∞ implies
→ 0 which implies p(θy ) → 0.
p(θ∗ y )
p(θ∗ y ) This holds for every θ = θ∗ , thus p(θ∗ y ) → 1.
Again, this theorem tells us that the posterior eventually becomes concen
trated on the value θ∗ . θ∗ corresponds to the likelihood model that is ‘closest’
in the KL sense to the true generating distribution q (·). If q (·) = p(·θ0 ) for
some θ0 ∈ Θ, then θ∗ = θ0 is the unique minimizer of (18) because p(·θ∗ ) is
closest to p(·θ0 ) when θ∗ = θ0 . The theorem tells us that the posterior will
become concentrated around the true value. This is only the case if in the
prior, p(θ∗ ) > 0, which shows the importance of choosing a prior that assigns
nonzero probability to every plausible value of θ.
We now present the continuous version of Theorem 3, but the proof is quite
technical and is omitted.
Theorem 4 (Posterior consistency in continuous parameter spa...
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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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