This time we will use a uniform prior p 2 0 12

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Unformatted text preview: → ∞).Thus as m → ∞, (19) obeys m p(θ|y ) p(θ) m p(yi |θ) log = log + log → −∞, ∗ |y ) ∗) p(θ p(θ p(yi |θ∗ ) i=1 and log p(θ|y ) p(θ|y ) → −∞ implies → 0 which implies p(θ|y ) → 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 non-zero 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.

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