MIT15_097S12_lec15

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

Ask a homework question - tutors are online