Monte Carlo Inference - STAT - EPFL

Birth□ We find thatlikelihood ratio =λ′ n ′j−1je −λ′ j (u∗ −u j ) × λ ′ j+1n ′ j+1e −λ′ j+1 (u j+1−u ∗ )λ n jj e−λ j(u j+1 −u j )prior ratio = p k+1 (2k + 3)(2k + 2) (u j+1 − u ∗ )(u ∗ − u j )p k L 2 (u j+1 − u j )× βαΓ(α)( λ′j λ ′ j+1λ j) α−1exp { −β(λ ′ j + λ ′ j+1 − λ j ) } ,proposal ratio = d k+1/(k + 1),b k /L∂(λJacobian =′ j ,λ′ j+1 )∣ ∂(λ j ,w) ∣ = (λ′ j + λ′ j+1 )2,λ j□ The proposal ratio for the birth move is the ratio of the probability for the corresponding death(k + 1) ↦→ k (which chooses one of the steps to remove at random) to the birth move (choosing arandom site for u ∗ )□ The Jacobian is obtained from the mapping (λ j ,w) ↦→ (λ ′ j ,λ′ j+1 ) on the previous slide. In termsof the general discussion leading to (10) we have u (1) ≡ λ j , u (2) ≡ (λ ′ j ,λ′ j+1 ).Monte Carlo Inference Spring 2009 – slide 210,Death□ The death move (k + 1) ↦→ k has to correspond to the birth move, so we sample a changepointj ∈ {1,... ,k + 1} at random, with probability (k + 1) −1 , and try to merge the values of λ j−1 , λ jto get λ ′ j−1 using the formula(u j+1 − u j )log λ j + (u j − u j−1 )log λ j−1 = (u j+1 − u j−1 )log λ ′ j−1 ,to correspond to the birth move.□ If the death move is accepted, then u 1 ,... ,u k+1 and λ 0 ,...,λ k+1 are modified by dropping u jand mapping (λ j ,λ j+1 ) ↦→ λ ′ j , with k ↦→ k − 1 and changepoints correspondingly relabelled.□ The acceptance probability for the death move (k + 1) ↦→ k is{}1min 1,(likelihood ratio) × (prior ratio) × (proposal ratio) × (Jacobian)where the terms here are those given on the previous slide for the birth move k ↦→ (k + 1), withonly minor notational changes to account for the fact that the inverse move is being proposed.Monte Carlo Inference Spring 2009 – slide 211204