Monte Carlo Inference - STAT - EPFL

UpdatingWe can now update the parameters λ 0 ,... ,λ k , u 1 ,... ,u k , β, α, giving posterior realisations of λ(u)like those below, for k = 0,1,2,3:Intensity0 1 2 3 4 5Intensity0 1 2 3 4 51880 1900 1920 1940 1960 19801880 1900 1920 1940 1960 1980Intensity0 1 2 3 4 5Intensity0 1 2 3 4 51880 1900 1920 1940 1960 19801880 1900 1920 1940 1960 1980Monte Carlo Inference Spring 2009 – slide 208Birth□ For the birth move, we first must propose a new location u ∗ , which will lie in the interval(u j ,u j+1 ), for j ∈ {0,... ,k}, and new heights for the corresponding interval.□ We take u ∗ ∼ U(0,L), and let λ ′ j ,λ′ j+1 denote the new heights, defined through(u j+1 − u ∗ )log λ ′ j+1 + (u∗ − u j )log λ ′ j = (u j+1 − u j )log λ j ,where w ∼ U(0,1). With a = (u ∗ − u j )/(u j+1 − u j ) this givesλ ′ j+1λ ′ j= 1 − ww ,λ j = (λ ′ j )a (λ ′ j+1 )1−a , w =λ ′ jλ ′ j + λ′ j+1and ensures that λ j lies between the proposed new heights.□ If the move is accepted we must set k ↦→ k + 1, relabel the new positionsu 1 ,... ,u j ,u ∗ ,u j+1 ,...,u k and the new heights λ 1 ,...,λ ′ j−1 ,λ′ j ,... ,λ k.□ We write the probability of acceptance asmin {1,(likelihood ratio) × (prior ratio) × (proposal ratio) × (Jacobian)}□ We put a prior distribution p j = Pr(k = j) on the number of steps j ∈ {0,... ,k max }.Monte Carlo Inference Spring 2009 – slide 209203