Monte Carlo Inference - STAT - EPFL

Heights of steps□ Conditional on k, u 1 ,... ,u k , we suppose that the heights satisfyλ 0 ,... ,λ k | u 1 ,... ,u k ,kiid ∼Γ(α,β)β ∼ Γ(e,f)α ∼ Γ(c,d),with c,d,e,f determined in advance (below we take e = f = 1, c = d = 2).□ This hierarchical model givesindλ j | rest ∼ Γ(α + n j ,β + u j+1 − u j ), j = 0,... ,k,⎧⎫⎨k∑ ⎬β | rest ∼ Γ⎩ e + (k + 1)α,f + λ j⎭j=0⎛ ⎞α c−1αk∏π(α | rest) ∝ ⎝e −dΓ(α) k+1 β k+1 ⎠ ,j=0□ Gibbs updates are possible for λ 0 ,...,λ k and β, and a random walk Metropolis step can be usedto update α, setting log α ′ ∼ N(log α,σ 2 ) (with σ = 0.5 below)Monte Carlo Inference Spring 2009 – slide 206Locations of steps□ To update the step locations u 1 ,... ,u k , we note that the joint density of the even order statisticsu 1 < · · · < u k from a random sample of size 2k + 1 from the U(0,L) distribution is(2k + 1)!L 2k+1 u 1 (u 2 − u 1 ) · · · (u k − u k−1 )(u k+1 − u k ),used to discourage changepoints from occurring too close together.□ We choose j ′ ∈ {1,... ,k} uniformly at random, and then propose to replace u j withu ′ j ∼ U(u j−1,u j+1 ), with the acceptance probability beingwhere L is the likelihood ratio{min 1,L × (u′ j − u j−1)(u j+1 − u ′ j ) }(u j − u j−1 )(u j+1 − u j )L = λn′ j−1j−1 e−λ j−1(u ′ j −u j−1) λ n′ jj e−λ j(u j+1 −u ′ j )λ n j−1j−1 e−λ j−1(u j −u j−1 ) λ n jj e−λ j(u j+1 −u j )and n ′ j−1 and n′ j are the numbers of events in the proposed new intervals [u j−1,u ′ j ) and[u ′ j ,u j+1).Monte Carlo Inference Spring 2009 – slide 207202