Posterior concentration rates for empirical Bayes procedures, with applications to Dirichlet Process mixtures Supplementary material
Sophie Donnet, Vincent Rivoirard, Judith Rousseau, Catia Scricciolo · 2014
1 Gibbs algorithm We detail the algorithm used to sample from the posterior distribution (λ,A, γ)|N in the Poisson process context, in the most complete case (with a hierarchical level on γ). In case where γ is set to a fixed value, then the corresponding part in the algorithm is removed. As a standard Gibbs algorithm, the simulation is decomposed into three steps: [1] λ|A, γ,N [2] A|λ, γ,N [3] γ|A, λ,N. where N is the observed Poisson process over [0, T], namely a number of jumps N(T) and jump instants (T1,..., TN(T)). In order to avoid an artificial truncation in λ, we use the slice sampler strategy proposed by Fall and Barat (2012). More precisely, we consider the stick breaking representation of λ. Let ci be the affectation variable of data Wi. The DPM model is written as: Wi|ci, θ ∗ ∼ hθ?ci, P (ci = k) = wk, ∀k ∈ N ∗ (wk)k∈N? ∼ Stick(A), (θ k)k∈N? ∼i.i.d Gγ. The slice sampler strategy consists in introducing a latent variable ui such that the joint dis-tribution of (Wi, ui) is p(Wi, ui|ω, θ k=1 wkhθ∗k(Wi) 1 ξk 1l[0,ξk](ui) with ξk = min(wk, ζ), which can be reformulated as: p(Wi, ui|ω, θ 1