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Up to normalization, the joint posterior isA Gibbs sampler cycle consists of:

Each proposal is the exact full conditional and is therefore accepted. If the cycle begins with density $p({\theta }^t\mid D)$, integrating the product of the current posterior and successive conditional kernels over all overwritten coordinates leavesso a full Gibbs sweep preserves the joint posterior.

A Gibbs sampler draws each variable from its full conditional distribution, so every proposed update is accepted. At the start of a sweep define $c_s={\tilde{c}}_s-E_s$, $y_s={\tilde{M}}_s-R{E}_s$, andFirst update the reddenings independently asRecompute $c_s$ and $y_s$, then make the Gaussian updatesandWiththe remaining full conditionals, in the parameterization given in the question, are the scaled inverse chi-squared lawsRepeating these updates in the displayed order gives a complete Gibbs sweep. The formulas assume $N>2$ and nondegenerate sampled predictors; posterior propriety must be checked because the hyperpriors are improper.

Put $z_i=\log Y_i$. The Weighted graph Laplacian $L_J$ of the tree is defined byMultiplication of the Gaussian likelihood by the prior shows that, conditionally on the precision parameter $\tau$,Completing the square therefore givesAs a function of $\tau$, the posterior density isso, in shape-rate notation,The precision matrix $M$ has the sparsity pattern of a tree. A sparse Cholesky decomposition and its triangular solves have $O(n)$ cost and storage on this graph, while the gamma update also costs $O(n)$. Hence each systematic-scan Gibbs sampler iteration costs $O(n)$.

Bayes theorem turns the categorical distribution prior probabilities and the component multivariate normal densities intoThese probabilities define the label update in the Gibbs sampler.

Choose an initial value $X_0$. One sweep of the Gibbs sampler consists ofAfter a burn-in period, the pairs $(Y_m,X_m)$ are retained as approximate samples. Standard irreducibility and recurrence conditions are needed for convergence from the chosen starting point.

The transition density from $(y,x)$ to $(y^{\prime },x^{\prime })$ isSuppose $(Y,X)$ currently has density $f_{XY}$. The density after one sweep isThis proves the stationarity of the two-coordinate Gibbs sampler: