Gibbs Sampling

Gibbs Sampler

Algorithm Gibbs Sampler Generates a sample from by sampling in turn from the conditional distributions and :

1. Arbitrarily choose .
2. For :
   1. Draw
   2. Draw
3. Produces In general -dimensional case, use the most up-to-date values of each parameter at each stage. Gibbs sampler works in an irreducible and aperiodic Markov chain that converges to a unique limiting distribution. Yet there would be more than one possible limiting distribution in general.

Thrm Hammersley-Clifford Theorem The following positivity condition is sufficient for the univariate conditionals to uniquely determine the joint distribution:

Prop The rate of convergence of Gibbs sampler depends on the degree of dependence between model parameters.

Assessing Convergence

Algorithm Burn In If convergence is slow, early chain values will not represent , so typically the first portion of each chain is discarded.

Algorithm Multiple Chains Start multiple chains at over dispersed starting points, and after convergence chains will be indistinguishable.

Convergence Diagnostics

Fact There are many convergence diagnostics, operating the principal that if chains have converged, then diagnostic tests will be passed. However, the reverse is not true. So are necessary, but not sufficient indicators of sampler convergence.

Def Gelman-Rubin Ratio Denote as -th sample from -th Markov chain. Each chain is length after discarding first half of sample path. The Gelman-Rubin ratio is given bywherewith the between chain variance be defined asand the within chain variance be defined as

Thrm Gelman-Rubin Diagnostic If chains have converged, then the Gelman-Rubin ratio decline to as .

Blocking

Fact If parameters and are highly correlated, then Gibbs sampling via and will converge slowly. If joint conditional distribution of is available, joint (block) updates can avoid effect of high correlation.