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## MCMC scheme

A data augmentation algorithm is formed by two successive probability transitions--one from to and the other from to . Given a current state we can generate the transition probability

Successively given a state we can update a state in from the Dirichlet distribution with parameter . Combined together,

becomes a Markov transition kernel on .

Gibbs sampler. Let

be the normalizing constant for the density  , and let

be the marginal density on  . Then we obtain the joint density by

Thus, the kernel is a Gibbs sampler using the conditional density and .

Alternative Markov chain. Combining the transition probabilities  and in the opposite order, we can introduce a Markov kernel  on the state space  by

where is a surface integral over . This Markov kernel  is also time-reversible with stationary distribution . Hence, we can instead materialize a Gibbs sampler for  in a discrete setting, and obtain the target observation from  by way of sampling it from