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

© TTU Mathematics