The general method for MCL analysis with missing data for statistical models specified by families of unnormalized densities was laid out by Gelfand and Carlin (1993); see also the introduction of Geyer (1994).
Suppose now that x is missing data and y is observed data for a model
specified by unnormalized joint densities 
for the complete
data (x,y). Then the normalizing function for the complete data model
is
Also recall from Section 1.3.2 that is also
an unnormalized conditional density of x given y and that the normalizing
function for the conditional family is
The likelihood is is the ratio of these normalizing functions
As usual, we write the log likelihood as a ratio against a fixed parameter
point 
The same argument (1.29) that gave us ratios of normalizing constants in the non-missing-data case now says
when applied to the complete data model and
when applied to the conditional family. Hence
In order to approximate this by Monte Carlo we need samples from both the
joint distribution of X and Y and the conditional distribution of X
given Y, both for the parameter value . We can use MCMC for both.
Both have the same unnormalized density 
. The only difference
is that we leave y fixed at its observed value in one of the samplers.
Hence suppose that 
, i = 1, 2, 
are samples from the
joint distribution and 
, i = 1, 2, are samples from the
conditional distribution and y is the observed value of Y. Then
is a Monte Carlo approximation to the exact log likelihood (1.45).
