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STS563 Davide Di Cecco et al.
where pyi|x indicates the conditional probability P(Yi = yi|X = x), or as in the
log-linear model notation

which reports only the higher order interactions (generators) of the model.
Any additional interaction term in (2) represents a relaxation of the CIA.

2.1 Prior distributions: The usual priors for log–linear models are based on
Multivariate Gaussian distributions. Here we propose a different prior based
on Dirichlet distributions. We find this approach easier in terms of elicitation
of prior knowledge, and also from a computational point of view, since it
allows us to develop a Gibbs sampler for obtaining a sample from the
posterior distribution of N1, so avoiding the use of a Metropolis–Hastings
algorithm. To illustrate our proposal we start with decomposable models. In
this case the prior distribution is simply the product of Dirichlet densities. In
Dawid et al (1993) it has been demonstrated that, if G is the dependence graph
of the decomposable model, { L1,. ..,Lg} are the maximal cliques of G, and { L1,.
..,Lg} are defined as

the joint distribution can be written as the product of conditional distributions:

where p over a (sub)graph is the (marginal) distribution over the variables
included in the (sub)graph. Let be the vector of parameters =
We define a prior distribution on as follows: for each

and for each value of we set a Dirichlet distribution defined for each
possible combination of values of the variables in The
Dirichlet densities are independent by construction, and this class of priors is
conjugate to (3). In the case of a general log–linear model, we made use of the
“Bayesian iterative proportional fitting” described in Schafer (1997) in order to
sample from a “Constrained Dirichlet”. That is, we generate samples from a
Dirichlet distribution which satisfies the constraints given by the log–linear
model. This prior has been rarely utilized in literature, and, as far as we know,
has never been utilized in capture–recapture analysis. Regarding N, in

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