  We examine the complexity of inference in Bayesian networks specified by
logical languages. We consider representations that range from fragments of
propositional logic to function-free first-order logic with equality; in doing
so we cover a variety of plate models and of probabilistic relational models.
We study the complexity of inferences when network, query and domain are the
input (the inferential and the combined complexity), when the network is fixed
and query and domain are the input (the query/data complexity), and when the
network and query are fixed and the domain is the input (the domain
complexity). We draw connections with probabilistic databases and liftability
results, and obtain complexity classes that range from polynomial to
exponential levels.
