  Identifiability of parameters is an essential property for a statistical
model to be useful in most settings. However, establishing parameter
identifiability for Bayesian networks with hidden variables remains
challenging. In the context of finite state spaces, we give algebraic arguments
establishing identifiability of some special models on small DAGs. We also
establish that, for fixed state spaces, generic identifiability of parameters
depends only on the Markov equivalence class of the DAG. To illustrate the use
of these results, we investigate identifiability for all binary Bayesian
networks with up to five variables, one of which is hidden and parental to all
observable ones. Surprisingly, some of these models have parameterizations that
are generically 4-to-one, and not 2-to-one as label swapping of the hidden
states would suggest. This leads to interesting difficulties in interpreting
causal effects.
