Parameter Identifiability of Discrete Bayesian Networks with Hidden Variables

Abstract

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 directed acyclic graphs (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 conflict in interpreting causal effects.

GraphicalMarkov Models, Unifying Results and Their Interpretation

“…Only the generating process in the middle of Fig.1 specifies a full ordering of all three variables as (1,2,3), while one cannot distinguish with the graph alone between (1,2,3) and (2,1,3) for the source V and between (1,2,3) and (1,3,2) for the sink V. More generally, a directed acyclic graph may be 'compatible with several orderings' of the variables such that the set of all independences, that is the 'independences structure' of a graph, remains unchanged. This poses problems for some machinelearning strategies.…”

Algebraic tools for the analysis of state space models

“…as in the introduction, with x(t) ∈ R N , y(t) ∈ R M , u(t) ∈ R R and p ∈ R n . The state space model (2) is called identifiable if the unknown parameter vector p can be recovered from observation of the input and output alone. The model is observable if the trajectories of the state space variables x(t) can be recovered from observation of the input and output alone.…”

“…Proposition 6.2. Consider a state space model of form (2). Let Π be the differential ideal generated the polynomials x −f (x, p, u), y −g(x, p).…”

“…Beltrametti and Robbiano [7] consider the ideal c 1 (p) − c 1 (p * ), c 2 (p) − c 2 (p * ), ..., c m (p) − c m (p * ) for detecting identifiability in the context of the Hough transform. Other areas include study of identifiability of graphical models [2,23,29,30,65] and identifability of phylogenetic models [1,43,45,57].…”

“…Consider a state space model of the form(2) where f and g are polynomial functions and where there are N state-space variables, M = 1 output variable, and R input variables. Let P be the idealx − f (x, p, u), .…”