INTRODUCTION TO NESTED MARKOV MODELS

抄録

Graphical models provide a principled way to take advantage of independence constraints for probabilistic and causal modeling, while giving an intuitive graphical description of “qualitative features” useful for these tasks. A popular graphical model, known as a Bayesian network, represents joint distributions by means of a directed acyclic graph (DAG). DAGs provide a natural representation of conditional independence constraints, and also have a simple causal interpretation. When all variables are observed, the associated statistical models have many attractive properties. However, in many practical data analyses unobserved variables may be present. In general, the set of marginal distributions obtained from a DAG model with hidden variables is a much more complicated statistical model: the likelihood of the marginal is often intractable; the model may contain singularities. There are also an infinite number of such models to consider.
It is possible to avoid these difficulties by modeling the observed marginal directly. One strategy is to define a model by means of conditional independence constraints induced on the observed marginal by the hidden variable DAG; we call this the ordinary Markov model. This model will be a supermodel that contains the set of marginal distributions obtained from the original DAG. Richardson and Spirtes (2002) and Evans and Richardson (2013a) gave parametrizations of this model in the Gaussian and discrete case, respectively.
However, it has long been known that hidden variable DAG models also imply nonparametric constraints which generalize conditional independences; these are sometimes called “Verma Constraints”. In this paper we describe a natural extension of the ordinary Markov approach, whereby both conditional independences and these generalized constraints are used to define a nested Markov model. The binary nested Markov model may be parametrized via a simple extension of the binary parametrization of the ordinary Markov model of Evans and Richardson (2013a). We also give evidence for a characterization of nested Markov equivalence for models with four observed variables. A consequence of this characterization is that, in some instances, most structural features of hidden variable DAGs can be recovered exactly when a single generalized independence constraint holds under the distribution of the observed variables.