Lattice Models for Conditional Independence in a Multivariate Normal Distribution

Abstract

The lattice conditional independence model $\mathbf{N}(\mathscr{K})$ is defined to be the set of all normal distributions on $\mathbb{R}^I$ such that for every pair $L, M \in \mathscr{K}, x_L$ and $x_M$ are conditionally independent given $x_{L \cap M}$. Here $\mathscr{K}$ is a ring of subsets of the finite index set $I$ and, for $K \in \mathscr{K}, x_K$ is the coordinate projection of $x \in \mathbb{R}^I$ onto $\mathbb{R}^K$. Statistical properties of $\mathbf{N}(\mathscr{K})$ may be studied, for example, maximum likelihood inference, invariance and the problem of testing $H_0: \mathbf{N}(\mathscr{K})$ vs. $H: \mathbf{N}(\mathscr{M})$ when $\mathscr{M}$ is a subring of $\mathscr{K}$. The set $J(\mathscr{K})$ of join-irreducible elements of $\mathscr{K}$ plays a central role in the analysis of $\mathbf{N}(\mathscr{K})$. This class of statistical models occurs in the analysis of nonnested multivariate missing data patterns and nonnested dependent linear regression models.