Marginal parameterizations of discrete models defined by a set of conditional independencies

Abstract

a b s t r a c tIt is well-known that a conditional independence statement for discrete variables is equivalent to constraining to zero a suitable set of log-linear interactions. In this paper we show that this is also equivalent to zero constraints on suitable sets of marginal log-linear interactions, that can be formulated within a class of smooth marginal log-linear models. This result allows much more flexibility than known until now in combining several conditional independencies into a smooth marginal model. This result is the basis for a procedure that can search for such a marginal parameterization, so that, if one exists, the model is smooth.

Marginal Log-Linear Parameters for Graphical Markov Models

“…This latter case is equivalent to the summary graphs of Wermuth (2011) and strictly includes all ancestral graphs (Richardson and Spirtes, 2002). Our approach may be seen as extending earlier work (Rudas et al, 2006(Rudas et al, , 2010Forcina et al, 2010) which described the conditional independence structure of certain MLL models. Richardson (2003) described local and global Markov properties for ADMGs, whereas Richardson (2009) gave a parameterization for discrete random variables via a collection of conditional probabilities of the form P.X H = 0|X T = x T /.…”

“…The next result relates MLL parameters to conditional independences; it is found as lemma 1 in Rudas et al (2010) and equation (6) of Forcina et al (2010). The special case of C = ∅ (giving marginal independence) was proved in the context of multivariate logistic parameters by Kauermann (1997).…”

“…Forcina et al . () provided an algorithm which gives a range of ‘admissible’ margins in which collections of conditional independence constraints may be defined.…”

“…This latter case is equivalent to the summary graphs of Wermuth () and strictly includes all ancestral graphs (Richardson and Spirtes, ). Our approach may be seen as extending earlier work ( Rudas et al ., , ; Forcina et al ., ) which described the conditional independence structure of certain MLL models.…”

“…Models defined by conditional independence constraints are central to many methods in multivariate statistics, and in particular to graphical models (Darroch et al, 1980;Whittaker, 1990). In the case of discrete data, marginal log-linear (MLL) parameters can be used to parameterize a broad range of models, including some graphical classes and models for conditional independence (Rudas et al, 2010;Forcina et al, 2010). These parameters are defined by considering an ordered sequence M 1 , M 2 , : : : , M k of margins of the distribution which respects inclusion (i.e.…”