Steffen L. Lauritzen

The EM algorithm for graphical association models with missing data

Abstract It is shown how the computational scheme of Lauritzen and Spiegelhalter (1988) can be exploited to perform the E-step of the EM algorithm when applied to finding maximum likelihood estimates or penalized maximum likelihood estimates in hierarchical log-linear models and recursive models for contingency tables with missing data. The generalization to mixed association models introduced in Lauritzen and Wermuth (1989) and Edwards (1990) is indicated.

Sequential updating of conditional probabilities on directed graphical structures

A directed acyclic graph or influence diagram is frequently used as a representation for qualitative knowledge in some domains in which expert system techniques have been applied, and conditional probability tables on appropriate sets of variables form the quantitative part of the accumulated experience. It is shown how one can introduce imprecision into such probabilities as a data base of cases accumulates. By exploiting the graphical structure, the updating can be performed locally, either approximately or exactly, and the setup makes it possible to take advantage of a range of well-established statistical techniques. As examples we discuss discrete models, models based on Dirichlet distributions and models of the logistic regression type.

Independence properties of directed markov fields

We investigate directed Markov fields over finite graphs without positivity assumptions on the densities involved. A criterion for conditional independence of two groups of variables given a third is given and named as the directed, global Markov property. We give a simple proof of the fact that the directed, local Markov property and directed, global Markov property are equivalent and – in the case of absolute continuity w. r. t. a product measure – equivalent to the recursive factorization of densities. It is argued that our criterion is easy to use, it is sharper than that given by Kiiveri, Speed, and Carlin and equivalent to that of Pearl. It follows that our criterion cannot be sharpened.

Propagation of Probabilities, Means, and Variances in Mixed Graphical Association Models

Abstract A scheme is presented for modeling and local computation of exact probabilities, means, and variances for mixed qualitative and quantitative variables. The models assume that the conditional distribution of the quantitative variables, given the qualitative, is multivariate Gaussian. The computational architecture is set up by forming a tree of belief universes, and the calculations are then performed by local message passing between universes. The asymmetry between the quantitative and qualitative variables sets some additional limitations for the specification and propagation structure. Approximate methods when these are not appropriately fulfilled are sketched. It has earlier been shown how to exploit the local structure in the specification of a discrete probability model for fast and efficient computation, thereby paving the way for exploiting probability-based models as parts of realistic systems for planning and decision support. The purpose of this article is to extend this computational s...

Stable local computation with conditional Gaussian distributions

This article describes a propagation scheme for Bayesian networks with conditional Gaussian distributions that does not have the numerical weaknesses of the scheme derived in Lauritzen (Journal of the American Statistical Association 87: 1098–1108, 1992).The propagation architecture is that of Lauritzen and Spiegelhalter (Journal of the Royal Statistical Society, Series B 50: 157– 224, 1988).In addition to the means and variances provided by the previous algorithm, the new propagation scheme yields full local marginal distributions. The new scheme also handles linear deterministic relationships between continuous variables in the network specification.The computations involved in the new propagation scheme are simpler than those in the previous scheme and the method has been implemented in the most recent version of the HUGIN software.

Graphical and recursive models for contingency tables

SUMMARY We discuss two classes of models for contingency tables, graphical and recursive models, both of which arise from restrictions that are expressible as conditional independencies of variable pairs. The first of these is a subclass of hierarchical log linear models. Each of its models can be represented by an undirected graph. In the second class each model corresponds to a particular kind of a directed graph instead and can be characterized by a nontrivial factorization of the joint distribution in terms of response variables. We derive decomposable or multiplicative models as the intersecting class. This result has useful consequences for exploratory types of analysis as well as for the model interpretation: we can give an aid for detecting well-fitting decomposable models in a transformation of the observed contingency table and each decomposable model may be interpreted with the help of an undirected or directed graph.

Chain graph models and their causal interpretations

Chain graphs are a natural generalization of directed acyclic graphs and undirected graphs. However, the apparent simplicity of chain graphs belies the subtlety of the conditional independence hypotheses that they represent. There are many simple and apparently plausible, but ultimately fallacious, interpretations of chain graphs that are often invoked, implicitly or explicitly. These interpretations also lead to flawed methods for applying background knowledge to model selection. We present a valid interpretation by showing how the distribution corresponding to a chain graph may be generated from the equilibrium distributions of dynamic models with feed-back. These dynamic interpretations lead to a simple theory of intervention, extending the theory developed for directed acyclic graphs. Finally, we contrast chain graph models under this interpretation with simultaneous equation models which have traditionally been used to model feed-back in econometrics.

Probabilistic expert systems for DNA mixture profiling

We show how probabilistic expert systems can be used to structure and solve complex cases of forensic identification involving DNA traces that might be mixtures of several DNA profiles. In particular, this approach can readily handle cases where the number of contributors to the mixture cannot be regarded as known in advance. The flexible modularity of the networks used also allows us to handle still more complex cases, for example where the finding of a mixed DNA trace is compounded by such features as missing individuals or the possibility of unobserved alleles.

Proper local scoring rules

We investigate proper scoring rules for continuous distributions on the real line. It is known that the log score is the only such rule that depends on the quoted density only through its value at the outcome that materializes. Here we allow further dependence on a finite number

Extremal families and systems of sufficient statistics

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