This paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all DAG independence models. This class of graphs, called maximal ancestral graphs, has two attractive features: there is at most one edge between each pair of vertices; every missing edge corresponds to an independence relation. These features lead to a simple parameterization of the corresponding set of distributions in the Gaussian case.

Summary. Formal rules governing signed edges on causal directed acyclic graphs are described and it is shown how these rules can be useful in reasoning about causality. Specifically, the notions of a monotonic effect, a weak monotonic effect and a signed edge are introduced. Results are developed relating these monotonic effects and signed edges to the sign of the causal effect of an intervention in the presence of intermediate variables. The incorporation of signed edges in the directed acyclic graph causal framework furthermore allows for the development of rules governing the relationship between monotonic effects and the sign of the covariance between two variables. It is shown that when certain assumptions about monotonic effects can be made then these results can be used to draw conclusions about the presence of causal effects even when data are missing on confounding variables.

The application of certain Bayesian techniques, such as the Bayes factor and model averaging, requires the specification of prior distributions on the parameters of alternative models. We propose a new method for constructing compatible priors on the parameters of models nested in a given DAG (Directed Acyclic Graph) model, using a conditioning approach. We define a class of parameterisations consistent with the modular structure of the DAG and derive a procedure, invariant within this class, which we name reference conditioning.

Robots must perform tasks efficiently and reliably while acting under uncertainty. One way to achieve efficiency is to give the robot commonsense knowledge about the structure of the world. Reliable robot behaviour can be achieved by modelling the uncertainty in the world probabilistically. We present a robot system that combines these two approaches and demonstrate the improvements in efficiency and reliability that result. Our first contribution is a probabilistic relational model integrating common-sense knowledge about the world in general, with observations of a particular environment. Our second contribution is a continual planning system which is able to plan in the large problems posed by that model, by automatically switching between decision-theoretic and classical procedures. We evaluate our system on object search tasks in two different real-world indoor environments. By reasoning about the trade-offs between possible courses of action with different informational effects, and exploiting the cues and general structures of those environments, our robot is able to consistently demonstrate efficient and reliable goal-directed behaviour. 1

Directed acyclic graph (DAG) models are popular tools for describing causal relationships and for guiding attempts to learn them from data. In particular, they appear to supply a means of extracting causal conclusions from probabilistic conditional independence properties inferred from purely observational data. I take a critical look at this enterprise, and suggest that it is in need of more, and more explicit, methodological and philosophical justification than it typically receives. In particular, I argue for the value of a clean separation between formal causal language and intuitive causal assumptions.

This paper unifies three complementary approaches to defining, identifying, and estimating causal effects: the classical structural equations approach of the Cowles Commision; the treatment effects framework of Rubin (1974) and Rosenbaum and Rubin (1983); and the Directed Acyclic Graph (DAG) approach of Pearl. The settable system framework nests these prior approaches, while affording significant improvements to each. For example, the settable system approach permits identification of causal effects without requiring exogenous instruments; instead, a weaker conditional exogeneity condition suffices. It removes the stable unit treatment value assumption of the treatment effect approach and provides significant insight into the selection of covariates. It generalizes the DAG ap-proach by accommodating mutual causality and attributes. We provide a variety of results ensuring structural identification of general covariate-conditioned average causal effects, laying the founda-tion for parametric and nonparametric estimation of effects of interest and new tests for structural identification.

We study the interrelations between (conditional) independence and causal relations in settable systems. We provide definitions in terms of functional dependence for direct, indirect, and total causality as well as for (indirect) causality via and exclusive of a set of variables. We then provide necessary and sufficient causal and stochastic conditions for (conditional) dependence among random vectors of interest in settable systems. Immediate corollaries ensure the validity of Reichenbach’s principle of common cause and its informative extension, the conditional Reichenbach principle of common cause. We relate our results to notions of d-separation and D-separation in the artificial intelligence literature.

We outline a representation for discrete multivariate distributions in terms of interventional potential functions that are globally normalized. This representation can be used to model the effects of interventions, and the independence properties encoded in this model can be represented as a directed graph that allows cycles. In addition to discussing inference and sampling with this representation, we give an exponential family parametrization that allows parameter estimation to be stated as a convex optimization problem; we also give a convex relaxation of the task of simultaneous parameter and structure learning using group ℓ1regularization. The model is evaluated on simulated data and intracellular flow cytometry data. 1

Recent advances in modeling directed acyclic graphs are used to sort-out causal patterns among a set of thirteen measures deemed relevant to the incidence of world poverty. Cross-section measures of the percent of population living on one and two dollars or less per day from eighty low income countries are exposed to a battery of tests of conditional independence with respect to measures of economic and political freedom, income inequality, income per person, agricultural income, child mortality, birth rate, life expectancy, relative size of rural population, illiteracy rate, foreign aid as a percentage of national income, international trade as a percentage of national income and percentage of population that is under-nourished. Motivation for the method of analysis precedes results. Results are presented as a graph that shows our measures of economic and political freedom, income inequality, illiteracy and agricultural income to be exogenous movers of poverty when measured as the percent of the population living on two dollars or less per day. Foreign aid and international trade are not connected to the other variables in the graph. Results on our measure of extreme poverty (people living on one dollar or less per day) show that such populations are immune from improvements in economic progress of the general economy. The “rising tide lifts all boats ” argument apparently doesn’t cover the extreme poor of our sample.

In recursive linear models, the multivariate normal joint distribution of all variables exhibits a dependence structure induced by recursive systems of linear structural equations. Such models appear in particular in seemingly unrelated regressions, structural equation modelling, simultaneous equation systems, and in Gaussian graphical modelling. We show that recursive linear models that are ‘bow-free’ are well-behaved statistical models, namely, they are everywhere identifiable and form curved exponential families. Here, ‘bow-free ’ refers to models satisfying the condition that if a variable x occurs in the structural equation for y, then the errors for x and y are uncorrelated. For the computation of maximum likelihood estimates in ‘bow-free ’ recursive linear models we introduce the Residual Iterative Conditional Fitting (RICF) algorithm. Compared to existing algorithms RICF is easily implemented requiring only least squares computations, has clear convergence properties, and finds parameter estimates in closed form whenever possible. KEY WORDS: Linear structural equation model; curved exponential family; maximum likelihood estimation; residual iterative conditional fitting; bow-free acyclic path diagrams; BAP. 1