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.

Introduction The introduction of Bayesian networks (Pearl 1986b) and associated local computation algorithms (Lauritzen and Spiegelhalter 1988, Shenoy and Shafer 1990, Jensen, Lauritzen and Olesen 1990) has initiated a renewed interest for understanding causal concepts in connection with modelling complex stochastic systems. It has become clear that graphical models, in particular those based upon directed acyclic graphs, have natural causal interpretations and thus form a base for a language in which causal concepts can be discussed and analysed in precise terms. As a consequence there has been an explosion of writings, not primarily within mainstream statistical literature, concerned with the exploitation of this language to clarify and extend causal concepts. Among these we mention in particular books by Spirtes, Glymour and Scheines (1993), Shafer (1996), and Pearl (2000) as well as the collection of papers in Glymour and Cooper (1999). Very briefly, but fundamentally,

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Consider a study whose design calls for the study subjects to be followed from enrollment (time t = 0) to time t = T,at which point a primary endpoint of interest Y is to be measured. The design of the study also calls for measurements on a vector V(t) of covariates to be made at one or more times t during the interval [0,T). We are interested in making inferences about the marginal mean µ0 of Y when some subjects drop out of the study at random times Q prior to the common fixed end of follow-up time T. The purpose of this article is to show how to make inferences about µ0 when the continuous drop-out time Q is modeled semiparametrically and no restrictions are placed on the joint distribution of the outcome and other measured variables. In particular, we consider two models for the conditional hazard of drop-out given ( ¯ V(T), Y), where ¯ V(t) denotes the history of the process V(t) through time t, t ∈ [0,T). In the first model, we assume that λQ(t | ¯ V(T), Y) = λ0(t | ¯ V(t)) exp(α0Y), where α0 is a scalar parameter and λ0(t | ¯ V(t)) is an unrestricted positive function of t and the process ¯ V(t). When the process ¯ V(t) is high dimensional, estimation in this model is not feasible with moderate sample sizes, due to the curse of dimensionality. For such situations, we consider a second model that imposes the additional restriction that λ0(t | ¯ V(t)) = λ0(t) exp(γ ′ 0W(t)), where λ0(t) is an unspecified baseline hazard function, W(t) = w(t, ¯ V(t)), w(·, ·) is a known function that maps (t, ¯ V(t)) to Rq, and γ0 is a q × 1 unknown parameter vector. When α0 � = 0, then drop-out is nonignorable. On account of identifiability problems, joint estimation of the mean µ0 of Y and the selection bias parameter α0 may be difficult or impossible. Therefore, we propose regarding the selection bias parameter α0 as known, rather than estimating it from the data. We then perform a sensitivity analysis to see how inference about µ0 changes as we vary α0 over a plausible range of values. We apply our approach to the analysis of ACTG 175, an AIDS clinical trial. KEY WORDS: Augmented inverse probability of censoring weighted estimators; Cox proportional hazards model; Identification;

... this paper is to use experimental or observational data to estimate decision regimes that result in a maximal mean response. To explicate our objective and state assumptions, we use the potential outcomes model. The proposed method makes smooth, parametric assumptions only on quantities directly relevant to the goal of estimating the optimal rules. We illustrate the proposed methodology via a small simulation.

this paper we consider two fundamental issues concerning Robins' theory. Firstly, do his assumed relations (between observed and unobserved---factual and counterfactual---random variables) place restrictions on the distribution of the observed variables. If the answer is yes, adopting his approach means making restrictive implicit assumptions---not very desirable. If however the answer is no, his approach is neutral. One can freely use it in modelling and estimation, exploring the consequences (for the unobserved variables) of the model. This follows the highly succesful tradition in all sciences of making thought experiments. In what philosophical sense counterfactuals actually exist seems to us less relevant. But it is important to know if a certain thought experiment is a priori ruled out by existing data

Counterfactual quantities representing path-specific effects arise in cases where we are interested in computing the effect of one variable on another only along certain causal paths in the graph (in other words by excluding a set of edges from consideration). A recent paper [7] details a method by which such an exclusion can be specified formally by fixing the value of the parent node of each excluded edge. In this paper we derive simple, graphical conditions for experimental identifiability of path-specific effects, namely, conditions under which path-specific effects can be estimated consistently from data obtained from controlled experiments. 1

ABSTRACT: I describe two new methods for estimating the optimal treatment regime (equivalently, protocol, plan or strategy) from very high dimesional observational and experimental data: (i) g-estimation of an optimal double-regime structural nested mean model (drSNMM) and (ii) g-estimation of a standard single regime SNMM combined with sequential dynamicprogramming (DP) regression. These methods are compared to certain regression methods found in the sequential decision and reinforcement learning literatures and to the regret modelling methods of Murphy (2003). I consider both Bayesian and frequentist inference. In particular, I propose a novel “Bayes-frequentist compromise ” that combines honest subjective non- or semiparametric Bayesian inference with good frequentist behavior, even in cases where the model is so large and the likelihood function so complex that standard (uncompromised) Bayes procedures have poor frequentist performance. 1

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This paper concerns the assessment of the effects of actions from a combination of nonexperimental data and causal assumptions encoded in the form of a directed acyclic graph in which some variables are presumed to be unobserved. We provide a procedure that systematically identifies cause effects between two sets of variables conditioned on some other variables, in time polynomial in the number of variables in the graph. The identifiable conditional causal effects are expressed in terms of the observed joint distribution. 1