Thomas S. Richardson

Bryan Andrews, Gregory F. Cooper, Thomas S. Richardson et al.

Directed acyclic graph (DAG) models have become widely studied and applied in statistics and machine learning -- indeed, their simplicity facilitates efficient procedures for learning and inference. Unfortunately, these models are not closed under marginalization, making them poorly equipped to handle systems with latent confounding. Acyclic directed mixed graph (ADMG) models characterize margins of DAG models, making them far better suited to handle such systems. However, ADMG models have not seen wide-spread use due to their complexity and a shortage of statistical tools for their analysis. In this paper, we introduce the m-connecting imset which provides an alternative representation for the independence models induced by ADMGs. Furthermore, we define the m-connecting factorization criterion for ADMG models, characterized by a single equation, and prove its equivalence to the global Markov property. The m-connecting imset and factorization criterion provide two new statistical tools for learning and inference with ADMG models. We demonstrate the usefulness of these tools by formulating and evaluating a consistent scoring criterion with a closed form solution.

Mario Beraha, Lorenzo Masoero, Stefano Favaro et al.

Accurately predicting the onset of specific activities within defined timeframes holds significant importance in several applied contexts. In particular, accurate prediction of the number of future users that will be exposed to an intervention is an important piece of information for experimenters running online experiments (A/B tests). In this work, we propose a novel approach to predict the number of users that will be active in a given time period, as well as the temporal trajectory needed to attain a desired user participation threshold. We model user activity using a Bayesian nonparametric approach which allows us to capture the underlying heterogeneity in user engagement. We derive closed-form expressions for the number of new users expected in a given period, and a simple Monte Carlo algorithm targeting the posterior distribution of the number of days needed to attain a desired number of users; the latter is important for experimental planning. We illustrate the performance of our approach via several experiments on synthetic and real world data, in which we show that our novel method outperforms existing competitors.

Thomas S. Richardson, James M. Robins

In this note we give proofs for results relating to the Instrumental Variable (IV) model with binary response $Y$ and binary treatment $X$, but with an instrument $Z$ with $K$ states. These results were originally stated in Richardson & Robins (2014), "ACE Bounds; SEMS with Equilibrium Conditions," arXiv:1410.0470.

Thomas S. Richardson

Acyclic directed mixed graphs, also known as semi-Markov models represent the conditional independence structure induced on an observed margin by a DAG model with latent variables. In this paper we present a factorization criterion for these models that is equivalent to the global Markov property given by (the natural extension of) d-separation.

Ilya Shpitser, Robin J. Evans, Thomas S. Richardson et al.

Hidden variables are ubiquitous in practical data analysis, and therefore modeling marginal densities and doing inference with the resulting models is an important problem in statistics, machine learning, and causal inference. Recently, a new type of graphical model, called the nested Markov model, was developed which captures equality constraints found in marginals of directed acyclic graph (DAG) models. Some of these constraints, such as the so called `Verma constraint', strictly generalize conditional independence. To make modeling and inference with nested Markov models practical, it is necessary to limit the number of parameters in the model, while still correctly capturing the constraints in the marginal of a DAG model. Placing such limits is similar in spirit to sparsity methods for undirected graphical models, and regression models. In this paper, we give a log-linear parameterization which allows sparse modeling with nested Markov models. We illustrate the advantages of this parameterization with a simulation study.

Peter L. Spirtes, Christopher Meek, Thomas S. Richardson

We show that there is a general, informative and reliable procedure for discovering causal relations when, for all the investigator knows, both latent variables and selection bias may be at work. Given information about conditional independence and dependence relations between measured variables, even when latent variables and selection bias may be present, there are sufficient conditions for reliably concluding that there is a causal path from one variable to another, and sufficient conditions for reliably concluding when no such causal path exists.

Thomas S. Richardson

Although the concept of d-separation was originally defined for directed acyclic graphs (see Pearl 1988), there is a natural extension of he concept to directed cyclic graphs. When exactly the same set of d-separation relations hold in two directed graphs, no matter whether respectively cyclic or acyclic, we say that they are Markov equivalent. In other words, when two directed cyclic graphs are Markov equivalent, the set of distributions that satisfy a natural extension of the Global Directed Markov condition (Lauritzen et al. 1990) is exactly the same for each graph. There is an obvious exponential (in the number of vertices) time algorithm for deciding Markov equivalence of two directed cyclic graphs; simply chech all of the d-separation relations in each graph. In this paper I state a theorem that gives necessary and sufficient conditions for the Markov equivalence of two directed cyclic graphs, where each of the conditions can be checked in polynomial time. Hence, the theorem can be easily adapted into a polynomial time algorithm for deciding the Markov equivalence of two directed cyclic graphs. Although space prohibits inclusion of correctness proofs, they are fully described in Richardson (1994b).

Thomas S. Richardson

Directed acyclic graphs have been used fruitfully to represent causal strucures (Pearl 1988). However, in the social sciences and elsewhere models are often used which correspond both causally and statistically to directed graphs with directed cycles (Spirtes 1995). Pearl (1993) discussed predicting the effects of intervention in models of this kind, so-called linear non-recursive structural equation models. This raises the question of whether it is possible to make inferences about causal structure with cycles, form sample data. In particular do there exist general, informative, feasible and reliable precedures for inferring causal structure from conditional independence relations among variables in a sample generated by an unknown causal structure? In this paper I present a discovery algorithm that is correct in the large sample limit, given commonly (but often implicitly) made plausible assumptions, and which provides information about the existence or non-existence of causal pathways from one variable to another. The algorithm is polynomial on sparse graphs.

Jacob A. Wegelin, Thomas S. Richardson

DAG models with hidden variables present many difficulties that are not present when all nodes are observed. In particular, fully observed DAG models are identified and correspond to well-defined sets ofdistributions, whereas this is not true if nodes are unobserved. Inthis paper we characterize exactly the set of distributions given by a class of one-dimensional Gaussian latent variable models. These models relate two blocks of observed variables, modeling only the cross-covariance matrix. We describe the relation of this model to the singular value decomposition of the cross-covariance matrix. We show that, although the model is underidentified, useful information may be extracted. We further consider an alternative parametrization in which one latent variable is associated with each block. Our analysis leads to some novel covariance equivalence results for Gaussian hidden variable models.

Mathias Drton, Thomas S. Richardson

Graphical models with bi-directed edges (<->) represent marginal independence: the absence of an edge between two vertices indicates that the corresponding variables are marginally independent. In this paper, we consider maximum likelihood estimation in the case of continuous variables with a Gaussian joint distribution, sometimes termed a covariance graph model. We present a new fitting algorithm which exploits standard regression techniques and establish its convergence properties. Moreover, we contrast our procedure to existing estimation methods.

Sanjay Chaudhari, Thomas S. Richardson

Pearls concept OF a d - connecting path IS one OF the foundations OF the modern theory OF graphical models : the absence OF a d - connecting path IN a DAG indicates that conditional independence will hold IN ANY distribution factorising according TO that graph. IN this paper we show that IN singly - connected Gaussian DAGs it IS possible TO USE the form OF a d - connection TO obtain qualitative information about the strength OF conditional dependence.More precisely, the squared partial correlations BETWEEN two given variables, conditioned ON different subsets may be partially ordered BY examining the relationship BETWEEN the d - connecting path AND the SET OF variables conditioned upon.

Rina Dechter, Thomas S. Richardson

This is the Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence, which was held in Cambridge, MA, July 13 - 16 2006.

Ilya Shpitser, Thomas S. Richardson, James M. Robins et al.

The constraints arising from DAG models with latent variables can be naturally represented by means of acyclic directed mixed graphs (ADMGs). Such graphs contain directed and bidirected arrows, and contain no directed cycles. DAGs with latent variables imply independence constraints in the distribution resulting from a 'fixing' operation, in which a joint distribution is divided by a conditional. This operation generalizes marginalizing and conditioning. Some of these constraints correspond to identifiable 'dormant' independence constraints, with the well known 'Verma constraint' as one example. Recently, models defined by a set of the constraints arising after fixing from a DAG with latents, were characterized via a recursive factorization and a nested Markov property. In addition, a parameterization was given in the discrete case. In this paper we use this parameterization to describe a parameter fitting algorithm, and a search and score structure learning algorithm for these nested Markov models. We apply our algorithms to a variety of datasets.

Mathias Drton, Thomas S. Richardson

Ancestral graph models, introduced by Richardson and Spirtes (2002), generalize both Markov random fields and Bayesian networks to a class of graphs with a global Markov property that is closed under conditioning and marginalization. By design, ancestral graphs encode precisely the conditional independence structures that can arise from Bayesian networks with selection and unobserved (hidden/latent) variables. Thus, ancestral graph models provide a potentially very useful framework for exploratory model selection when unobserved variables might be involved in the data-generating process but no particular hidden structure can be specified. In this paper, we present the Iterative Conditional Fitting (ICF) algorithm for maximum likelihood estimation in Gaussian ancestral graph models. The name reflects that in each step of the procedure a conditional distribution is estimated, subject to constraints, while a marginal distribution is held fixed. This approach is in duality to the well-known Iterative Proportional Fitting algorithm, in which marginal distributions are fitted while conditional distributions are held fixed.

Ayesha R. Ali, Thomas S. Richardson, Peter L. Spirtes et al.

It is well known that there may be many causal explanations that are consistent with a given set of data. Recent work has been done to represent the common aspects of these explanations into one representation. In this paper, we address what is less well known: how do the relationships common to every causal explanation among the observed variables of some DAG process change in the presence of latent variables? Ancestral graphs provide a class of graphs that can encode conditional independence relations that arise in DAG models with latent and selection variables. In this paper we present a set of orientation rules that construct the Markov equivalence class representative for ancestral graphs, given a member of the equivalence class. These rules are sound and complete. We also show that when the equivalence class includes a DAG, the equivalence class representative is the essential graph for the said DAG

Robin J. Evans, Thomas S. Richardson

Acyclic directed mixed graphs, also known as semi-Markov models represent the conditional independence structure induced on an observed margin by a DAG model with latent variables. In this paper we present the first method for fitting these models to binary data using maximum likelihood estimation.

Ilya Shpitser, Thomas S. Richardson, James M. Robins

Probabilistic inference in graphical models is the task of computing marginal and conditional densities of interest from a factorized representation of a joint probability distribution. Inference algorithms such as variable elimination and belief propagation take advantage of constraints embedded in this factorization to compute such densities efficiently. In this paper, we propose an algorithm which computes interventional distributions in latent variable causal models represented by acyclic directed mixed graphs(ADMGs). To compute these distributions efficiently, we take advantage of a recursive factorization which generalizes the usual Markov factorization for DAGs and the more recent factorization for ADMGs. Our algorithm can be viewed as a generalization of variable elimination to the mixed graph case. We show our algorithm is exponential in the mixed graph generalization of treewidth.