This paper describes a Bayesian method for combining an arbitrary mixture of observational and experimental data in order to learn causal Bayesian networks. Observational data are passively observed. Experimental data, such as that produced by randomized controlled trials, result from the experimenter manipulating one or more variables (typically randomly) and observing the states of other variables. The paper presents a Bayesian method for learning the causal structure and parameters of the underlying causal process that is generating the data, given that (1) the data contains a mixture of observational and experimental case records, and (2) the causal process is modeled as a causal Bayesian network. This learning method was applied using as input various mixtures of experimental and observational data that were generated from the ALARM causal Bayesian network. In these experiments, the absolute and relative quantities of experimental and observational data were varied systematically. For each of these training datasets, the learning method was applied to predict the causal structure and to estimate the causal parameters that exist among randomly selected pairs of nodes in ALARM that are not confounded. The paper reports how these structure predictions and parameter estimates compare with the true causal structures and parameters as given by the ALARM network. 1

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,

We propose a decision theoretic approach for deciding which interventions to perform so as to learn the causal structure of a model as quickly as possible. Without such interventions, it is impossible to distinguish between Markov equivalent models, even given infinite data. We perform online MCMC to estimate the posterior over graph structures, and use importance sampling to find the best action to perform at each step. We assume the data is discrete-valued and fully observed.

We propose a new method of discovering causal structures, based on the detection of local, spontaneous changes in the underlying data-generating model. We analyze the classes of structures that are equivalent relative to a stream of distributions produced by local changes, and devise algorithms that output graphical representations of these equivalence classes. We present experimental results, using simulated data, and examine the errors associated with detection of changes and recovery of structures. 1

We propose a new method of discovering causal structures, based on the detection of local, spontaneous changes in the underlying data-generating model. We derive expressions for the Bayesian score that a causal structure should obtain from streams of data produced by locally changing distributions. Simulation experiments indicate that dynamic information may improve the power of discovery up to the theoretical limits set by statistical indistinguishability. 1

Abstract. This paper analyzes the problem of learning the structure of a Bayes net (BN) in the theoretical framework of Gold’s learning paradigm. Bayes nets are one of the most prominent formalisms for knowledge representation and probabilistic and causal reasoning. We follow constraint-based approaches to learning Bayes net structure, where learning is based on observed conditional dependencies between variables of interest (e.g., “X is dependent on Y given any assignment to variable Z”). Applying learning criteria in this model leads to the following results. (1) The mind change complexity of identifying a Bayes net graph over variables V from dependency data is � � |V|, the maximum number of 2 edges. (2) There is a unique fastest mind-change optimal Bayes net learner; convergence speed is evaluated using Gold’s dominance notion of “uniformly faster convergence”. This learner conjectures a graph if it is the unique Bayes net pattern that satisfies the observed dependencies with a minimum number of edges, and outputs “no guess ” otherwise. Therefore we are using standard learning criteria to define a natural and novel Bayes net learning algorithm. We investigate the complexity of computing the output of the fastest mind-change optimal learner, and show that this problem is NP-hard (assuming P=RP). To our knowledge this is the first NP-hardness result concerning the existence of a uniquely optimal Bayes net structure. 1

This short report, compiled upon request from Dave Siegrist and Ted Senator, surveys the spectrum of technologies that can help with Biosurveillance. We indicate which we have chosen, so far, to use in our development of analysis methods and our reasons. 1 Time-weighted averaging This is directly applicable to a scalar signal (such as “number of respiratory cases today”. This method, more commonly used in computational finance, simply compares the count during the current time period with the weighted average of the counts of recent days. Exponential weighting is typically used, where the half-life is known as the “time window ” parameter. This time-window parameter is typically chosen by hand. We prefer the Serfling and Univariate HMM methods described below. 2 Serfling method This method (Serfling, 1963) is a cyclic regression model, and is the standard CDC algorithm for flu detection. It is, again, applicable to scalar signals. It assumes that the signal follows a sinusoid with a period of one year, and thus finds the four parameters ¢¤£¦¥¨ § and © in where the parameters are chosen to minimize the sum of squares of residuals. It is an easy matter of regression analysis to determine, on any date, whether

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Abstract. The term “changes in structure, ” originating from work in econometrics, refers to structural modifications invoked by actions on a causal model. In this paper we formalize the representation of reversibility of a mechanism in order to support modeling of changes in structure in systems that contain reversible mechanisms. Causal models built on our formalization can answer two new types of queries: (1) When manipulating a causal model (i.e., making an endogenous variable exogenous), which mechanisms are possibly invalidated and can be removed from the model? (2) Which variables may be manipulated in order to invalidate and, effectively, remove a mechanism from a model? 1

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