Causal models defined in terms of structural equations have proved to be quite a powerful way of representing knowledge regarding causality. However, a number of authors have given examples that seem to show that the Halpern-Pearl (HP) definition of causality (Halpern & Pearl 2005) gives intuitively unreasonable answers. Here it is shown that, for each of these examples, we can give two stories consistent with the description in the example, such that intuitions regarding causality are quite different for each story. By adding additional variables, we can disambiguate the stories. Moreover, in the resulting causal models, the HP definition of causality gives the intuitively correct answer. It is also shown that, by adding extra variables, a modification to the original HP definition made to deal with an example of Hopkins and Pearl (2003) may not be necessary. Given how much can be done by adding extra variables, there might be a concern that the notion of causality is somewhat unstable. Can adding extra variables in a "conservative" way (i.e., maintaining all the relations between the variables in the original model) cause the answer to the question "Is X = x a cause of Y =  y ?" to alternate between "yes" and "no"? Here it is shown that adding an extra variable can change the answer from "yes' to "no", but after that, it cannot cannot change back to "yes".

The causal assumptions, the study design and the data are the elements required for scientific inference in empirical research. The research is adequately communicated only if all of these elements and their relations are described precisely. Causal models with design describe the study design and the missing data mechanism together with the causal structure and allow the direct application of causal calculus in the estimation of the causal effects. The flow of the study is visualized by ordering the nodes of the causal diagram in two dimensions by their causal order and the time of the observation. Conclusions whether a causal or observational relationship can be estimated from the collected incomplete data can be made directly from the graph. Causal models with design offer a systematic and unifying view scientific inference and increase the clarity and speed of communication. Examples on the causal models for a case-control study, a nested case-control study, a clinical trial and a two-stage case-cohort study are presented.

Sample size determination for a data set is an important statistical process for analyzing the data to an optimum level of accuracy and using minimum computational work. The applications of this process are credible in every domain which deals with large data sets and high computational work. This study uses Bayesian analysis for determination of minimum sample size of vibration signals to be considered for fault diagnosis of a bearing using pre-defined parameters such as the inverse standard probability and the acceptable margin of error. Thus an analytical formula for sample size determination is introduced. The fault diagnosis of the bearing is done using a machine learning approach using an entropy-based J48 algorithm. The following method will help researchers involved in fault diagnosis to determine minimum sample size of data for analysis for a good statistical stability and precision.

One of Judea Pearl's many, many important contributions to the study of causality was the first attempt to use the mathematical tools of causal modeling to give an account of "actual causation", a notion that has been of considerable interest among philosophers and legal theorists (Pearl, 2000, Chapter 10). Pearl later revised his account of actual causation in joint work with Halpern (Halpern & Pearl, 2005). A number of authors (Hall, 2007; Halpern, 2008; Hitchcock, 2007; Menzies, 2004) have suggested that an account of actual causation must be sensitive to considerations of normality, as well as to causal structure. In (Halpern & Hitchcock, 2011), we suggest a way of incorporating considerations of normality into the Halpern-Pearl theory, and show how to extend the account to illuminate features of the psychology of causal judgment, as well as features of causal reasoning in the law. Our account of actual causation makes use of "extended causal models", which include both structural equations among a set of variables, and a partial preorder on possible worlds, which represents the relative "normality" of those worlds. We actually want to think of people as working with the structural equations and normality order to evaluate actual causation. However, consideration of even simple examples immediately suggests a problem. A direct representation of the equations and normality order is too cumbersome for cognitively limited agents to use effectively. If our account of actual causation is to be at all realistic as a model of human causal judgment, some form of compact representation will be needed.

Statistical estimates can often be improved by fusion of data from several different sources. One example is so-called ensemble methods which have been successfully applied in areas such as machine learning for classification and clustering. In this paper, we present an ensemble method to improve community detection by aggregating the information found in an ensemble of community structures. This ensemble can found by re-sampling methods, multiple runs of a stochastic community detection method, or by several different community detection algorithms applied to the same network. The proposed method is evaluated using random networks with community structures and compared with two commonly used community detection methods. The proposed method when applied on a stochastic community detection algorithm performs well with low computational complexity, thus offering both a new approach to community detection and an additional community detection method.

Complex systems are naturally hybrid: their dynamic behavior is both continuous and discrete. For these systems, maintenance and repair are an increasing part of the total cost of final product. Efficient diagnosis and prognosis techniques have to be adopted to detect, isolate and anticipate faults. This paper presents an original integrated theoretical framework for diagnosis and prognosis of hybrid systems. The formalism used for hybrid diagnosis is enriched in order to be able to follow the evolution of an aging law for each fault of the system. The paper presents a methodology for interleaving diagnosis and prognosis in a hybrid framework.

Counterfactual reasoning arises in broad array of fields, from statistics to economics to law. Not surprisingly, there has been a great deal of work on giving semantics to counterfactuals. Perhaps the best-known approach is due to Lewis [1973] and Stalnaker [1968], and involves possible worlds. The idea is that a counterfactual of the form "ifAwere the case thenB would be the case", typically written A B, is true at a worldwifB is true at all the worlds closest tow whereAis true. Of course, making this precise requires having some notion of "closeness" among worlds. More recently, Pearl [2000] proposed the use of causal models based on structural equations for reasoning about causality. In causal models, we can examine the effect of interventions, and answer questions of the form "if random variable X were set to x, what would the value of random variable Y be". This suggests that causal models can also provide semantics for (at least some) counterfactuals. The relationship between the semantics of counterfactuals in causal models and in counterfactual structures (i.e., possible-worlds structures where the semantics of counterfactuals is given in terms of A preliminary version of this paper appears in the Proceedings of the Twelfth International Conference on Principles of Knowledge Representation and Reasoning (KR 2010), 2010.

We analyze the foundations of cyclic causal models for discrete variables, and compare structural equation models (SEMs) to an alternative semantics as the equilibrium (stationary) distribution of a Markov chain. We show under general conditions, discrete cyclic SEMs cannot have independent noise; even in the simplest case, cyclic structural equation models imply constraints on the noise. We give a formalization of an alternative Markov chain equilibrium semantics which requires not only the causal graph, but also a sample order. We show how the resulting equilibrium is a function of the sample ordering, both theoretically and empirically.

Diagnosis, i.e., the identification of root causes for failing or unexpected system behavior, is an important task in practice. Within the last three decades, many different AI-based solutions for solving the diagnosis problem have been presented and have been gaining in attraction. This leaves us with the question of which algorithm to prefer in a certain situation. In this paper we contribute to answering this question. In particular, we compare two classes of diagnosis algorithms. One class exploits conflicts in their search, i.e., sets of system components whose correct behavior contradicts given observations. The other class ignores conflicts and derives diagnoses from observations and the underlying model directly. In our study we use different reasoning engines ranging from an optimized Horn-clause theorem prover to general SAT and constraint solvers. Thus we also address the question whether publicly available general reasoning engines can be used for an efficient diagnosis.

Counting the models of a propositional formula is a key issue for a number of AI problems, but few propositional languages offer the possibility to count models efficiently. In order to fill the gap, we introduce the language EADT of (extended) affine decision trees. An extended affine decision tree simply is a tree with affine decision nodes and some specific decomposable conjunction or disjunction nodes. Unlike standard decision trees, the decision nodes of an EADT formula are not labeled by variables but by affine clauses. We study EADT, and several subsets of it along the lines of the knowledge compilation map. We also describe a CNF-to-EADT compiler and present some experimental results. Those results show that the EADT compilation-based approach is competitive with (and in some cases is able to outperform) the model counter Cachet and the d-DNNF compilation-based approach to model counting.