Data containing deterministic relations cannot be handled by current constraint-based causal learning algorithms; they entail conditional independencies that cannot be represented by a faithful graph. Violation of the faithfulness property is characterized by an information equivalence of two sets of variables with respect to a reference variable. The conditional independencies do not provide information about which set should be connected to the reference variable. We propose to use the complexity of the relationships as criterion to determine adjacency. Correct decisions are made under the assumption that the complexity of relations does not increase along a causal path. This paper defines an augmented Bayesian network which explicitly models deterministic relations. The faithfulness property is redefined by using a generalized definition of the d-separation criterion, which also gives the conditional independencies following from deterministic relations, and by limiting the conditional independencies that are graphically described with the simplicity condition. Based on this, an extension to the PC learning algorithm is developed that allows the construction of minimal augmented Bayesian networks from observational data. Correct models are learned from data generated by a set of structural equations.

Bayesian networks (BNs) are a compact representation ofa joint probability distribution. In this paper we show how they can beused for modeling other agents in the environment. More precisely we willhave special attention to the problem of large state spaces and incompleteinformation. For our experiments we will consider the robotic soccersimulation for several reasons explained. Robotic soccer clients will learnthrough Q-learning, a form of reinforcement learning. The long-term...

Data containing deterministic relations entail conditional independencies that cannot be represented by a faithful graph, due to violations of the intersection condition. Such data can not be handled by current constraint-based learning algorithms. More generally, these violations are characterized by information equivalence of two sets of variables with respect to a target variable. We argue that deterministically related variables contain valuable information and should not be eliminated from the data. This paper proposes augmented Bayesian networks that explicitly model such information equivalences. For attaining minimality, only the set which has the simplest relation with the target variable is connected to it. Under the assumption that complexity does not increase along a causal path, this selection criterion results in consistent models. Under weak transitivity, faithfulness of the graph is reestablished by using the generalized definition of the d-separation criterion, called Deq-separation, and by limiting the conditional independencies that are graphically described with the simplicity condition. Based on this, an extension to the PC learning algorithm is developed that allows the construction of minimal augmented Bayesian networks from observational data. Correct models are learned from data generated by a set of structural equations.

In this article, we demonstrate the usefulness of causal Bayesian networks as probabilistic reasoning systems. The biggest advantage of causal Bayesian networks over traditional probabilistic Bayesian networks is that they sometimes allow to perform causal inference, i.e. the calculation of the causal effect of one variable on other variables. We treat a state-of-the-art algorithm for performing causal inference that is based on a new factorization of the joint probability distribution and is a systematic approach for the calculation due to Tian and Pearl.

Semi-Markovian causal models (SMCMs) are an extension of causal Bayesian networks for modeling problems with latent variables. However, there is a big gap between the SMCMs used in theoretical studies and the models that can be learned from observational data alone. The result of standard algorithms for learning from observations, is a complete partially ancestral graph (CPAG), representing the Markov equivalence class of maximal ancestral graphs (MAGs). In MAGs not all edges can be interpreted as immediate causal relationships. In order to apply state-of-the-art causal inference techniques we need to completely orient the learned CPAG and to transform the result into a SMCM by removing non-causal edges. In this paper we combine recent work on MAG structure learning from observational data with causal learning from experiments in order to achieve that goal. More specifically, we provide a set of rules that indicate which experiments are needed in order to transform a CPAG to a completely oriented SMCM and how the results of these experiments have to be processed. We will propose an alternative representation for SMCMs that can easily be parametrised and where the parameters can be learned with classical methods. Finally, we show how this parametrisation can be used to develop methods to efficiently perform both probabilistic and causal inference.

In this paper we discuss multi-agent causal models, which are an extension of causal Bayesian networks to the multi-agent case. In this paper we illustrate how these recently introduced models could prove useful for dependability analysis. Their main difference with other graphical modeling techniques that have been applied to the problem is that multi-agent causal models allow for multi-agent, privacy-preserving quantitative causal inference in models with hidden variables.

From causal theory it is known that the indepen- dencies entailed by deterministic relations in a stochastic distribution cannot be represented by a faithful causal model. Deterministic relations lead to situations in which either of two vari- ables X and Y become conditionally indepen- dent from a third variable Z by conditioning on the other variable. More generally, this occurs when X and Y contain the same information about Z, they are called information-equivalent. The joint distribution defines an equivalent par- titioning of the domains of X and Y by which only the states are related for which the condi- tional distribution of target Z is the same, hence P(Z | X) = P(Z | Y ). We propose to select the relation with the target variable containing the least complexity. Under the assumption that complexity does not increase along a Markov chain, this selection criterion results in consistent models. Faithfulness of the graph can be reestab- lished by limiting the conditional independencies by the simplicity criterion in cases of equivalent information. On the other hand, all conditional independencies among the variables can be re- trieved from the graph by a generalized definition of the d-separation property. Finally, the PC al- gorithm was extended to learn models containing information-equivalent variables from data.

This article discusses graphical models that can handle latent variables without explicitly modeling them quantitatively. There exist several paradigms for such problem domains. Two of them are semi-Markovian causal models and maximal ancestral graphs. Applying these techniques to a problem domain consists of several steps, typically: structure learning from observational and experimental data, parameter learning, probabilistic inference, and, quantitative causal inference. A problem is that research in each of the existing approaches only focuses on one or a few of all the steps involved in the process of modeling a problem including latent variables. In other work we have investigated the integral process from observational and experimental data unto different types of efficient inference. The goal of this article is to focus on learning the structure of causal models in the presence of latent variables from a combination of observational and experimental data. Semi-Markovian causal models (SMCMs) are an approach developed by Pearl and Tian [3, 6]. They are specifically suited for performing quantitative causal inference in the presence of latent variables. However, at this time no efficient parametrisation of such models is provided and there are no techniques for performing efficient probabilistic inference. Furthermore there are no techniques to learn these models from data issued from observations, experiments or both. Maximal ancestral graphs (MAGs) are an approach developed by Richardson et al. [4]. They are specifically suited for structure learning in the presence of latent variables from observational data. However, the techniques only learn up to Markov equivalence and provide no clues on which additional experiments to perform in order to obtain the fully oriented causal graph. See [1, 2] for that type of results for Bayesian networks without latent variables. Furthermore, as of yet no parametrisation for discrete variables is provided for MAGs and no techniques for probabilistic inference have been developed. There is some work on algorithms for causal inference, but it is restricted to causal inference

In this paper we introduce the paradigm of multi-agent causal models (MACM), which are an extension of causal graphical models to a setting where there is no longer one single computational entity (agent) observing or not observing all the domain variables V. Instead there are several agents each having access to non-disjoint subsets of V. The incentive for introducing cooperative multiagent modeling is, as domains become larger, more complex, open and inherently distributed, building and maintaining a model by a single agent would prove to be very costly or even impossible. In our approach every agent has a semi-Markovian causal model over its local domain variables, determined by an acyclic causal diagram with directed and bi-directed edges and a joint probability distribution over its observed variables. After defining MACMs we introduce an algorithm for calculating the causal effect of variable X on another variable Y from purely observational data if possible in a bi-agent model. This is the effect of manipulating variable X on variable Y. In the algorithm the only communication between agents concerns variables they share in their models or variables X and Y and thus protects the privacy of the individual agent models. Our algorithm is an extension of a single agent algorithm due to Tian and Pearl. As it can happen that the set variables that two agents share are not sufficient to answer some multiagent causal queries, we have developed a negotiation algorithm for agents to cooperatively extend the set of variables they share, while maximally preserving the privacy of the individual agent models.

In this paper we introduce chain multi-agent causal models which are an extension of causal Bayesian networks to a multi-agent setting. Instead of 1 single agent modeling the entire domain, there are several agents organised in a chain, each modeling non-disjoint subsets of the domain. Every agent has a causal model over the variables in his domain, determined by an acyclic causal diagram and a joint probability distribution over its observed variables. See Figure 1 for an example.

In this paper we propose a distributed structure learning algorithm for the recently introduced Multi-Agent Causal Models (MACMs). MACMs are an extension of Causal Bayesian Networks (CBN) to a distributed domain. In this setting it is assumed that there is no single database containing all the information of the domain. Instead, there are several sites holding non-disjoint subsets of the domain variables. At each site there is an agent capable of learning a local causal model. We study the possibility of combining the information of the local models into one globally consistent model. We propose an algorithm that yields the possibility to learn new local structures that can be combined to perform globally consistent causal inference.

In this paper we introduce multi-agent causal models (MACMs) which are an extension of causal Bayesian networks to a multi-agent setting. Instead of 1 single agent modeling the entire domain, there are several agents each modeling non-disjoint subsets of the domain. Every agent has a causal model, determined by an acyclic causal diagram and a joint probability distribution over its observed variables. We study the identification of causal effects, which is the calculation of the effect of manipulating a variable on other variables from purely observational data. More specifically, we extend an existing single agent identification algorithm to multi-agent causal models. Given some assumptions, we provide a technique to calculate the effect of manipulating a variable in agent A on some variables in another agent B, while only communicating informati on concerning variables that are shared by agents A and B and variables that are being studied in that specific query.