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This chapter introduces Bayesian networks and probabilistic independence, and shows how Bayesian nets are used to represent probability functions. Inference in Bayesian nets is discussed and the problem of constructing Bayesian nets is introduced. One of the simplest methods for constructing Bayesian nets — sequentially adding arrows — is explored in some detail in order to highlight some of the key features of the construction problem.

Methods for performing complex probabilistic reasoning tasks, often based on masses of different forms of evidence obtained from a variety of different sources, are being sought by, and developed for, persons in many important contexts including law, medical diagnosis, and intelligence analysis. The complexity of these tasks can often be captured and represented by graphical structures now called inference networks. These networks are directed acyclic graphs (DAGs), consisting of nodes, representing relevant hypotheses, items of evidence, and unobserved variables, and arcs (arrows) joining some of the nodes, representing dependency relations among them. This chapter describes and comments on two different approaches to inference network construction. In the first approach, a DAG network structure is explicitly constructed as a vehicle for probabilistic analyses. Since the associated computations can be regarded as generalising the use of Bayes' rule, such networks are commonly called Bayesian networks. The second approach stems from the work of the American jurist John H. Wigmore who was the very first person to attempt a systematic study of inference networks.

A convenient way of modelling complex interactions is by employing graphs or networks which correspond to conditional independence structures in an underlying statistical model. One main class of models in this regard are Bayesian networks, which have the drawback of making parametric assumptions. Bayesian nonparametric mixture models offer a possibility to overcome this limitation, but have hardly been used in combination with networks. This manuscript bridges this gap by introducing nonparametric Bayesian network models. We review (parametric) Bayesian networks, in particular Gaussian Bayesian networks, from a Bayesian perspective as well as nonparametric Bayesian mixture models. Afterwards these two modelling approaches are combined into nonparametric Bayesian networks. The new models are compared both to Gaussian Bayesian networks and to mixture models in a simulation study, where it turns out that the nonparametric network models perform favourably in non‐Gaussian situations. The new models are also applied to an example from systems biology, namely finding modules within the MAPK cascade.

This chapter seeks to explain hierarchical models and how they differ from simple Bayesian models and to illustrate building hierarchical models using mathematically correct expressions. It begins with the definition of hierarchical models. Next, the chapter introduces four general classes of hierarchical models that have broad application in ecology. These classes can be used individually or in combination to attack virtually any research problem. Examples are used to show how to draw Bayesian networks that portray stochastic relationships between observed and unobserved quantities. The chapter furthermore shows how to use network drawings as a guide for writing posterior and joint distributions.

The principle of Kolmogorov minimal sufficient statistic (KMSS) states that the meaningful information of data is given by the regularities in the data. The KMSS is the minimal model that describes the regularities. The meaningful information given by a Bayesian network is the directed acyclic graph (DAG) which describes a decomposition of the joint probability distribution into conditional probability distributions (CPDs). If the description given by the Bayesian network is incompressible, the DAG is the KMSS and is faithful. The chapter proves that if a faithful Bayesian network exists, it is the minimal Bayesian network. Moreover, if a Bayesian network gives the KMSS, modularity of the CPDs is the most plausible hypothesis, from which the causal interpretation follows. On the other hand, if the minimal Bayesian network is compressible and is thus not the KMSS, the above implications cannot be guaranteed. When the non‐minimality of the description is due to the compressibility of an individual CPD, the true causal model is an element of the set of minimal Bayesian networks and modularity is still plausible. Faithfulness cannot be guaranteed though. When the concatenation of the descriptions of the CPDs is compressible, the true causal model is not necessarily an element of the set of minimal Bayesian networks. Also modularity may become implausible. This suggests that either there is a kind of meta‐mechanism governing some of the mechanisms or a wrong model class is considered.

It is well‐known that Bayesian networks are so‐called because of their use of Bayes theorem for probabilistic inference. However, since Bayesian networks commonly use frequentist probabilities exclusively, is this sense they are not Bayesian. In this chapter it is argued that Bayesian networks that are objectively Bayesian, in other words those whose prior distribution is based on all and only the available information, have certain desirable properties and strengths over and above those based solely on the frequentist approach to probability. It is demonstrated, through an example, that these specially constructed graphical models may be used in otherwise intractable situations where data is unavailable or scarce and decisions need to be made.

This chapter provides solutions to the problems presented in the preceding chapter. It presents the diagrams for each problem as well as some explanations on how the solutions are arrived at. As has been advised in the previous chapter, each solution is best referred to individually, as each problem in the preceding chapter has been solved. The problems this chapter addresses involve the mean number of ticks per sheep assuming the counts are distributed as a Poisson random variable, the portrayal of light limitation for a single tree species, the landscape occupancy of Swiss breeding birds, the allometry of savanna trees, and the movement of seals in the North Atlantic.

This chapter compares and contrasts the account with three other major formal accounts of belief revision: AGM-theory; Justified Truth-Maintenance Systems; and Bayesian networks. There are both critical and constructive things to say about these competing accounts. This discussion should serve to situate our work for the reader in mathematical logic, AI and/or computing science.

Introduces different interpretations of witness reliability into the models and constructs Bayesian-Network representations. Applies the models to Condorcet-style jury voting and Tversky and Kahneman’s Linda puzzle.

The aim of this chapter is to offer an advanced tutorial to scientists with no background or no deep background on probabilistic graphical models. To readers more familiar with these models, this chapter is to be used as a compendium of definitions and general methods, to browse through at will. Intentionally self-contained, this chapter first begins with reminders of essential definitions such as the distinction between marginal independence and conditional independence. Then the chapter briefly surveys the most popular classes of probabilistic graphical models: Markov chains, Bayesian networks, and Markov random fields. Next probabilistic inference is explained and illustrated in the Bayesian network context. Finally parameter and structure learning are presented.

In a segregating population, quantitative trait loci (QTL) mapping can identify QTLs with a causal effect on a phenotype. A common feature of these methods is that QTL mapping and phenotype network reconstruction are conducted separately. As both tasks have to benefit from each other, this chapter presents an approach which jointly infers a causal phenotype network and causal QTLs. The joint network of causal phenotype relationships and causal QTLs is modeled as a Bayesian network. In addition, a prior distribution on phenotype network structures is adjusted by biological knowledge, thus extending the former framework, QTLnet, into QTLnet-prior. This integrative approach can incorporate several sources of biological knowledge such as protein-protein interactions, gene ontology annotations, and transcription factor and DNA binding information. A Metropolis-Hastings scheme is described that iterates between accepting a network structure and accepting k weights corresponding to the k types of biological knowledge.

The causal power of C over E is (roughly) the degree to which changes in C cause changes in E. A formal measure of causal power would be very useful, as an aid to understanding and modelling complex stochastic systems. Previous attempts to measure causal power, such as those of Good (1961), Cheng (1997), and Glymour (2001), while useful, suffer from one fundamental flaw: they only give sensible results when applied to very restricted types of causal system, all of which exhibit causal transitivity. Causal Bayesian networks, however, are not in general transitive. The chapter develops an information‐theoretic alternative, causal information, which applies to any kind of causal Bayesian network. Causal information is based upon three ideas. First, the chapter assumes that the system can be represented causally as a Bayesian network. Second, the chapter uses hypothetical interventions to select the causal from the non‐causal paths connecting C to E. Third, we use a variation on the information‐theoretic measure mutual information to summarize the total causal influence of C on E. The chapter's measure gives sensible results for a much wider variety of complex stochastic systems than previous attempts and promises to simplify the interpretation and application of Bayesian networks.

This chapter provides a set of structured problems to hone the reader's skills in model building. Each problem requires the reader to draw a Bayesian network and write the posterior and joint distributions. Computational syntax is de-emphasized because there are many sources that teach the details of writing code for MCMC software using ecological examples and a few that show how to construct one's own MCMC algorithms. The examples offered in this chapter are drawn from several ecological sub-disciplines, though the intent of these problems is to be instructive regardless of the reader's focus of research. The chapter also offers some pointers for approaching these problems and how to apply their lessons to one's own research.

Most Bayesian network applications to gene network reconstruction assume a single distributional model across all the samples and treatments analyzed. This assumption is likely to be unrealistic especially when describing the relationship between genes across a range of treatments with potentially different impacts on the networks. To address this limitation, a mixture Bayesian network approach has been developed. Besides, the equivalence between Bayesian networks and regression approaches has been demonstrated. Here, two strategies are compared: the mixture Bayesian network approach and the mixture regression approach, when used for the purpose of gene network inference. The finite mixture model that is integrated into both strategies allows the characterization of gene relationships unique to particular conditions as well as the identification of interactions shared across conditions. The chapter reviews performances on real data describing a pathway analyzed under up to nine different experimental conditions, and highlights the strengths of the approaches evaluated.

The relative scarcity of the results reported by genetic association studies (GAS) prompted many research directions. Despite the centrality of the concept of association in GASs, refined concepts of association are missing; meanwhile, various feature subset selection methods became de facto standards for defining multivariate relevance. On the other hand, probabilistic graphical models, including Bayesian networks (BNs) are more and more popular, as they can learn nontransitive, multivariate, nonlinear relations between complex phenotypic descriptors and heterogeneous explanatory variables. To integrate the advantages of Bayesian statistics and BNs, the Bayesian network based Bayesian multilevel analysis of relevance (BN-BMLA) was proposed. This approach allows the processing of multiple target variables, while ensuring scalability and providing a multilevel view of the results of multivariate analysis. This chapter discusses the use of Bayesian BN-based analysis of relevance in exploratory data analysis, optimal decision and study design, and knowledge fusion, in the context of GASs.

Many different, seemingly mutually exclusive, theories of categorization have been proposed in recent years. The most notable theories have been those based on prototypes, exemplars, and causal models. This chapter provides “representation theorems” for each of these theories in the framework of probabilistic graphical models. More specifically, it shows for each of these psychological theories that the categorization judgments predicted and explained by the theory can be wholly captured using probabilistic graphical models. In other words, probabilistic graphical models provide a lingua franca for these disparate categorization theories, and so we can quite directly compare the different types of theories. These formal results are used to explain a variety of surprising empirical results, and to propose several novel theories of categorization.

This chapter sets the stage for what follows, introducing the reader to the philosophical principles and the mathematical formalism behind Bayesian inference and its scientific applications. We explain and motivate the representation of graded epistemic attitudes (“degrees of belief”) by means of specific mathematical structures: probabilities. Then we show how these attitudes are supposed to change upon learning new evidence (“Bayesian Conditionalization”), and how all this relates to theory evaluation, action and decision-making. After sketching the different varieties of Bayesian inference, we present Causal Bayesian Networks as an intuitive graphical tool for making Bayesian inference and we give an overview over the contents of the book.