Part of Allen et al.’s criticism of Hamilton’s rule makes sense only if we are interested in social adaptation rather than merely social selection. Under the assumption that we are interested in casually modeling social adaptation, I illustrate how graphical causal models of social adaptation can be useful for predicting evolution by adaptation. I then argue for two consequences of this approach given some of the recent philosophical literature. I argue Birch’s claim that the proper way to understand (...) Hamilton’s rule is as providing an organizational framework for causal models is incorrect. I provide an account of a causally adequate decomposition of evolutionary change due to social adaptation and show that my account is superior to Okasha’s. (shrink)

These are chapters from a book forthcoming from MIT Press. Comments to the author at [email protected] would be most welcome. Still time for changes.

Effective recommendations on healthy food choice need to be personalized and sent out on a large scale. In this paper, we present a model of automatic message selection tailored on the characteristics of the recipient and focused on the reduction of red meat consumption. This model is obtained through the collaboration between social psychologists and artificial intelligence experts. Starting from selected psychosocial models on food choices and the framing effects of recommendation messages, we involved a sample of Italian (...) participants in an experiment in which they: filled out a first questionnaire, which was aimed at detecting the psychosocial antecedents of the intention to eat red/processed meat; read messages differing as to the framing of the hypothetical consequences of reducing versus not reducing red/processed meat consumption; filled out a second questionnaire, which was aimed at detecting participants’ reaction to the messages, as well as any changes in their intention to consume red/processed meat. Data collected were then employed to learn both the structure and the parameters of a Graphical Causal Model based on a Dynamic Bayesian Network, aimed to predicting the potential effects of message delivery from the observation of the psychosocial antecedents. Such probabilistic predictor is intended as the basis for developing automated interactions strategies using Deep Reinforcement Learning techniques. Discussion focuses on how to develop automatic interaction strategies able to foster mindful eating, thanks to considering the psychosocial characteristics of the people involved; sending messages tailored on these characteristics; adapting interaction strategies according to people’s reactions. (shrink)

Peter Spirtes, Clark Glymour, Richard Scheines, Christopher Meek, S. Fineberg, E. Slate. Prediction and Experimental Design with Graphical Causal Models.

Causality offers the first comprehensive coverage of causal analysis in many sciences, including recent advances using graphical methods. Pearl presents a unified account of the probabilistic, manipulative, counterfactual and structural approaches to causation, and devises simple mathematical tools for analyzing the relationships between causal connections, statistical associations, actions and observations. The book will open the way for including causal analysis in the standard curriculum of statistics, artificial intelligence, business, epidemiology, social science and economics.

Graphical models are a powerful tool for causal model specification. Besides allowing for a hierarchical representation of variable interactions, they do not require any a priori specification of the functional dependence between variables. The construction of such graphs hence often relies on the mere testing of whether or not model variables are marginally or conditionally independent. The identification of causal relationships then solely requires some general assumptions on the relation between stochastic and causal independence, (...) such as the Causal Markov Condition and the Faithfulness Condition (Spirtes et al. 2000; Pearl 2000). However, a procedure would require further assumptions to hold. Namely those the independence tests themselves are based on. In continuous settings, Spirtes et al. (2000) suggest causal inference based on a very restrictive formulation of independence, that is, vanishing partial correlations. Such a measure does, however, limit the applicability of causal inference to linear systems. This constitutes a serious drawback especially for the social sciences where an a priori specification of the functional form proves difficult or at odds with linearity. In short: graphical models theoretically reduce specification uncertainty regarding functional dependence, but their implementation in practice deprives them of this virtue. In this paper we investigate how causal structures in continuous settings can be identified when both functional forms and probability distributions of the variables remain unspecified. We focus on tests exploiting the fact that if X and Y are conditionally independent given a set of variables Z, the two conditional densities f (X|Y, Z) and f (X|Z) must coincide. We start by estimating the conditional densities f(X|Y, Z) and f(X|Z) via nonparametric techniques (kernel methods). We proceed by testing if some metric expressing the distance between these very conditional densities is sufficiently close to zero. Out of several metrics available in the literature to express such distance we choose two, the Euclidean, and the Hellinger distance. We investigate in a Monte Carlo study how different tests involving either measure are able to detect statistical independence, conditioned on a small set of variables. One limitation may result from nonparametric density estimation being subject to the curse of dimensionality. As the number of variables increases, the estimated empirical density converges at a slower rate to its population value. To compensate this drawback we use a local bootstrap procedure which consists of resampling the data for each test. While local bootstrap strongly increases the computational time of the test, it succeeds in counterbalancing the curse of dimensionality. (shrink)

The verbs cause, enable, and prevent express beliefs about the way the world works. We offer a theory of their meaning in terms of the structure of those beliefs expressed using qualitative properties of causal models, a graphical framework for representing causal structure. We propose that these verbs refer to a causal model relevant to a discourse and that “A causes B” expresses the belief that the causal model includes a link from A (...) to B. “A enables/allows B” entails that the model includes a link from A to B, that A represents a category of events necessary for B, and that an alternative cause of B exists. “A prevents B” entails that the model includes a link from A to B and that A reduces the likelihood of B. This theory is able to account for the results of four experiments as well as a variety of existing data on human reasoning. (shrink)

A graphical model is a graph that represents a set of conditional independence relations among the vertices (random variables). The graph is often given a causal interpretation as well. I describe how graphical causal models can be used in an algorithm for constructing partial information about causal graphs from observational data that is reliable in the large sample limit, even when some of the variables in the causal graph are unmeasured. I also describe (...) an algorithm for estimating from observational data (in some cases) the total effect of a given variable on a second variable, and theoretical insights into fundamental limitations on the possibility of certain causal inferences by any algorithm whatsoever, and regardless of sample size. (shrink)

Causal reasoning is primarily concerned with what would happen to a system under external interventions. In particular, we are often interested in predicting the probability distribution of some random variables that would result if some other variables were forced to take certain values. One prominent approach to tackling this problem is based on causal Bayesian networks, using directed acyclic graphs as causal diagrams to relate post-intervention probabilities to pre-intervention probabilities that are estimable from observational data. However, such (...) causal diagrams are seldom fully testable given observational data. In consequence, many causal discovery algorithms based on data-mining can only output an equivalence class of causal diagrams. This paper is concerned with causal reasoning given an equivalence class of causal diagrams, represented by a ancestral graph. We present two main results. The first result extends Pearl 's celebrated do-calculus to the context of ancestral graphs. In the second result, we focus on a key component of Pearl's calculus---the property of invariance under interventions, and give stronger graphical conditions for this property than those implied by the first result. The second result also improves the earlier, similar results due to Spirtes et al. (shrink)

Peter Spirtes. Discovering Causal Relations Among Latent Variables in Directed Acyclical Graphical Models.

Erratum to: Synthese 191:1925–1930 DOI:10.1007/s11229-013-0380-3 The authors were unaware that points in their article appeared in “Caveats for Causal Reasoning with Equilibrium Models,” by Denver Dash and Marek Druzdzel, published in S. Benferhat and P. Besnard : European Conferences on Symbolic and Quantitative Approaches to Reasoning with Uncertainty 2001, Lecture Notes in Artificial Intelligence 2143, pp. 192–203. The authors were unaware of this essay and would like to apologize to the authors for failing to cite their excellent work.

Causal models provide a framework for precisely representing complex causal structures, where specific models can be used to efficiently predict, infer, and explain the world. At the same time, we often do not know the full causal structure a priori and so must learn it from data using a causal model search algorithm. This chapter provides a general overview of causal models and their uses, with a particular focus on causal graphical models (...) (the most commonly used causal modeling framework) and methods for learning such models from observational and experimental data. The chapter concludes with two examples of productive causal search and modeling in experimental philosophy. (shrink)

Questions concerning mediated causal effects are of great interest in psychology, cognitive science, medicine, social science, public health, and many other disciplines. For instance, about 60% of recent papers published in leading journals in social psychology contain at least one mediation test (Rucker, Preacher, Tormala, & Petty, 2011). Standard parametric approaches to mediation analysis employ regression models, and either the “difference method” (Judd & Kenny, 1981), more common in epidemiology, or the “product method” (Baron & Kenny, 1986), more common (...) in the social sciences. In this article, we first discuss a known, but perhaps often unappreciated, fact that these parametric approaches are a special case of a general counterfactual framework for reasoning about causality first described by Neyman (1923) and Rubin (1924) and linked to causal graphical models by Robins (1986) and Pearl (2006). We then show a number of advantages of this framework. First, it makes the strong assumptions underlying mediation analysis explicit. Second, it avoids a number of problems present in the product and difference methods, such as biased estimates of effects in certain cases. Finally, we show the generality of this framework by proving a novel result which allows mediation analysis to be applied to longitudinal settings with unobserved confounders. (shrink)

Humans often attribute the things that happen to one or another actual cause. In this chapter, we survey some recent philosophical and psychological research on causal attribution. We pay special attention to the relation between graphical causal modeling and theories of causal attribution. We think that the study of causal attribution is one place where formal and experimental techniques nicely complement one another.

We study the selection of adjustment sets for estimating the interventional mean under an individualized treatment rule. We assume a non-parametric causal graphical model with, possibly, hidden variables and at least one adjustment set composed of observable variables. Moreover, we assume that observable variables have positive costs associated with them. We define the cost of an observable adjustment set as the sum of the costs of the variables that comprise it. We show that in this setting there (...) exist adjustment sets that are minimum cost optimal, in the sense that they yield non-parametric estimators of the interventional mean with the smallest asymptotic variance among those that control for observable adjustment sets that have minimum cost. Our results are based on the construction of a special flow network associated with the original causal graph. We show that a minimum cost optimal adjustment set can be found by computing a maximum flow on the network, and then finding the set of vertices that are reachable from the source by augmenting paths. The optimaladj Python package implements the algorithms introduced in this article. (shrink)

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 (...) class='Hi'>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. (shrink)

In this article, we propose a new method to learn the underlying acyclic mixed graph of a linear non-Gaussian structural equation model with given observational data. We build on an algorithm proposed by Wang and Drton, and we show that one can augment the hidden variable structure of the recovered model by learning multidirected edges rather than only directed and bidirected ones. Multidirected edges appear when more than two of the observed variables have a hidden common cause. We (...) detect the presence of such hidden causes by looking at higher order cumulants and exploiting the multi-trek rule. Our method recovers the correct structure when the underlying graph is a bow-free acyclic mixed graph with potential multidirected edges. (shrink)

How do we make causal judgments? Many studies have demonstrated that people are capable causal reasoners, achieving success on tasks from reasoning to categorization to interventions. However, less is known about the mental processes used to achieve such sophisticated judgments. We propose a new process model—the mutation sampler—that models causal judgments as based on a sample of possible states of the causal system generated using the Metropolis–Hastings sampling algorithm. Across a diverse array of tasks and (...) conditions encompassing over 1,700 participants, we found that our model provided a consistently closer fit to participant judgments than standard causal graphical models. In particular, we found that the biases introduced by mutation sampling accounted for people's consistent, predictable errors that the normative model by definition could not. Moreover, using a novel experimental methodology, we found that those biases appeared in the samples that participants explicitly judged to be representative of a causal system. We conclude by advocating sampling methods as plausible process‐level accounts of the computations specified by the causal graphical model framework and highlight opportunities for future research to identify not just what reasoners compute when drawing causal inferences, but also how they compute it. (shrink)

Existing research suggests that people's judgments of actual causation can be influenced by the degree to which they regard certain events as normal. We develop an explanation for this phenomenon that draws on standard tools from the literature on graphical causal models and, in particular, on the idea of probabilistic sampling. Using these tools, we propose a new measure of actual causal strength. This measure accurately captures three effects of normality on causal judgment that have been (...) observed in existing studies. More importantly, the measure predicts a new effect ("abnormal deflation"). Two studies show that people's judgments do, in fact, show this new effect. Taken together, the patterns of people's causal judgments thereby provide support for the proposed explanation. (shrink)

Within the causal modeling literature, debates about the Causal Faithfulness Condition have concerned whether it is probable that the parameters in causal models will have values such that distinct causal paths will cancel. As the parameters in a model are fixed by the probability distribution over its variables, it is initially puzzling what it means to assign probabilities to these parameters. I propose that to assign a probability to a parameter in a model is (...) to treat that parameter as a function of a variable in an augmented model. By combining this proposal with widely adopted principles regarding which variables must be included in a model, I argue that the various proposed counterexamples to CFC involving coordinated parameters are not genuine counterexamples. I then consider the cases in which CFC fails due not to coordination, but by coincidence, and propose explanatory and predictive bases for ruling out such coincidences without presuming that they are improbable. The aim of the proposed defenses is not to show that CFC never fails, but rather to argue that its use in a particular context may be defended using general modeling assumptions rather than by relying on claims about how often it fails. (shrink)

We propose that children employ specialized cognitive systems that allow them to recover an accurate “causal map” of the world: an abstract, coherent, learned representation of the causal relations among events. This kind of knowledge can be perspicuously understood in terms of the formalism of directed graphical causal models, or “Bayes nets”. Children’s causal learning and inference may involve computations similar to those for learning causal Bayes nets and for predicting with them. Experimental results (...) suggest that 2- to 4-year-old children construct new causal maps and that their learning is consistent with the Bayes net formalism. (shrink)

This paper studies the causal interpretation of counterfactual sentences using a modifiable structural equation model. It is shown that two properties of counterfactuals, namely, composition and effectiveness, are sound and complete relative to this interpretation, when recursive (i.e., feedback-less) models are considered. Composition and effectiveness also hold in Lewis's closest-world semantics, which implies that for recursive models the causal interpretation imposes no restrictions beyond those embodied in Lewis's framework. A third property, called reversibility, holds in nonrecursive (...) class='Hi'>causal models but not in Lewis's closest-world semantics, which implies that Lewis's axioms do not capture some properties of systems with feedback. Causal inferences based on counterfactual analysis are exemplified and compared to those based on graphical models. (shrink)

We outline a cognitive and computational account of causal learning in children. We propose that children employ specialized cognitive systems that allow them to recover an accurate “causal map” of the world: an abstract, coherent representation of the causal relations among events. This kind of knowledge can be perspicuously represented by the formalism of directed graphical causal models, or “Bayes nets”. Human causal learning and inference may involve computations similar to those for learnig (...) class='Hi'>causal Bayes nets and for predicting with them. Preliminary experimental results suggest that 2- to 4-year-old children spontaneously construct new causal maps and that their learning is consistent with the Bayes-Net formalism. (shrink)

Does y obtain under the counterfactual supposition that x? The answer to this question is famously thought to depend on whether y obtains in the most similar world in which x obtains. What this notion of ‘similarity’ consists in is controversial, but in recent years, graphical causal models have proved incredibly useful in getting a handle on considerations of similarity between worlds. One limitation of the resulting conception of similarity is that it says nothing about what would obtain (...) were the causal structure to be different from what it actually is, or from what we believe it to be. In this paper, we explore the possibility of using graphical causal models to resolve counterfactual queries about causal structure by introducing a notion of similarity between causal graphs. Since there are multiple principled senses in which a graph G* can be more similar to a graph G than a graph G**, we introduce multiple similarity metrics, as well as multiple ways to prioritize the various metrics when settling counterfactual queries about causal structure. (shrink)

The past century has witnessed numerous methodological innovations in probabilistic and statistical methods of causal inference (e.g., the graphical modelling and the potential outcomes frameworks, as introduced in Chapter 1). These innovations have not only enhanced the methodologies by which scientists across diverse domains make causal inference, but they have also made a profound impact on the way philosophers think about causation. The philosophical issues discussed in this thesis are stimulated and inspired by these methodological innovations. Chapter (...) 2 addresses the question of how the holding of screening-off conditions for a causal model depends on the choice of variables. As bridge principles between probability and causation, screening-off conditions (especially the Causal Markov Condition) play a key role in causal inference. However, it has been known that these conditions may fail due to poor variable choice. My aim in this chapter is to further examine those constraints on variable choice that are deemed necessary for the satisfaction of screening-off conditions. The idea of a well-defined (hypothetical) intervention is also crucial for reliable causal inference. Chapter 3 explores the question of when interventions invoked in causal inference are “well-defined” or unambiguous, and how this requirement constrains the choice of cause-variables. I propose that an intervention is well-defined just in case the effect of interest is well-defined (under ideal interventions), and that the intervention can serve as a suitable means to identify that effect. Based on this proposal, several distinct types of ambiguous interventions are identified. Methodological progress in causal inference also poses the following question: Can such progress shed light on the ontology of causation? My answer is yes. In Chapter 4, I develop a Carnapian-pragmatist approach to the ontology of causation as an alternative to existing metaphysical approaches. I argue that, compared to traditional metaphysics, the pragmatist approach provides a superior picture of how the ontology and methodology of causation interact with each other in scientific practice. I conclude in Chapter 5 that the thing we call “causation” consists in both the right worldly infrastructure (e.g., screening-off patterns and possibilities for interventions) and appropriate ways of framing this infrastructure. (shrink)

THE ART OF CAUSAL CONJECTURE Glenn Shafer Table of Contents Chapter 1. Introduction........................................................................................ ...........1 1.1. Probability Trees..........................................................................................3 1.2. Many Observers, Many Stances, Many Natures..........................................8 1.3. Causal Relations as Relations in Nature’s Tree...........................................9 1.4. Evidence............................................................................................ ...........13 1.5. Measuring the Average Effect of a Cause....................................................17 1.6. Causal Diagrams..........................................................................................20 1.7. Humean Events............................................................................................23 1.8. Three Levels of Causal Language................................................................27 1.9. An Outline of the Book................................................................................27 Chapter 2. Event Trees............................................................................................... .....31 2.1. Situations and Events...................................................................................32 2.2. The Ordering of Situations and Moivrean Events.......................................35 2.3. Cuts................................................................................................ ..............39 (...) 2.4. Humean Events............................................................................................43 2.5. Moivrean Variables......................................................................................49 2.6. Humean Variables........................................................................................53 2.7. Event Trees for Stochastic Processes...........................................................54 2.8. Timing in Event Trees.................................................................................56 2.9. Intersecting Event Trees...............................................................................60 2.10. Notes on the Literature...............................................................................61 Chapter 3. Probability Trees...........................................................................................63 3.1. Some Types of Probability Trees.................................................................64 ii 3.2. Axioms for the Probabilities of Moivrean Events.......................................68 3.3. Zero Probabilities....................................................................................... ..70 3.4. A Sample-Space Analysis of the Event-Tree Axioms.................................72 3.5. Probabilities and Expected Values for Variables.........................................74 3.6. Martingales......................................................................................... .........79 3.7. The Expectation of a Variable in a Cut........................................................83 3.8. Conditional Expected Value and Conditional Expectation.........................87 Chapter 4. The Meaning of Probability...........................................................................91 4.1. The Interpretation of Expected Value..........................................................92 4.2. The Interpretation of Expectation................................................................95 4.3. The Long Run..............................................................................................98 4.4. Changes in Belief.........................................................................................101 4.5. The Empirical Validation of Probability......................................................106 4.6. The Diversity of Uses of Probability...........................................................108 4.7. Notes on the Literature.................................................................................110 Chapter 5. Independent Events.......................................................................................113 5.1. Independence........................................................................................ .......114 5.2. Weak Independence.....................................................................................118 5.3. The Principle of the Common Cause...........................................................121 5.4. Conditional Independence............................................................................128 5.5. Notes on the Literature.................................................................................133 Chapter 6. Events Tracking Events.................................................................................135 6.1. Tracking............................................................................................ ...........137 6.2. Tracking and Conditional Independence.....................................................142 6.3. Stochastic Subsequence...............................................................................143 6.4. Singular Diagrams for Stochastic Subsequence...........................................147 6.5. Conjunctive and Interactive Forks...............................................................149 Chapter 7. Events as Signs of Events..............................................................................153 iii 7.1. Sign................................................................................................ ..............154 7.2. Weak Sign................................................................................................ ....159 7.3. The Ethics of Causal Talk............................................................................160 7.4. Screening Off...............................................................................................16 2 Chapter 8. Independent Variables...................................................................................167 8.1. Unconditional Independence........................................................................170 8.2. Conditional Independence............................................................................175 8.3. Independence for Partitions.........................................................................177 8.4. Independence for Families of Variables......................................................182 8.5. Individual Properties of the Independence Relations...................................186 Chapter 9. Variables Tracking Variables........................................................................189 9.1. Tracking and Conditional Independence: A Summary...............................190 9.2. Strong Tracking............................................................................................ 192 9.3. Strong Tracking and Conditional Independence..........................................198 9.4. Stochastic Subsequence...............................................................................201 9.5. Functional Dependence................................................................................203 9.6. Tracking in Mean.........................................................................................204 9.7. Linear Tracking............................................................................................ 207 9.8. Tracking by Partitions..................................................................................210 9.9. Tracking by Families of Variables...............................................................212 Chapter 10. Variables as Signs of Variables...................................................................215 10.1. Sign................................................................................................ ............219 10.2. Linear Sign................................................................................................ .222 10.3. Scored Sign................................................................................................ 225 10.4. Families of Variables.................................................................................227 Chapter 11. AnTheory of Event Trees.............................................................229 11.1. Event Trees as Sets of Sets........................................................................230 11.2. Event Trees as Partially Ordered Sets........................................................232 iv 11.3. Regular Event Trees...................................................................................240 11.4. The Resolution of Moivrean Variables......................................................244 11.5. Humean Events and Variables...................................................................246 Chapter 12. Martingale Trees..........................................................................................247 12.1. Examples of Decision Trees......................................................................249 12.2. The Meaning of Probability in a Decision Tree.........................................253 12.3. Martingales......................................................................................... .......257 12.4. The Structure of Martingale Trees.............................................................261 12.5. Probability and Causality...........................................................................265 12.6. Lower and Upper Probability.....................................................................269 12.7. The Law of Large Numbers.......................................................................272 12.8. Notes on the Literature...............................................................................274 Chapter 13. Refining............................................................................................ ...........275 13.1. Examples of Refinement............................................................................277 13.2. A Constructive Definition of Finite Refinement.......................................281 13.3. Axioms for Refinement..............................................................................282 13.4. Lifting Moivrean Events and Variables.....................................................288 13.5. Refining Martingale Trees.........................................................................288 13.6. Grounding........................................................................................... .......294 Chapter 14. Principles of Causal Conjecture..................................................................299 14.1. The Diversity of Causal Explanation.........................................................302 14.2. The Mean Effect of the Happening of a Moivrean Event..........................305 14.3. The Effect of a Humean Variable..............................................................311 14.4. Attribution and Generality.........................................................................316 14.5. The Statistical Measurement of the Effect of a Cause...............................319 14.6. Measurement by Experiment.....................................................................320 14.7. Using Our Knowledge of How Things Work............................................322 v 14.8. Sampling Error...........................................................................................329 14.9. The Sampling Frame..................................................................................329 14.10. Notes on the Literature.............................................................................330 Chapter 15. Causal Models.............................................................................................3 31 15.1. The Causal Interpretation of Statistical Prediction....................................333 15.2. Generalizing to a Family of Exogenous Variables....................................337 15.3 Some Joint Causal Diagrams......................................................................339 15.4. Causal Path Diagrams................................................................................342 15.5. Causal Relevance Diagrams.......................................................................346 15.6. The Meaning of Latent Variables..............................................................352 15.7. Notes on the Literature...............................................................................357 Chapter 16. Representing Probability Trees...................................................................359 16.1. Three Graphical Representations...............................................................361 16.2. Skeletal Simplifications.............................................................................368 16.3. Martingale Trees in Type Theory...............................................................371 Appendix A. Huygens’s Probability Trees.....................................................................379 Huygens’s Manuscript in Translation..................................................................380 Appendix B. Some Elements of Graph Theory..............................................................385 B1. Undirected Graphs........................................................................................385 B2. Directed Graphs............................................................................................38 6 Appendix C. Some Elements of Order Theory...............................................................393 C1. Partial and Quasi Orderings.........................................................................393 C2. Singular and Joint Diagrams for Binary Relations.......................................394 C3. Lattices............................................................................................ .............395 C4. The Lattice of Partitions of a Set..................................................................396 Appendix D. The Sample-Space Framework for Probability.........................................399 D1. Probability Measures....................................................................................399 D2. Variables........................................................................................... ...........400 vi D3. Families of Variables...................................................................................401 D4. Expected Value............................................................................................402 D5. The Law of Large Numbers.........................................................................405 D6. Conditional Probability................................................................................406 D7. Conditional Expected Value........................................................................407 Appendix E. Prediction in Probability Spaces................................................................409 E1. Conditional Distribution...............................................................................411 E2. Regression on a Single Variable...................................................................412 E3. Regression on a Partition or a Family of Variables......................................415 E4. Linear Regression on a Single Variable.......................................................418 E5. Linear Regression on a Family of Variables................................................422 Appendix F. Sample-Space Concepts of Independence.................................................425 F1. Overview............................................................................................ ...........426 F2. Independence Proper.....................................................................................432 F3. Unpredictability in Mean..............................................................................434 F4. Simple Uncorrelatedness..............................................................................437 F5. Mixed Uncorrelatedness...............................................................................438 F6. Partial Uncorrelatedness...............................................................................440 F7. Independence for Partitions..........................................................................442 F8. Independence for Families of Variables.......................................................445 F9. The Basic Role of Uncorrelatedness.............................................................448 F10. Dawid’s Axioms.........................................................................................449 Appendix G. Prediction Diagrams..................................................................................453 G1. Path Diagrams............................................................................................ ..454 G2. Generalized Path Diagrams..........................................................................462 G3. Relevance Diagrams.....................................................................................466 G4. Bubbled Relevance Diagrams......................................................................475 Appendix H. Abstract Stochastic Processes...................................................................479 vii H1. Probability Conditionals and Probability Distributions...............................477 H2. Abstract Stochastic Processes......................................................................479 H3. Embedding Variables and Processes in a Sample Space.............................480 References.......................................................................................... ..............................491. (shrink)

In everyday life, as well as in science, we have to deal with and act on the basis of partial (i.e. incomplete, uncertain, or even inconsistent) information. This observation is the source of a broad research activity from which a number of competing approaches have arisen. There is some disagreement concerning the way in which partial or full ignorance is and should be handled. The most successful approaches include both quantitative aspects (by means of probability theory) and qualitative aspect (by (...) means of graphical or causal models or logic). Some of these approaches have important impacts on various disciplines including philosophy, computer science, statistics, mathematics, physics, and social science. Most notably, the relation between causal and probabilistic information is a topic of interest in all of these fields. (shrink)

Curie’s Principle says that any symmetry property of a cause must be found in its effect. In this article, I consider Curie’s Principle from the point of view of graphical causal models, and demonstrate that, under one definition of a symmetry transformation, the causal modeling framework does not require anything like Curie’s Principle to be true. On another definition of a symmetry transformation, the graphical causal modeling formalism does imply a version of Curie’s Principle. These (...) results yield a better understanding of the logical landscape with respect to the relationship between Curie’s Principle and graphical causal modeling. (shrink)

Objects have dispositions. Dispositions are normally analyzed by providing a meaning to disposition ascriptions like ‘This piece of salt is soluble’. Philosophers like Carnap, Goodman, Quine, Lewis and many others have proposed analyses of such disposition ascriptions. In this paper we will argue with Quine that the proper analysis of ascriptions of the form ‘x is disposed to m ’, where ‘x’ denotes an object, ‘m’ a manifestation, and ‘C’ a condition, goes like this: ‘x is of natural kind k’, (...) and the generic ‘ks are m ’ is true. For the analysis of the generic, we propose an analysis in terms of causal powers: ‘ks have the causal power to m’. The latter, in turn, is analyzed in a very precise way, making use of Pearl’s probabilistic graphical causal models. We will show how this natural kind-analysis improves on standard conditional analyses of dispositions by avoiding the standard counterexamples, and that it gives rise to precise observable criteria under which the disposition ascription is true. (shrink)

By combining experimental interventions with search procedures for graphical causal models we show that under familiar assumptions, with perfect data, N - 1 experiments suffice to determine the causal relations among N > 2 variables when each experiment randomizes at most one variable. We show the same bound holds for adaptive learners, but does not hold for N > 4 when each experiment can simultaneously randomize more than one variable. This bound provides a type of ideal for (...) the measure of success of heuristic approached in active learning methods of casual discovery, which currently use less informative measures. (shrink)

This article assesses how people reason with categories whose features are related in causal cycles. Whereas models based on causal graphical models have enjoyed success modeling category-based judgments as well as a number of other cognitive phenomena, CGMs are only able to represent causal structures that are acyclic. A number of new formalisms that allow cycles are introduced and evaluated. Dynamic Bayesian networks represent cycles by unfolding them over time. Chain graphs augment CGMs by allowing the (...) presence of undirected links that model feedback relations between variables. Unfolded chain graphs are chain graphs that unfold over time. An existing model of causal cycles is also evaluated. Four experiments in which subjects reason about categories with cyclically related features provided evidence against DBNs and alpha centrality and for the two types of chain graphs. Chain graphs—a mechanism for representing the equilibrium distribution of a dynamic system—may thus be good candidates for modeling how people reason causally with complex systems. Applications of chain graphs to areas of cognition other than category-based judgments are discussed. (shrink)

Perfect adaptation in a dynamical system is the phenomenon that one or more variables have an initial transient response to a persistent change in an external stimulus but revert to their original value as the system converges to equilibrium. With the help of the causal ordering algorithm, one can construct graphical representations of dynamical systems that represent the causal relations between the variables and the conditional independences in the equilibrium distribution. We apply these tools to formulate sufficient (...) graphical conditions for identifying perfect adaptation from a set of first-order differential equations. Furthermore, we give sufficient conditions to test for the presence of perfect adaptation in experimental equilibrium data. We apply this method to a simple model for a protein signalling pathway and test its predictions in both simulations and using real-world protein expression data. We demonstrate that perfect adaptation can lead to misleading orientation of edges in the output of causal discovery algorithms. (shrink)

We describe a unification of old and recent ideas for formulating graphical models to explain time series data, including Granger causality, semi-automated search procedures for graphical causal models, modeling of contemporaneous influences in times series, and heuristic generalized additive model corrections to linear models. We illustrate the procedures by finding a structure of exogenous variables and mediating variables among time series of remote geospatial indices of ocean surface temperatures and pressures. The analysis agrees with known exogenous (...) drivers of the indices, not assumed in the analysis. Automated search applied to the residuals after regressing each series on its lags and the lags of its Granger causes yields a graphical model of “comtemporaneous” causal relations identical with the qualitative graphical relations from the time series. A similar analysis produces reasonable results when applied to candidate climate indices obtained by a clustering method from sea surface temperature and sea level pressure data. (shrink)

Discovering high-level causal relations from low-level data is an important and challenging problem that comes up frequently in the natural and social sciences. In a series of papers, Chalupka etal. (2015, 2016a, 2016b, 2017) develop a procedure forcausal feature learning (CFL) in an effortto automate this task. We argue that CFL does not recommend coarsening in cases where pragmatic considerations rule in favor of it, and recommends coarsening in cases where pragmatic considerations rule against it. We propose a new (...) technique, pragmatic causal feature learning (PCFL), which extends the original CFL algorithm in useful and intuitive ways. We show that PCFL has the same attractive measure-theoretic properties as the original CFL algorithm. We compare the performance of both methods through theoretical analysis and experiments. (shrink)

We describe a unification of old and recent ideas for formulating graphical models to explain time series data, including Granger causality, semi-automated search procedures for graphical causal models, modeling of contemporaneous influences in times series, and heuristic generalized additive model corrections to linear models. We illustrate the procedures by finding a structure of exogenous variables and mediating variables among time series of remote geospatial indices of ocean surface temperatures and pressures. The analysis agrees with known exogenous (...) drivers of the indices, not assumed in the analysis. Automated search applied to the residuals after regressing each series on its lags and the lags of its Granger causes yields a graphical model of “comtemporaneous” causal relations identical with the qualitative graphical relations from the time series. A similar analysis produces reasonable results when applied to candidate climate indices obtained by a clustering method from sea surface temperature and sea level pressure data. (shrink)

Adjusting for covariates is a well-established method to estimate the total causal effect of an exposure variable on an outcome of interest. Depending on the causal structure of the mechanism under study, there may be different adjustment sets, equally valid from a theoretical perspective, leading to identical causal effects. However, in practice, with finite data, estimators built on different sets may display different precisions. To investigate the extent of this variability, we consider the simplest non-trivial non-linear (...) class='Hi'>model of a v-structure on three nodes for binary data. We explicitly compute and compare the variance of the two possible different causal estimators. Further, by going beyond leading-order asymptotics, we show that there are parameter regimes where the set with the asymptotically optimal variance does depend on the edge coefficients, a result that is not captured by the recent leading-order developments for general causal models. As a practical consequence, the adjustment set selection needs to account for the relative magnitude of the relationships between variables with respect to the sample size and cannot rely on purely graphical criteria. (shrink)

An important component in the interventionist account of causal explanation is an interpretation of counterfactual conditionals as statements about consequences of hypothetical interventions. The interpretation receives a formal treatment in the framework of functional causal models. In Judea Pearl’s influential formulation, functional causal models are assumed to satisfy a “unique-solution” property; this class of Pearlian causal models includes the ones called recursive. Joseph Halpern showed that every recursive causal model is Lewisian, in the sense (...) that from the causal model one can construct a possible worlds model in David Lewis’s well-known semantics that satisfies the exact same formulas in a certain language. Moreover, he demonstrated that some Pearlian (non-recursive) models are not Lewisian in this sense. This raises the question regarding the exact contour of Lewisian causal models. In this paper, we provide a characterization of the class of Lewisian causal models and a complete axiomatization with respect to this class. Our results have philosophically interesting consequences, two of which are especially worth noting. First, the class of Stalnakerian causal models, a subclass of Lewisian causal models, is precisely the class of Pearlian models that do not contain any cycle of counterfactual dependence (in a sense of counterfactual dependence akin to Lewis’s famous relation between distinct events). Second, a more natural class of causal models is actually a superclass of Lewisian causal models, the logic of which respects only weak centering rather than centering. (shrink)

Cause and effect relationships play a central role in how we perceive and make sense of the world around us, how we act upon it, and ultimately, how we understand ourselves. Almost two decades ago, computer scientist Judea Pearl made a breakthrough in understanding causality by discovering and systematically studying the “Ladder of Causation” [Pearl and Mackenzie 2018], a framework that highlights the distinct roles of seeing, doing, and imagining. In honor of this landmark discovery, we name this the Pearl (...) Causal Hierarchy (PCH). In this chapter, we develop a novel and comprehensive treatment of the PCH through two complementary lenses, one logical-probabilistic and another inferential-graphical. Following Pearl’s own presentation of the hierarchy, we begin by showing how the PCH organically emerges from a well-specified collection of causal mechanisms (a structural causal model, or SCM). We then turn to the logical lens. Our first result, the Causal Hierarchy Theorem (CHT), demonstrates that the three layers of the hierarchy almost always separate in a measure theoretic sense. Roughly speaking, the CHT says that data at one layer virtually always underdetermines information at higher layers. Since in most practical settings the scientist does not have access to the precise form of the underlying causal mechanisms – only to data generated by them with respect to some of PCH’s layers – this motivates us to study inferences within the PCH through the graphical lens. Specifically, we explore a set of methods known as causal inference that enable inferences bridging PCH’s layers given a partial specification of the SCM. For instance, one may want to infer what would happen had an intervention been performed in the environment (second-layer statement) when only passive observations (first-layer data) are available. We introduce a family of graphical models that allows the scientist to represent such a partial specification of the SCM in a cognitively meaningful and parsimonious way. Finally, we investigate an inferential system known as docalculus, showing how it can be sufficient, and in many cases necessary, to allow inferences across PCH’s layers. We believe that connecting with the essential dimensions of human experience as delineated by the PCH is a critical step towards creating the next generation of AI systems that will be safe, robust, human-compatible, and aligned with the social good. (shrink)

Mechanist philosophers have examined several strategies scientists use for discovering causal mechanisms in neuroscience. Findings about the anatomical organization of the brain play a central role in several such strategies. Little attention has been paid, however, to the use of network analysis and causal modeling techniques for mechanism discovery. In particular, mechanist philosophers have not explored whether and how these strategies incorporate information about the anatomical organization of the brain. This paper clarifies these issues in the light of (...) the distinction between structural, functional and effective connectivity. Specifically, we examine two quantitative strategies currently used for causal discovery from functional neuroimaging data: dynamic causal modeling and probabilistic graphical modeling. We show that dynamic causal modeling uses findings about the brain’s anatomical organization to improve the statistical estimation of parameters in an already specified causal model of the target brain mechanism. Probabilistic graphical modeling, in contrast, makes no appeal to the brain’s anatomical organization, but lays bare the conditions under which correlational data suffice to license reliable inferences about the causal organization of a target brain mechanism. The question of whether findings about the anatomical organization of the brain can and should constrain the inference of causal networks remains open, but we show how the tools supplied by graphical modeling methods help to address it. (shrink)

In everyday life, as well as in science, we have to deal with and act on the basis of partial (i.e. incomplete, uncertain, or even inconsistent) information. This observation is the source of a broad research activity from which a number of competing approaches have arisen. There is some disagreement concerning the way in which partial or full ignorance is and should be handled. The most successful approaches include both quantitative aspects (by means of probability theory) and qualitative aspect (by (...) means of graphical or causal models or logic). Some of these approaches have important impacts on various disciplines including philosophy, computer science, statistics, mathematics, physics, and social science. Most notably, the relation between causal and probabilistic information is a topic of interest in all of these fields. The general question of how uncertainty is related to ignorance, and the connection thereof to causality, have been the key topics of the international summer school at the University of Konstanz, Germany, in August 2004. This was the third event in a series of summer schools organized by the interdisciplinary research group “Philosophy, Probability, and Modeling”. The group was in operation from September 1st, 2001, until August 31, 2005. The first summer school, held in September 2002, focused on the application of probability theory to topics of philosophical interest. Problems and questions that arise when probabilistic models and techniques are being applied in the special sciences were then discussed during the second summer school, held in July 2003. The fourth and final event in the series, in August 2005, was devoted to philosophical questions related to the application and interpretation of probabilities in physics. Each of the four summer schools attracted 40–50 people from many different countries. It was an inspiring forum for an interdisciplinary discussion among researchers and students from many different areas. This special issue collects a selection of the invited and contributed papers presented at the summer school in 2004. We acknowledge the generous support of the Center for Junior Research Fellows at the University of Konstanz, the Gesellschaft für analytische Philosophie e.V., the Centre for Philosophy of Natural and Social Science at the London School of Economics, the Alexander von Humboldt Foundation, the Federal Ministry of Education and Research and the Program for the Investment in the Future (ZIP) of the German Government through a Sofja Kovalevskaja Award. We thank the referees of the papers for their work, and James Moor for his interest in dedicating a special issue of this journal to Causality, Uncertainty and Ignorance. (shrink)

Recent interest in the external validity of prediction models has produced many methods for finding predictive distributions that are invariant to dataset shifts and can be used for prediction in new, unseen environments. However, these methods consider different types of shifts and have been developed under disparate frameworks, making it difficult to theoretically analyze how solutions differ with respect to stability and accuracy. Taking a causal graphical view, we use a flexible graphical representation to express various types (...) of dataset shifts. Given a known graph of the data generating process, we show that all invariant distributions correspond to a causal hierarchy of graphical operators, which disable the edges in the graph that are responsible for the shifts. The hierarchy provides a common theoretical underpinning for understanding when and how stability to shifts can be achieved, and in what ways stable distributions can differ. We use it to establish conditions for minimax optimal performance across environments, and derive new algorithms that find optimal stable distributions. By using this new perspective, we empirically demonstrate that that there is a tradeoff between minimax and average performance. (shrink)

Since the late nineties there has been an increased interested in probabilistic logic learning, an area within AI that combines machine learning with logic-based knowledge representation and uncertainty reasoning. Several different formalisms for combining first-order logic with probability reasoning have been proposed, and it has been studied how models in these formalisms can be automatically learned from data. -/- This talk starts with a brief introduction to probabilistic logic learning, after which we will focus on a relatively new formalism known (...) as CP-logic. CP-logic stands for “causal probabilistic logic”. It is a knowledge representation formalism that allows us to write down rules that indicate that a certain combination of conditions may cause certain effects with a particular probability (e.g., tossing a coin may cause a result of heads or tails, each with 50% probability). Besides the fact that this formalism is interesting for knowledge representation in itself, it also offers interesting opportunities from the machine learning point of view. Indeed, given the semantics of these CP-logic programs, learning them from data amounts to extracting probabilistic causal influences from data. We will discuss recent research on learning CP-logic programs, including: algorithms for learning them; how they relate to graphical models; and applications of learning CP-logic programs. (shrink)

This chapter explains the screening‐off rule in the psychological laboratory. The Markov assumption states that any variable in a set is independent in probability of all its ancestors in the set conditional on its own parents. The screening‐off rule is also critical to allow Bayes nets to make an inference of the state of an unknown variable in a causal structure from the states of other variables in that structure. The chapter examines which causal representations people use to (...) make predictions and whether people conform to the screening‐off rule with respect to a causal model they have in mind. It analyzes what leads them to modify their causal model and whether or not people are causally oriented when making predictions. Making probability judgments is hard and requires careful deliberation. People are capable of such deliberation although they avoid it until they cannot, until the facts require more careful thought. (shrink)

Materialism, epiphenomenalism, dualism, idealism, and dual-aspect theories may all be represented by an appealing abstract mathematical device called a commutative diagram. Properties of the components of such diagrams characterize and, to some extent, even parameterize these systems and attendant metaphysical concepts (such as causal closure and supervenience) in a unified framework; process thought is of particular interest in this connection. In many cases we can even exemplify the theories typified by these diagrams in explicit graphical models. All of (...) this tends to clarify the relationships among key philosophical positions and to sharpen our sense of the effective domain and principal limitations of each. Systematic variation of these abstract diagrams may even suggest cogent metaphysical systems yet to be examined. (shrink)