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Probabilistic causality

Two-week summer school

Central European University

Budapest, Hungary

July 21-August 1, 2008

1. Syllabus



a) Statement of the purpose of the course


The aim of the summer school is to teach and discuss current results and recent trends in probabilistic causation. Probabilistic causality emerged during the second half of the 20th century as a truly interdisciplinary field, involving concepts and methods of philosophy (metaphysics and philosophy of science), classical and non-classical probability theory and the special sciences, especially physics and economics. A recent outgrowth of the theory is Bayes nets, investigated not only in philosophy of science but also in computer science and utilized in causal modeling. After a review of the basic notions and classical results on probabilistic causation the course focuses on discussing problems that are open and debated in the literature at this time. The course intends to bring together advanced graduate students and young researchers from different disciplines (philosophy, physics, economics and computer science) and provides an opportunity for cross-disciplinary discussion. Such discussions will be facilitated by organizing sessions of short presentations by participants; in the short talks participants can formulate their research topics and solicit comments by faculty and other participants during the summer school. Faculty of the summer school have taught different aspects of probabilistic causality in regular courses at LSE, ELTE, and CEU, and the summer school is based on a combination of these different courses.


b) Pre-requisites for the course



Participants are assumed to be familiar with the basic philosophical literature on classical analysis of causation and are expected to possess basic knowledge of classical probability theory.


c) Brief overview of the course



Probabilistic theories of causation have emerged during the second half of the 20th century with the development of mathematical probability theory and with the great success of probabilistic methods in both the exact and social sciences. One can discern two, complementary trends in this development: motivated by the sciences and relying on their concepts and techniques, abstract, philosophical theories of probabilistic causation were worked out, enriching analytic metaphysics by creating new approaches to the classical problem of causation. Examples of such theories include H. Reichenbach’s notion of common cause to explain probabilistic correlations and D. Lewis’ theory of counterfactual probabilistic causation. In turn, these philosophically elaborated and technically sharpened new concepts and tools have found their way back to the sciences: interesting new questions about causation in the particular sciences have been raised and new developments in special sciences were triggered. Examples of this latter include the problem of whether quantum correlations can have causal explanations in terms of Reichenbachian common causes, and the theory of causal (Bayes) nets, which are now popular also in computer science. This mutual fertilization of philosophy and sciences is very characteristic of probabilistic causality and makes probabilistic causation a truly interdisciplinary field. The course reviews the recent results in the theory of probabilistic causation, putting the emphasis on the open problems and on the currently debated issues.


The course is divided into six major sections:


  1. Introductory lectures. A general review of the current status of probabilistic causation is given, and, since notions of probability theory will be extensively used in the subsequent lectures and discussions, one lecture will review briefly the relevant concepts of both classical and non-classical (quantum) probability theory.

  2. The Common Cause Principle. In this block Reichenbach’s classical notion of common cause and the related Common Cause Principle is recalled and analyzed. The problem of causal (in)completeness and common cause completability of classical and quantum probability spaces is defined, and results on common cause (in)completeness and common cause completability are presented and analyzed. Generalizations of the notion of common cause to common cause systems will be given and, after proving existence theorems about Reichenbachian Common Cause systems, properties of these common cause systems will be investigated. Causal nets and the causal Markov condition used in Bayes nets are generalizations of the Common Cause Principle. The basic definitions of and the recent debates about the causal Markov condition will be reviewed and discussed.

  3. Probabilistic causality in economics.

There is a long tradition of concepts of causality in economics, particularly in the sub-discipline of econometrics, which aims to identify causal relations from non-experimental economic data. In this part of the course, the main approaches to causality in econometrics will be presented, critically compared and contrasted to each other. In particular, Granger causality (where causal order is based on time order) and structural approach (where causal order is based on invariance-to-interventions) will be presented. These two key approaches represent the two dominant strands of causality in econometrics, and both have affinities with philosophical treatments of probabilistic causality. Granger causality is close to Suppes’ probabilistic causality, while the structural approach, relates closely to work on causality by Cartwright, Woodward and others. This course will be concluded by analysis of how methods of causal inference work in econometrics for both kinds of causality presented above.



  1. Causal explanations in physics. EPR correlations predicted by quantum mechanics (and observed in Nature) are a special challenge for the Common Cause Principle because to explain these correlations in terms of common causes, the common causes need to satisfy additional locality conditions that express the no-action-at-a-distance principle of relativistic physics. The lectures in this section review the possible formulations of the locality conditions and the new No-go theorems that have been recently obtained, which indicate that EPR correlations cannot be explained by local Reichenbachian common causes. Local (relativistic) quantum field theory also predicts correlations between causally disjoint (spacelike) entities; hence the status of the Common Cause Principle arises in quantum field theory as well, this (largely) open problem will be analyzed.



I.Learning Causal Influences Using Bayesian Networks Bayesian networks are graphical structures for representing the probabilistic relationships among a large number of variables and doing probabilistic inference with those variables. The 1990's saw the emergence of excellent algorithms for learning Bayesian networks from passive data. By making certain assumptions about the probabilistic causal relationships among the observed variables, we can learn something about causal influences in Bayesian networks learning algorithms. This course will discuss both the constraint-based and Bayesian approaches to learning these causal influences. Furthermore, the course will discuss the identification of causal effects using a causal graph.





  1. Presentation by participants. In this section, planned to take place at the end of the course, participants will give short (20 min.) presentations of their research topic, which will be commented on by faculty and participants of the summer school.


d) Bibliography and Reader


  • Background reading

Literature used by faculty in designing the course is listed below


  • Reader

A Reader containing a selection of the most important papers the lectures and seminars rely on will be created and made available for participants at the e-Learning page link


Background reading:


G.E.M Anscombe (1993) “Causality and Determination” in E. Sosa – M. Tooley (eds.) Causation (Oxford University Press, 1993) 88-104.


N. Cartwright (2007): Hunting Causes and Using Them: Approaches in Philosophy and Economics (Cambridge University Press, 2007)


N. Cartwright (2006): “From metaphysics to method: Comments on manipulability and the causal Markov condition” British Journal for the Philosophy of Science 57 (2006) 197-218


N. Cartwright (2000): “Measuring Causes: Invariance, Modularity and the Causal Markov Condition”, Measurement in Physics and Economics, Discussion Paper Series Monograph DP MEAS 9/00, London: Centre for Philosophy of Natural and Social Science, 2000


N. Cartwright (2001): “What is wrong with Bayes nets?” The Monist (2001) 242-264


N. Cartwright (1999): The Dappled World, Cambridge: Cambridge University Press. (Esp. Ch. 5, 'Causal Diversity, Causal Stability')


N. Cartwright (1993): "Causality and realism in the EPR experiment" Erkenntnis 38 (1993) 269-190


N. Cartwright (1989): Nature’s Capacities and their Measurement (Oxford, Claredon Press, 1989)


N. Cartwright (1987): "How to tell a common cause: generalization of the conjunctive cause criterion," in: J. H. Fetzer (ed.) Probability and Causality (Reidel, 1987) 181-188


R. Engle, D. Hendry and J. Richard (1983): ‘Exogeneity’, Econometrica, 51(2), 277-304.


D. Fennell (2005): A Philosophical Analysis of Causality in Econometrics, PhD Dissertation, University of London.


J.H. Fetzer (ed.) (1988): Probability and Causality (Reidel Pub. Co., Boston, 1988)


C. Granger, (1980) ‘Testing for Causality: A Personal Viewpoint’, Journal of Economic Dynamics and Control, 2, 4, 329-352.


G. Graßhoff, S. Portmann and A. Wüthrich (2005): “Minimal Assumption Derivation of a Bell-type Inequality” British Journal for the Philosophy of Science 56 (2005) 663 – 680


B. Gyenis, M. Redei (2004): "When can statistical theories be causally closed?" Foundations of Physics 34 (2004) 1285-1303


J. Halpern, and J. Pearl (2005): “Causes and Explanations: A Structural-Model Approach,” British Journal of Philosophy of Science, Vol. 56


C. Hitchcock (2005): “Of Humean Bondage,” British Journal for the Philosophy of Science 54(1): 1-25


G. Hofer-Szabo (2007): Separate- versus common-common-cause-type derivations of the Bell inequalities" Synthese (forthcoming)


G. Hofer-Szabo, M. Redei (2006): "Reichenbachian Common Cause Systems of arbitrary finite size exist" Foundations of Physics Letters 35 (2006) 745-746


G. Hofer-Szabo, M. Redei, L. Szabo (2002): "Common-causes are not common common-causes" Philosophy of Science 69 (2002) 623-636


G. Hofer-Szabo, M. Redei, L. Szabo (2000): "Reichenbach's Common Cause Principle: Recent results and open problems" Reports on Philosophy, No. 20 (2000)


G. Hofer-Szabo, M. Redei, L. Szabo (1999): "On Reichenbach's common cause principle and Reichenbach's notion of common cause" The British Journal for the Philosophy of Science 50 (1999) 377-399


K. Hoover (2001) Causality in Macroeconomics, Cambridge University Press.


D. Lewis (1986): “Causation” and “Chancy Causation” in Philosophical Papers Vol. II (Oxford University Press, 1986) 159-172 and 175-184.


H. D. Mellor (1995): “On Raising the Chances of Effects”, The Facts of Causation (Routledge, 1995)


P. Menzies (1989): “Probabilistic Causation and Causal Processes’ in Philosophy of Science, LVI (1989) 642-63.


Neapolitan, R.E., Learning Bayesian Networks, Prentice Hall, Upper Saddle River, NJ, 2003.


R.E. Neapolitan, and X. Jiang, “A Tutorial on Learning Causal Influences,” in Holmes, D. and L. Jain (Eds.): Innovations in Machine Learning, Springer-Verlag, New York, 2006.


J. Pearl (2000): Causality: Models, Reasoning, and Inference, Cambridge University Press, Cambridge, UK (esp. pp. 65-85, 173-176)


P. Tetlock and A. Belkin (1996): Counterfactual Thought Experiments in World Politics, Princeton: Princeton University Press. (esp. Ch. 1, 'Counterfactual Thought Experiments in World Politics: Logical, Methodological, and Psychological Perspectives')


T. Placek (ed.) (2000), Reports on Philosophy, Special Issue on the Common Cause Principle, No. 20 (2000)


M. Redei (2002): "Reichenbach's Common Cause Principle and quantum correlations" in Modality, Probability and Bell's Theorems, NATO Science Series, II. Vol. 64. T. Placek and J. Butterfield (eds.), (Kluwer Academic Publishers, Dordrecht, Boston, London, 2002) 259-270


M. Redei, S.J. Summers (2007): "Quantum probability theory" Studies in the History and Philosophy of Modern Physics (forthcoming in June 2007)


H. Reichenbach (1956): The Direction of Time (University of California Press, Los Angeles, 1956)


W.C. Salmon (1984): Scientific Explanation and the Causal Structure of the World (Princeton University Press, Princeton, 1984)


H. Simon, (1953) ‘Causal Ordering and Identifiability’ reprinted in Herbert Simon, Models of Man, New York: John Wiley and Sons.


P.C. Spirtes, C. Glymour and R. Scheines (2000): Causation, Prediction, and Search, Springer-Verlag, New York, 1993; 2nd ed.: MIT Press, Cambridge, Massachusetts


E. Sober (2001): “Venetian sea levels, British bread prices, and the principle of common cause” The British Journal for the Philosophy of Science 52 331-346


L. E. Szabo (2007): “The Einstein--Podolsky--Rosen Argument and the Bell Inequalities” (To be published in the Internet Encyclopedia of Philosophy)


L.E. Szabo (2000): “On an attempt to resolve the EPR-Bell paradox via Reichenbachian concept of common cause” International Journal of Theoretical Physics 39 (2000) 911


J. Tian and J. Pearl (2002): “General Identification Condition for Causal Effects,” In Proceedings of the Eighteenth Conference on Artificial Intelligence, AAAI/The MIT Press: Menlo Park, August 


B. C. Van Fraassen (1982): "Rational Belief and Common Cause Principle," in: R. McLaughlin (ed.), What? Where? When? Why?, Reidel, 193-209.


B.C. Van Fraassen (1989). "The charybdis of realism: epistemological implications of Bell's inequality," in: J. T. Cushing and E. McMullin (eds.), Philosophical Consequences of Quantum Theory (University of Notre Dame Press, Ind., 1989) 97-113.


M. Weber (2001 [1905]): "Objective Possibility and Adequate Causation in Historical Explanation", in Michael Martin and Lee McIntyre (eds), Readings in the Philosophy of Social Science, Cambridge (MA): MIT Press


J. Williamson (2006): “Causal pluralism versus epistemic causality,” Philosophica 77(1): 69-96


2. Course schedule:

Day 1. Monday, July 21, 2008


Introductory lectures (3 lectures + 2 seminar discussions)




Day 1







Title/topic

Lecturer


Lecture 1

50 min


Seminar 1

50 min

Pluralism in the Philosophy of Causation




Literature: Williamson (2006), additional: Hitchcock (2005)

Reiss

Lecture 2

50 min

Seminar 2

50 min

Causation and Probability



Traditional accounts of causation were deterministic in the sense that undetermined events (provided that any such event exists) were taken to be uncaused. Almost all of the contemporary theories of causation admit, however, the possibility of probabilistic causation. The lecture and the seminar will address the following questions. 1. What are our conceptual reasons for introducing the notion of probabilistic causation? 2. How can some of the extant theories of causation accommodate the idea of probabilistic causation?



Literature: Mellor (1995), Menzies (1989)

Huoranszki

Lecture 3

50 min

Basic notions of classical and non-commutative probability theory


Classical probability theory in measure theoretic form: Boolean algebras, classical probability measures, random variables, conditional probability, independence, correlations. Non-classical probability theory as non-commutative measure theory: Hilbert lattices and von Neumann lattices, quantum states as probability measures, operators as random variables. Types of classical and non-classical probability spaces.


Literature: Redei and Summers, 2007

Redei



Day 2. Tuesday, July 22, 2008


The Common Cause Principle I. (2 lectures + 2 seminar discussions)


Day 2






Title/topic

Lecturer


Lecture 1

50 min


Seminar 1

50 min

Reichenbach’s notion of common cause and the Common Cause Principle


In the history of probabilistic causation Reichenbach's definition of the common cause was one of the first technically explicit formulations of a causal concept. The definition is a combination of statistical relevance conditions relating the cause to its two effects and of the so called screening-off property. Reichenbach's definition has become central in the philosophy of causation and in particular in the interpretation of the Common Cause Principle: no correlation without causation. In the Lecture Reichenbach's definition of the common cause will be recalled, some elementary examples for correlations and common causes will be presented, and various interpretations of the Reichenbachian Common Cause Principle will be distinguished and discussed.


Literature: Sober, 2001; Hofer-Szabo, Redei and Szabo, 2000; additional: Reichenbach, 1956; Van Fraassen, 1982; Cartwright, 1987

Hofer

Lecture 2

50 min


Seminar 2

50 min

Common cause completability and causal completeness of probability theories


A probabilistic theory is common cause completable with respect of a correlation if it can be extended in such a way that the extension contains a common cause of the correlation. Proof of common cause completability of classical and quantum probability spaces. The implications of common cause completability for the problem of falsifiability of the Common Cause Principle. A probability space is causally complete if it contains a common cause of every correlation between causally independent events. Examples of causally complete and incomplete probabilistic theories.


Literature: Hofer-Szabo, Redei and Szabo, 1999; Gyenis and Redei, 2004

Redei


Day 3. Wednesday, July 23, 2008


The Common Cause Principle II. (3 lectures + 1 seminar discussion)


Day 3






Title/topic

Lecturer


Lecture 1

50 min


Seminar 1

50 min

Reichenbachian common cause systems



The Reichenbachian common cause system is a natural generalization of Reichenbach's original definition of the common cause to the case when more than one single factor contribute to the correlation: The Reichenbachian common cause system of a correlation is a partition such that every element of the partition screens off the correlation and any two elements in the partition behave like a Reichenbachian common cause and its complement. It is shown that given any finite size and any correlation in a classical probability measure space, the space can be extended in such way that there exists a Reichenbachian common cause system of the given size in the extension. It will also be shown that every chain in the partially ordered set of all partitions of an algebra contains only one Reichenbachian common cause system for a given correlation.


Literature: Hofer-Szabo and Redei, 2006;

Hofer

Lecture 2

50 min


Lecture 3

50 min


Some Comments on Recent Work on the Causal Markov Condition



Counterfactuals, Thought Experiments and Singular Causal Inference in History


Literature: Sober (2001), additional: Cartwright (1999)

Literature: Weber (2001), additional: Tetlock-Belkin (1996)

Reiss




Day 4. Thursday, July 24, 2008


Probabilistic causation in economics (2 lectures + 2 seminars)




Day 4







Title/topic

Lecturer


Lecture 1

50 min


Lecture 2

50 min

Introduction to Causality in Econometrics




To understand causality in economics requires a basic understanding of econometrics. These two lectures will present some simple econometric models, to show how the statistical inference in econometrics takes place. Given this, the lecture will introduce the two key approaches to causality in econometrics, Granger causality and the structural approach. These will be related briefly to some counterparts in the philosophical literature. Important aspects will be emphasized, such as the strong counterfactual assumptions of the econometric approach. Finally, some of the different ways econometric methods can be used to find out about causes will be set out.



Required reading: Chap 7, Hoover 2001

Fennell

Lecture 3

50 min

Seminar 1

50 min

Causal Inference and Issues in Econometrics



How do concepts of causality relate to the methods of inference used in econometrics? In this lecture, three key problems will be presented: the problem of exogeneity, the use of instrumental variables and the problem of identification. The lecture will emphasize how bringing a philosophical perspective to econometric methods helps to show its strengths and limits, while conversely, econometric methods show how probabilistic causality can be put into practice in difficult circumstances.


The seminar will be used to go over the concepts and issues discussed in the lectures, and critically discuss the Hoover 2001 reading.


Required reading: Chap 7, Hoover 2001.

Fennell



Day 5. Friday, July 25, 2008


Presentations by participants



Presentations are scheduled to take place in one-hour sessions, one session accommodating 2 presentations, on the following model of the schedule of the first session of the day:


Day 5.




Time

Title of presentation

Speaker

9.00-9.20

9.20-9.30 discussion







9.30-9.50

9.50-10.00 discussion








3 such sessions (altogether 6 presentations) are planned for Day 9. of the summer school.


Saturday, Sunday, July 26 and 27 – Free time


Day 6. Monday, July 28, 2008


Causal explanation of correlations in physics II. (3 lectures + 2 seminar discussions)





Day 6.






Title/topic

Lecturer


Lecture 1

50 min


Lecture 2

50 min


Seminar 1

50 min

EPR correlations and the Common Cause Principle



EPR experiments, the Reality Criterion and the original EPR argument. To complicate matter: non-Kolmogorovity of quantum probability, locality, non-conspiracy. What would it mean to explain EPR correlations in terms of common causes? Bell’s inequalities. Further discussions: Are "quantum probabilities" probabilities? What do we actually observe in an EPR correlation experiment? What does locality actually mean?


Literature: Szabo, 2007

Szabo

Lecture 3

50 min


Seminar 2

Recent No-go theorems on local Reichenbachian common cause explanations of EPR correlations


Standard derivations of the Bell’s inequalities assume, besides locality and no-conspiray, a common common cause system that is a common screener-off for all correlations featuring in Bell’s inequalities. However, replacing the strong assumption of a common common cause system for the correlations by the weaker assumption of separate common cause systems (= a set of separate screener-offs explaining the correlations separately) some Bell-like inequalities can still be derived. The violation of these Bell-like inequalities entails the non-existence of a local, non-conspiratorial, separate-common-cause-model of both perfect and imperfect EPR correlations.


Literature: Szabo, 2000; Grasshoff, Portman and Wüthrich, 2005; Hofer-Szabo, 2007

Hofer



Day 7. Tuesday, July 29, 2008


Causal explanation of correlations in physics II. (3 lectures + 1 seminar discussion)





Day 7.






Title/topic

Lecturer


Lecture 1

50 min


Lecture 2

50 min

Spacelike correlations in local relativistic quantum field theory


The idea of a local quantum physics: associating explicitly physical observables with definite spatiotemporal locations. Locality and local observables in local relativistic quantum field theory. The notion of maximal Bell correlation between observables localized in spacelike separated local spacetime regions. Propositions characterizing maximal and non-maximal violations of Bell’s inequality for observables pertaining to typical tangent and strictly spacelike separated regions. The difference between violations of Bell’s inequality in ordinary and relativistic quantum theory.


Literature: Redei and Summers, 2002

Redei

Lecture 3

50 min


Seminar

50 min

The status of Reichenbach’s Common Cause Principle in quantum field theory


The notion of a local Reichenbachian common cause in relativistic quantum field theory and the open problem of the status of the Common Cause Principle: Can spacelike correlations be explained by a common cause localized in the common causal past of the correlated observables? A Reichanbchian common cause in quantum field theory is called weakly localized if it is in the union of the causal pasts of the correlated observables. Proof of existence of weakly localized common causes of spacelike correlations. Is the problem of status of the Common Cause Principle decidable by the axioms of local quantum field theory?


Literature: Redei and Summers, 2002

Redei



Day 8. Wednesday, July 30, 2008


Learning Causal Influences Using Bayesian Networks (2 lectures +2 seminar discussions)





Day 8.







Title/topic

Lecturer


Lecture 1

1.5 hours


Seminar

.5 hours


Lecture 2

1.5 hours


Seminar

.5 hours

Learning Causes Using Manipulation.

Causal Graphs.

Causal Markov Assumption.

Causal Faithfulness Assumption.

Constraint-Based Causal Learning Assuming Faithfulness.


Causal Embedded Faithfulness Assumption.

Constraint-Based Learning Assuming Embedded Faithfulness.

Causal Embedded Faithfulness Assumption with Selection Bias.

Constraint-Based Learning Assuming Embedded Faithfulness with Selection Bias.


Literature: Neapolitan, 2003; Spirtes, Glymour, and Scheines, 2000

Neapolitan



Day 9. Thursday, July 31, 2008


Learning Causal Influences Using Bayesian Networks II. (2 lectures +2 seminar discussions)





Day 9.







Title/topic

Lecturer


Lecture 3

1.5 hours


Seminar

.5 hours


Lecture 4

1.5 hours


Seminar

.5 hours

Bayesian Method for Causal Learning.

Causal Learning from Data on Two Variables.


Identifying Causal Influences Using a Causal Graph.


Literature: Halpern and Pearl, 2005; Neapolitan, 2003; Pearl, 2000; Tian and Pearl, 2002. 

Neapolitan



Day 10. Friday, August 1, 2008


Presentations by participants



Presentations are scheduled to take place in one-hour sessions, one session accommodating 2 presentations, on the following model of the schedule of the first session of the day:


Day 10.




Time

Title of presentation

Speaker

9.00-9.20

9.20-9.30 discussion







9.30-9.50

9.50-10.00 discussion








3 such sessions (altogether 6 presentations) are planned for Day 10. of the summer school.





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