Alternative Probability Theories for Cognitive Psychology

Transcription

1 Topics in Cognitive Science 6 (2014) Copyright 2013 Cognitive Science Society, Inc. All rights reserved. ISSN: print / online DOI: /tops Alternative Probability Theories for Cognitive Psychology Louis Narens Department of Cognitive Sciences, University of California, Irvine Received 25 February 2013; accepted 15 February 2013 Abstract Various proposals for generalizing event spaces for probability functions have been put forth in the mathematical, scientific, and philosophic literatures. In cognitive psychology such generalizations are used for explaining puzzling results in decision theory and for modeling the influence of context effects. This commentary discusses proposals for generalizing probability theory to event spaces that are not necessarily boolean algebras. Two prominent examples are quantum probability theory, which is based on the set of closed subspaces of a Hilbert space, and topological probability theory, which is based on the set of open sets of a topology. Both have been applied to a variety of cognitive situations. This commentary focuses on how event space properties can influence probability concepts and impact cognitive modeling. Experimental psychology has developed elaborate procedures for avoiding the impact of context, for example, the impact of a response on subsequent responses. Such procedures, while effective at eliminating some prominent context effects, usually do not, on careful examination, eliminate all of them. Instead, they tend to hold enough of the contexts in check so that the sought-after experimental results will be significant and replicate. The need for such procedures is testament to the ingrained nature of context in psychological experimentation. This suggests that context should be studied and modeled in psychology as well as controlled by experimental paradigms. To accomplish this, theories and methodologies are needed for modeling what is invariant across context as well as relationships among contexts. Recently, quantum probability theory has been used by some psychologists for this purpose (e.g., see Busemeyer & Bruza, 2012). While a useful first step, quantum probability theory appears to me to be based too much on geometrical and algebraic considerations designed to fit in with concepts of quantum physics. This fit was a natural consequence of the co-evolution of Hilbert space mathematics with quantum mechanics. However, to me at least, many of the fundamental geometric and Correspondence should be sent to Louis Narens, Department of Cognitive Sciences, 2201 Social & Behavioral Sciences Gateway Building, University of California, Irvine, CA lnarens@uci.edu.

2 L. Narens / Topics in Cognitive Science 6 (2014) 115 algebraic considerations of probabilistic Hilbert space do not appear to make sense or fit in with psychological phenomena. What is needed is a foundational probability theory that captures the important uses of quantum probability theory in psychology in a way that is designed to mesh with cognitive psychological experimentation and theory. As discussed later, a good place to start is to develop a probability theory that has many of the properties of quantum probability theory but does not assume an underlying Hilbert space. Current cognitive psychology has produced an abundance of experimental results demonstrating irrational human behavior involving uncertainty. Many of these results can be interpreted as having to do with context, for example, the participant is not holding the probability of an event or the utility of an outcome constant across context, where rationality holds that they should remain constant. The economic and philosophical literatures present various arguments for the rationality of traditional probability theory and the SEU model of decision making. These arguments contradict numerous experiment findings involving human participants. However, it is noteworthy that the arguments assume an underlying boolean algebra of events. I have not seen justifications for this assumption. Narens (2011, 2012b) looks at decision situations with an underlying non-boolean algebra of events, called a topological lattice (described later in this Commentary). He shows that many decision situations considered irrational in the literature satisfies rationality arguments if a topological event space is used in place of a boolean algebra of events. This is because the non-boolean case is modeled in a manner so that contexts become part of the events. For example, in traditional theory an event appearing in multiple contexts is modeled as a single event belonging to an underlying boolean algebra of events; in the non-boolean case it is modeled as two or more related events. Narens uses the non-boolean formulation to argue that many kinds of violations of traditional probability theory and subjective expected utility theory (SEU) should be considered rational. This manner of viewing the relationship of event and its contexts for its occurrence allows better insight into deciding which irrational probabilistic psychological phenomena can be potentially be incorporated into economic decision theory. A good probability concept requires its event space to have a reasonable logical structure. Following Birkhoff and von Neumann (1936), I will assume a lattice algebra as a minimal, natural logic for a good probability concept: L ¼hL; d; e; X; i is said to be a lattice algebra of sets or just lattice of sets for short if and only if L is a nonempty set of subsets of X, X and are elements of L, and for all A and B in L, AdB and AeB are in L, where 1. AdB = the least -upper bound in L of A and B, and 2. AeB = the greatest -lower bound in L of A and B. There are various kinds of lattice of sets. For example, let T be a topology with universal set X. Then T ¼hT; [; \; X; i is a lattice of sets. It is called a topological lattice. Topological lattices satisfy two important algebraic lattice properties: distributivity and pseudocomplementation:

3 116 L. Narens / Topics in Cognitive Science 6 (2014) The lattice L ¼hL; d; e; X; i is said to satisfy distributivity if and only if for all A, B and C in L, AeðBdCÞ ¼ðAeBÞdðAeCÞ, and L is said to satisfy pseudocomplementation if and only if for all A in L there exists B in L called the pseudocomplement of A such that A \ B ¼ and for all C in L, ifaec ¼ then C B. For a topology T with universal set X, the pseudocomplement of an open set A in T is the largest open set B in T that is disjoint from A. Because T is a topology, such a B exists. It is immediate that the lattice T ¼hT; [; \; X; i is distributive. Thus, the topological lattice T is both pseudocomplemented and distributive. Let L ¼hL; d; e; X; i be a lattice of sets and A 2L. Then B is said to be a complement of A if and only if B 2L, AeB ¼, and AdB ¼ X. L is said to be complemented if and only if each element of L has a complement. L is said to be a boolean lattice of sets if and only if it is complemented and distributive. An important special case of a boolean lattice of sets is called a boolean algebra of events, when L is complemented, d ¼[, and e ¼\. These two concepts of boolean need to be distinguished. It is a well-known theorem of lattice theory the Stone Representation Theorem that each boolean lattice of sets is isomorphic to a boolean algebra of events. Thus, to obtain a lattice of sets that is algebraically different from a boolean algebra of events, the condition complemented and distributive must be generalized. There are several ways to accomplish this. This commentary focuses on generalizing the complementation operation of a boolean lattice of sets. Let L ¼hL; d; e; _ ; X; i be a boolean lattice of sets, where _ is a complementation operation of L. It is well known that _ has the following two properties: (a) _ is the unique complementation operation on L, and (b) _ is a pseudocomplementation operation on L. The following theorem can be shown: Theorem1 Suppose L ¼hL; d; e; X; i is a lattice of sets and _ is both a complementation operation of L and a pseudocomplementation of L. Then L is boolean. Proof Narens (2012a). An interesting consequence of Theorem 1 is that distributivity in lattices can be expressed purely in terms of complementation. The complementation operation of boolean lattices has other characteristic features. The most important of these are orthocomplementation and De Morgan Laws: Let L ¼hL; d; e; X; i be a lattice of sets. Then is said to be an orthocomplementation operation of L if and only if is a complementation operation of L such that for each A and B in L,

4 L. Narens / Topics in Cognitive Science 6 (2014) 117 A B iff B? A? ; ð1þ and A?? ¼ A: ð2þ It is well known that each orthocomplemented lattice satisfies the De Morgan Laws, ðadbþ? ¼ðAÞ? eðbþ? and ðaebþ? ¼ðAÞ? dðbþ? : The following is a theorem of lattice theory. Theorem 2 Suppose L ¼hL; d; e; X; i is a lattice of sets where _ is an orthocomplementation operation of L and _ is the only complementation operation of L. Then L is boolean. Proof Narens (2012a). It is desirable to generalize traditional probability in a manner so that the probability functions of the generalized theory retain many of the algebraic characteristics of the probability functions of traditional probability theory. This commentary proposes that this can be accomplished through generalizing the concept of a boolean algebra of events. Theorems 1 and 2 suggest a means for doing this. Theorem 1 suggests deleting the complementation condition for a boolean algebra of events and replacing it with the condition that the lattice is pseudocomplemented. This produces a lattice sets of the form L ¼hL; [; \; _ ; X; i, where _ is the operation of pseudocomplementation. This lattice is, of course, distributive. An example of it is a topological lattice with L being a topology. In general, it can be shown that every pseudocomplemented distributive lattice is isomorphic to a sublattice of L with L being a topology (Narens, 2007). For the pseudocomplemented case, the notion of probability function of L needs to be changed a little. This is because the finite additivity of traditional probability functions, PðA [ BÞ ¼PðAÞþPðBÞ for A \ B ¼ ; ð3þ is effective because boolean algebras of events are endowed with a sufficiently rich collection of disjoint events. In the presence of the other conditions for a boolean algebra of events, this sufficient richness of disjoint events follows from complementation. This is not necessarily the case for a pseudocomplemented, distributive lattice of sets. Because of this, Eq. 3 needs to be strengthened to Eq. 4 below.

5 118 L. Narens / Topics in Cognitive Science 6 (2014) Let L ¼hL; [; \; _ ; X; i be a pseudocomplemented, distributive lattice. Then P is said to be a generalized probability function on L if and only if 1. P is a function from L into [0,1] such that PðXÞ ¼1 and Pð Þ ¼0; 2. for all A and B in L, PðA [ BÞ ¼PðAÞþPðBÞ PðA \ BÞ: ð4þ Note that traditional probability functions on boolean algebras of events satisfy Eq. 4. It is known that each pseudocomplemented distributive lattice has a generalized probability function on it (Narens, 2007). Narens (2007, 2009, 2011) employed pseudocomplemented distributive lattices to provide a new kind of mathematical foundation for modeling decision making and judgments of probability. Theorem 2 suggests a different approach to generalizing traditional probability theory: Look at lattices that are complemented but not uniquely complemented. Such lattices necessarily fail to satisfy the lattice distributive law (because, as mentioned earlier, all boolean lattices of sets are uniquely complemented). To make the generalization closer to traditional probability theory, additionally assume that the lattice is orthocomplemented. The following defines probability functions for orthocomplemented lattices. Suppose L ¼hL; [; \;? ; X; i is an orthocomplemented lattice. Then P is said to be a orthoprobability function on L if and only if P is a function from L into [0,1] such that PðXÞ ¼1 and Pð Þ ¼0, and for all A and B in L, ifb A, then PðA [ BÞ ¼PðAÞ þpðbþ. Dirac (1930) and von Neumann (1932) employed an orthocomplemented lattice algebra of closed subspaces of a Hilbert space to provide a mathematical foundation for quantum mechanics. Von Neumann attempted to describe this lattice algebraically by generalizing the lattice distributive law, to the following: For all A, B, and C in a lattice algebra, AeðBdCÞ ¼ðAeBÞdðAeCÞ; 1 Modular law : If A B then AeðBdCÞ ¼ ðaebþdðaecþ: 1 However, the modular law applies only finite dimensional Hilbert space. A more general law is required for general Hilbert space. For general Hilbert spaces, Husimi (1937) suggested the following: For all A, B, and C in the orthocomplemented lattice algebra, Orthomodular law : If A B then AdðA? ebþ ¼B; where is the lattice s orthocomplementation operation. In a Hilbert space, is the operation that assigns to each closed subspace its orthogonal complement. The orthomodular

6 L. Narens / Topics in Cognitive Science 6 (2014) 119 law holds for the lattice of all closed subspaces of a Hilbert space. It is implied by the modular law, but it does not imply the modular law. Husimi suggested that for the lattice theory development of quantum mechanics that the orthomodular law be used instead of the modular law. This practice has been followed by physicists, mathematicians, and philosophers, and produced a field today called quantum logic, which generalizes the logical structure inherent in Hilbert space. A problem with basing scientific applications on just quantum logic is that not every quantum logic has an orthoprobability function (Greechie, 1971). Thus, in particular, for cognitive applications additional conditions need to be assumed. Busemeyer and Bruza (2012) and others have applied quantum logic in the form of closed subspaces of a Hilbert space to a variety of cognitive phenomena. The problem I see in using Hilbert space for this kind of application is that it contains very strong assumptions that do not appear to be abstractions of the kinds of properties and structures one finds in the cognitive phenomena to be modeled. Even the weaker system consisting of just quantum logic is problematic: What in the modeling of cognitive phenomena suggests that orthomodularity should be used for modeling events in situations where distributivity fails? As far as I can see, nothing directly. Thus, for a quantum logic to be justifiably useful in cognitive modeling, orthomodularity should be derivable from other justifiable psychological-experimental assumptions. I do not see this as a particularly difficult problem, but it is outside the scope of this commentary to go into details. In conclusion, mathematics tells us there are a very limited range of event spaces for generalizing traditional probability theory in manners potentially useful for cognitive psychology. Two of these topological lattices and lattice of closed subspaces of a Hilbert space have been recently used in the modeling of cognitive phenomena. They appear to be interesting first steps that bring together and provide common explanations for a variety of disparate and puzzling cognitive phenomena. Hilbert space modeling appears to me to provide a richer and more applicable basis for the experimental and theoretical study of contexts than topological lattice modeling. For the modeling of context in cognitive psychology, the next step should be tailoring the underlying mathematics to reflect the structure inherent in cognitive phenomena and experimental procedures to replace the assumption of a Hilbert space. I believe it is unlikely that such tailoring would extend to producing lattices isomorphic to lattices of closed subspaces of a Hilbert space. This is because the uses of Hilbert space lattices of closed subspaces in quantum probability theory were developed for and co-evolved with quantum mechanics. As a consequence, they incorporate in essential ways aspects of particle physics and its dynamics that appear to me to currently lack justification for application in psychology. Acknowledgments Discussions with Jeffrey Barrett and Brian Skyrms were helpful in formulating ideas for this commentary. This research was support by grant FA from AFOSR.

7 120 L. Narens / Topics in Cognitive Science 6 (2014) References Birkhoff, G. and von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37(4), Busemeyer, J. R. & Bruza, P. D. (2012). Quantum models of cognition and decision. Cambridge, UK: Cambridge University Press. Dirac, P. A. M. (1930). The principles of quantum mechanics. Oxford, UK: Oxford University Press. Greechie, R. J. (1971). Orthomodular lattices admitting no states. Journal of Combinatorial Theory, 10, Husimi, K. (1937). Studies in the foundations of quantum mechanics. Proceedings of the Physico- Mathematical of Japan, 19, Narens, L. (2007). Theories of probability: An examination of logical and qualitative foundations. London: World Scientific. Narens, L. (2009). A foundation for support theory based on a non-boolean event space. Journal of Mathematical Psychology, 53, Narens L. (2011). Probabilistic lattices: Theory with an application to decision theory. In E. N. Dzhafarov & L. Perry (Eds.), Descriptive and normative approaches to human behavior ( ). Singapore: World Scientific. Narens, L. (2012a). Probabilistic lattices with applications to the behavioral sciences. Unpublished book manuscript. Narens, L. (2012b). Multimode utility theory. Unpublished manuscript. Von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechank. Berlin: Springer