Quantum Cognition

There are two mathematical approaches to constructing probabilisitc systems: classic Kolmogorov probabilities and quantum von Neumann probabilities. The majority of models in cognitive science and psychology use the classical probability system. However, classic probability theory imposes a restrictive set of assumptions on the representation of complex systems such as the human cognitive system. Quantum probability theory postulates a more general method for representing and analyzing these types of complex systems.

Quantum probability theory is the noncommutative analog of classical probability theory derived from quantum mechanics. As formalized by von Neumann (1932), quantum probability theory is a geometric approach to probability where events are represented as subspaces of a vector space, and an additive measure assigns probabilities to events. By using subspaces as events rather than subsets, quantum probability theory entails a different logic than classical probability theory. The logic of quantum probability theory is the logic of subspaces which relaxes some of the assumptions of Boolean logic. In particular, quantum probability theory does not always have to obey the closure, commutative, and distributive properties. Similarly, human judgments do not always obey these three properties.

Quantum probability theory is a viable new research area in social and behavioral sciences that has been able to account for numerous findings in cognition and decision-making. These include violations of the sure thing principle (Pothos & Busemeyer, 2009), conjunction and disjunction fallacies (Busemeyer, Pothos, Franco, & Trueblood, 2011), interference effects in perception (Conte, Khrennikov, Todarello, Federici, & Zbilut, 2009), interference of categorization on decision-making (Busemeyer, Wang, & Lambert-Mogiliansky, 2009), violations of dynamic consistency (Busemeyer, Wang, & Trueblood, 2012), order effects in survey questions (Wang & Busemeyer, 2014), and causal reasoning (Trueblood & Pothos, 2014).

Short Introductory Chapter (pdf)

Introductory Materials from the CogSci 2015 Tutorial

Introductory Slides (pdf)

Quantum Probability Slides (pdf)

Quantum Dynamics Slides (pdf)

Advanced Topics Slides (pdf)

Advanced Topics Notes (pdf)

Applications of Quantum Probability Theory to Causal Reasoning

There is considerable variety in the types of situations were people reason about causes and effects. For example, individuals might experience the variation and covariation of events through observation such as learning the effect of a chemical on DNA mutations (Lober & Shanks, 2000). Alternatively, people might be asked to make causal inferences from descriptions that lack statistical information, e.g., "Imagine you exercise hard in April. How likely is it that you weigh less in May?" (Fernbach et al., 2010). A challenge for cognitive modelers is developing a comprehensive framework for modeling causal reasoning across different types of tasks. Models based on classical probability theory (e.g., Jenkins & Ward, 1965; Cheng, 1997) provide good accounts of causal reasoning in both experienced situations and those described by statistical information. However, there is evidence that causal inferences from descriptions without statistical information often deviate from the normative prescription of classical probability (Sloman & Fernbach, 2011). As a result, it is difficult to use classical probability models in this domain, thereby producing a disparate account of casual reasoning across tasks.

This research proposes a unified account of human causal reasoning using quantum probability theory. We postulate a hierarchy of mental representations, from fully quantum to fully classical, that could be adopted for different situations. Different levels in the hierarchy are associated with models of different dimensionality.

Classical probability models assume a single space that provides a complete and exhaustive description of all events. Due to the closure property of Boolean logic, these models are typically high dimensional. They also represent the highest level in our hierarchy, which corresponds to the most complete (and also most complex) types of mental representations. These representations might be adopted in a number of different situations including when an individual has extensive knowledge of events or the description of events is very clear.

Unlike classical probability, quantum probability allows for multiple sample spaces, which are related geometrically by rotations. When two events are described by two different sample spaces, they are called incompatible and their joint event does not exist. Psychologically, this implies that individuals do not have a mental representation of the joint event. That is, they cannot think about these two events simultaneously. Rather, the events are processed sequentially. Models with multiple sample spaces typically have low dimensionality. These models represent lower levels in our hierarchy, which correspond to simple mental representations. We hypothesize that these representations are adopted when individuals do not have a wealth of past experience or when information is vague. Our hypothesis is motivated by the idea that causal learning is structurally local (Fernbach & Sloman, 2009). That is, when people are faced with a complex problem, they break it up by focusing on individual parts. Inferences about the full causal structure are constructed by combining local inferences piece by piece.

Consider the situation where there are two binary causes X and Y for a binary effect E. There are at least three possible approaches to modeling this causal structure using two, four, or eight dimensions. The 2-dimensional model (see below) assumes that individuals consider one event at a time and do not have mental representations of joint events. This is the simplest model in our hierarchy. The 4-dimensional model assumes that individuals form mental representations for single cause and effect relationships, but cannot think about multiple causal relationships simultaneously. The 8-dimensional model assumes individuals have mental representations of all joint events. This is equivalent to a classical probability model with a single sample space and represents the highest level in the hierarchy.

In the 2-dimensional model, the three events X, Y, and E are represented as different bases (i.e., coordinates) for a 2-dimensional vector space where the two dimensions correspond to the two possible values (0 and 1). The vector psi (in black) is a knowledge state that represents an individual's beliefs about the different events. This vector has unit length and is represented within a unit circle. It is projected (dotted line) onto the effect E. The squared length of this projection (solid black bar) is the probability that an individual believes that effect E occurs pr(e1).

Learn More! (pdf)

Relevant Links

Jerome Busemeyer's Quantum Page

Quantum Models of Cognition and Decision Book

Quantum Interaction Conference

This material is based upon work supported by National Science Foundation Grant SES-1326275.