Updating: A psychologically basic situation of probability revision

Abstract

The Bayesian model has been used in psychology as the standard reference for the study of probability revision. In the first part of this paper we show that this traditional choice restricts the scope of the experimental investigation of revision to a stable universe. This is the case of a situation that, technically, is known as focusing. We argue that it is essential for a better understanding of human probability revision to consider another situation called updating (Katsuno & Mendelzon, 1992), in which the universe is evolving. In that case the structure of the universe has definitely been transformed and the revision message conveys information on the resulting universe. The second part of the paper presents four experiments based on the Monty Hall puzzle that aim to show that updating is a natural frame for individuals to revise their beliefs.

Keywords:

• Bayes' rule
• Focusing
• Probability revision
• Updating

Notes

¹There is also another revision situation in a stable universe (not considered in this paper) that is called in the literature revising (Alchourron, Gärdenfors, & Makinson, 1985). When revising you do not know precisely the class of reference (your beliefs are erroneous or incomplete) and the message you learn specifies or disconfirms your initial belief.

²There are several specifications of “Bayes' rule”. Although these specifications are equivalent, in this paper it appears important to distinguish two major forms. Let H and ^¬H be two alternative hypotheses, let M be one event such that M ≠∅, P(H) the prior probability of H, P(H|M) the posterior probability, P(M|H) and P(M|^¬H) the likelihoods and finally P(M) the probability of message M.

Form 1, “conditional probability”: ,

Form 2, “Bayes' identity”: .

Bayes's rule is a revision rule that is defined only for a stable universe. It is the only possible rule in a focusing situation. It operates a change of reference class—it leads from a prior to a posterior probability (de Finetti, 1974). Bayes' rule also proves an adequate rule (among others) to change degrees of belief in a situation of revising (Walliser & Zwirn, 2002).

³From a theoretical point of view, this rule is a particular instantiation of Lewis's General Imaging rule (Cross, 2000; Dubois & Prade, 1993; Gärdenfors, 1988; Lepage, 1997; Lewis, 1976; Perea, 2009).

⁴The MHP is usually presented in two versions. One standard version originates from a popular television game show, Let's Make A Deal, aired in 1963 in America. It is set up as a problem of choice where participants decide to switch or keep the door initially chosen (vos Savant, 1990). An older version called the Three Prisoners Problem is cast in a probabilistic format (Gardner, 1959). In this paper we analyse the puzzle in a probabilistic format.

⁵This solution of the MHP was first suggested by Dubois and Prade (1992).

⁶Numerous studies of probability judgement have supported the hypothesis that participants infer a different representation of the task than experimenters do, based on various pragmatic considerations (Dulany & Hilton, 1988; Krosnick, Li, & Lehman, 1990; Macchi, 1995; Politzer & Macchi, 2005; Politzer & Noveck, 1991).

⁷We borrow here the term naive from the theorists who differentiate a Bayesian conditioning on a naive set and a Bayesian conditioning on a more sophisticated set in the MHP (Grünwald & Halpern, 2003; Jeffrey, 1988). Independently, it was also used by some psychologists in a similar sense (Fox & Levay, 2004; Johnson-Laird et al., 1999).

⁸Unlike the first two experiments, in this and the next experiment the numbers were relatively small. The first two experiments were run with larger numbers because we used novel materials, so that the base rate of incorrect answers and the order of magnitude of the effects to be demonstrated were unknown.

⁹A number of academic people, including a world-famous mathematician, wrote to Vos Savant (1991) to express their disagreement that 1/3 was the correct answer to the MHP (see also Vazsonyi, 1999).

¹⁰The solutions 1/5 in the focusing situation and 1/3 in the updating situation are obvious by changing the prior probabilities (1/3, 1/3, 1/3) to (2/4, 1/4, 1/4) in Figure A1 in the Appendix.

¹¹A very small minority of studies deals with the situation of revising (see footnote 1) (Baratgin & Politzer, 2007). Updating has already been considered explicitly in a deductive framework, in connection with the problem of belief revision for knowledge bases (Elio & Pelletier, 1997; Politzer & Carles, 2001), and implicitly in the field of counterfactual reasoning: In effect, a counterfactual statement does express a modification of the universe, albeit virtual. Finally, the situation called intervention in the field of causal reasoning (Pearl, 2000) can be considered as a special case of updating. A few experimental studies on causal reasoning have shown that the situation of intervention seems natural for participants (Lagnado & Sloman, 2004, 2006; Sloman & Lagnado, 2005; Steyvers, Tenenbaum, Wagenmakers, & Blum, 2003).