Is the mind Bayesian? The case for agnosticism

Abstract

This paper aims to make explicit the methodological conditions that should be satisfied for the Bayesian model to be used as a normative model of human probability judgment. After noticing the lack of a clear definition of Bayesianism in the psychological literature and the lack of justification for using it, a classic definition of subjective Bayesianism is recalled, based on the following three criteria: an epistemic criterion, a static coherence criterion and a dynamic coherence criterion. Then it is shown that the adoption of this framework has two kinds of implications. The first one regards the methodology of the experimental study of probability judgment. The Bayesian framework creates pragmatic constraints on the methodology that are linked to the interpretation of, and the belief in, the information presented, or referred to, by an experimenter in order for it to be the basis of a probability judgment by individual participants. It is shown that these constraints have not been satisfied in the past, and the question of whether they can be satisfied in principle is raised and answered negatively. The second kind of implications consists of two limitations in the scope of the Bayesian model. They regard (1) the background of revision (the Bayesian model considers only revising situations but not updating situations), and (2) the notorious case of the null priors. In both cases Lewis’ rule is an appropriate alternative to Bayes’ rule, but its use faces the same operational difficulties.

Appendices

Appendix 1

One can put into question the extensively used term of “Bayes’ rule.” The literature indeed offers several specifications of Bayes’ rule. Although these specifications are equivalent, it appears important to distinguish three major forms. Given two alternative hypotheses H and $^{\neg}H$, one event D such that $D \ne \emptyset$, let P(H) be the prior probability of H, P(H|D) the posterior probability, P(D|H) and $P(D | ^{\neg}H)$ the likelihood and finally P(D) the probability of the data D.

• Form 1. “Conditional probability”:

  $$P(H | D) = \frac{{P(H \cap D)}}{{P(D)}},$$

• Form 2. “Bayes’ identity”:

  $$P(H | D) = \frac{{P(H) P(D | H)}}{{P(D)}},$$

• Form 3. “Derived Bayes’ identity”:

  $$O(H | D) = \frac{{P(H | D)}}{{P(^{\neg}H | D)}} = \frac{{P(H)P(D | H)}}{{P(^{\neg}H)P(D |^{\neg}H)}} = O(H) \frac{{P(D | H)}}{{P(D |^{\neg}H)}} = O(H)L(D | H).$$

The vast majority of studies have exclusively used the second form of Bayes’ rule (b). Few studies have used the third form (c), considering more specifically the phenomenon of conservatism (for example Phillips and Edwards 1966) or the base rate neglect (for example Kahneman and Tversky 1973). Pieces of research quoting the first form of Bayes’ rule (form a) are much less numerous (see nevertheless Gigerenzer and Hoffrage 1995; Lewis and Keren 1999; Mellers and McGraw 1999).

Appendix 2

In the 1960s, several studies in various fields (computer science, psychology and medicine) strove to define a rule that would hold in the case where the new evidence D is not certain (let it be D*). Dodson (1961) addressed this problem. An uncertain message D* can lead one to elaborate subjective judgment on the probability of D being true, thus yielding P(D|D ^*) and its complementary $P(^{\neg}D|D^{*})$. His idea was to incorporate P(D|D ^*) and $P(^{\neg}D|D^{*})$ in the conditioning of H based on the notion of expectation. Dodson suggested that posterior probability must be equal to the sum of the probabilities conditionally to D and $^{\neg}D$ multiplied by the probabilities of being in the situation where D is true or where $^{\neg}D$ is true. Dodson (1961) established the following rule:

$$P(H | D^{*}) = P(D | D^{*})P(H | D) + P(^{\neg}D | D^{*})P(H |^{\neg}D).$$

One can notice that in the degenerated cases where P(D|D ^*)=1 and $P(^{\neg}D|D^{*}) = 0$, the rule reduces to simple conditioning $P(H | D^{*}) = P(H | D) \, {\rm and} \, P(H | D^{*}) = P(H |^{\neg}D)$ when P(D|D ^*)=0 and $P(^{\neg}D|D^{*}) = 1$, respectively. In other words, Dodson’s rule can be viewed as the generalization of Bayes’ identity. Independently, Jeffrey (1965) proposed the same rule of revision. However, in the subjective interpretation of the Bayesian model as specified in the present paper, an individual can revise his/her probability directly when learning D*, and it can be shown that, conversely, Bayes’ identity is equivalent to the Dodson–Jeffrey rule (under the assumption that D* depends on D but not on H; see for example Gettys and Wilke 1969; Jaynes 1994; Schaefer and Borcherding 1973; Schum and du Charme 1971).

Appendix 3

Gärdenfors (1988) showed that Lewis’ (1976) rule (imaging) can be analyzed as a rule for probabilistic revision. In this case, the methodology of revision is different from the one provided by Bayes’ rule. We simply intend here to give an intuitive idea of it (for a detailed technical analysis, see Gärdenfors 1992; Pearl 2000; Walliser and Zwirn 2002). In order to explain this rule, it seems convenient to use Kripke’s (1962) possible worlds semantics. An agent considers a distribution of probability on possible worlds. Following a message invalidating a possible world, degrees of belief concerning this world are redistributed on the other possible worlds that the agent considers to be the closest to the invalidated world. Let us consider the following example (Dubois and Prade 1994; Walliser and Zwirn 2002). A basket may be described by four worlds depending on whether it contains an apple (a) or not $(\neg a)$ together with a banana (b) or not $(\neg b)$. Someone believes at t ₀ that the basket contains at least one fruit: $K = (a \wedge b) \vee (a \wedge \neg b) \vee (\neg a \wedge b)$. Let us assume that the person’s prior probabilities associated to each possible world are 1/2 for $(a \wedge b)$, 1/3 for $(\neg a \wedge b)$, 1/6 for $(a \wedge \neg b)$ and 0 for $(\neg a \wedge \neg b)$. A revising message brought up by a reliable direct witness informs the person that there is no banana: $A = (a \wedge \neg b) \vee (\neg a \wedge \neg b)$. The revised belief K*A is now $(a \wedge \neg b)$, so that the posterior probability allocated to $(a \wedge \neg b)$ is 1. This is what Bayes’ identity gives:

$${\left(\begin{aligned} & P_{A} (a \wedge \neg b) = P((a \wedge \neg b) | A),\\ & P_{A} (a \wedge \neg b) = \frac{{P(a \wedge \neg b)P(A | (a \wedge \neg b))}}{{P(a \wedge \neg b) P(A | (a \wedge \neg b)) + P(a \wedge b)P(A | (a \wedge b)) + P(\neg a \wedge b)P(A | (\neg a \wedge b))}},\\ & P_{A} (a \wedge \neg b) = \frac{{1/3 \times 1}}{{1/3 \times 1 + 1/2 \times 0 + 1/6 \times 0}} = 1.\\ \end{aligned} \right)}$$

But with Lewis’s rule (or imaging) one will end up with a different result. An updating message says at t ₁ that there is no more banana. In this updating situation, the revised belief K*A is $(a \wedge \neg b) \vee (\neg a \wedge \neg b)$ because the reasoner starts afresh a distribution over all possible worlds. In order to do so, the prior probabilities (1/2 and 1/3) attached to $(a \wedge b) \, {\rm and} \, (\neg a \wedge b)$ are distributed over other close worlds. One can think that the world $(a \wedge b)$ is closer to $(a \wedge \neg b)$ than to $(\neg a \wedge \neg b)$ (choosing arbitrarily the intuitive physical action as a basis to define the distance) and therefore the degree of belief 1/2 attached to $(a \wedge b)$ is attributed to

$$(a \wedge \neg b): {\left( {P_{A} (a \wedge \neg b) = 1/6 + 1/2 = 2/3} \right)}.$$

Alternatively, one can think that the world $(\neg a \wedge b)$ is closer to $(\neg a \wedge \neg b)$ than to $(a \wedge \neg b)$ and therefore the degree of belief 1/3 attached to $(\neg a \wedge b)$ is attributed to $(\neg a \wedge \neg b)$:

$${\left( {P_{A} (\neg a \wedge \neg b) = 1/3} \right)}.$$

This example illustrates the two situations of revision using, of course, a distance arbitrarily chosen in the updating case. In the latter case (Lewis’ rule), the initial belief can be thoroughly modified, since one could expect to have no fruit anymore in the basket following the message.

Appendix 4

Kolmogorv’s axioms contain the following two “convexity rules” (the impossible event is noted $\emptyset$ and the set of possible results Ω):

• 1. $P(\emptyset) = 0 \; {\rm and} \; P(\Omega) = 1$;

• 2. H is said to be “significant” if and only if 0<P(H)<1;

and the additivity rule:

• 3. ${\rm If} \, H \cap G = \emptyset, \; {\rm then} \; P(H \cup G) = P(H) + P(G)$

together with its corollary, the complementarity constraint $P(H) + P(^{\neg}H) = 1$.

The conditional probabilities must satisfy the rule of compound probabilities:

• 4. $P(H \cap G) = P(H)P(G | H)$;

and its corollary, namely Bayes’ identity (Appendix 1).