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Confirmation in the Cognitive Sciences: The Problematic Case of Bayesian Models

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Abstract

Bayesian models of human learning are becoming increasingly popular in cognitive science. We argue that their purported confirmation largely relies on a methodology that depends on premises that are inconsistent with the claim that people are Bayesian about learning and inference. Bayesian models in cognitive science derive their appeal from their normative claim that the modeled inference is in some sense rational. Standard accounts of the rationality of Bayesian inference imply predictions that an agent selects the option that maximizes the posterior expected utility. Experimental confirmation of the models, however, has been claimed because of groups of agents that “probability match” the posterior. Probability matching only constitutes support for the Bayesian claim if additional unobvious and untested (but testable) assumptions are invoked. The alternative strategy of weakening the underlying notion of rationality no longer distinguishes the Bayesian model uniquely. A new account of rationality—either for inference or for decision-making—is required to successfully confirm Bayesian models in cognitive science.

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Notes

  1. For a partial list of phenomena and references see the “Appendix”.

  2. P(E) need not be computed if we are interested only in the relative probabilities of the various hypotheses. It is necessary to model the impact of rare (i.e., surprising) data.

  3. Examples of Bayesian models with mechanistic commitments include Rao (2005), Lee and Mumford (2003), and Doya et al. (2007).

  4. Bayesian models and rational analyses are not coextensive in principle (Danks 2008); there can be non-Bayesian rational analyses and Bayesian models that are not rational analyses. For example, standard reinforcement learning models are generally not Bayesian, but are rational in a wide range of environments; Bayesian models using limited hypothesis spaces are (in the absence of arguments about memory or computational limits) not necessarily rational. In practice, though, almost all Bayesian models are rational analyses, and the nature of the normative claim of rationality in these models will turn out to be one of the main sticking points for their confirmation.

  5. For our purposes here the relevant point is that the match of the response distribution with the model prediction is considered the relevant criterion to assess fit. We do not intend to argue here about whether or not the model actually constitutes a good fit. Visually there are some quite clear discrepancies between model prediction and data in the cross domain condition, but the authors claim their “model accurately predicted the responses […] with a Pearson product-moment correlation coefficient of r(9) = .85.”

  6. There are some notable exceptions that do actually try to compare each individual participant’s responses with the predictions of a Bayesian model. For example, Körding and Wolpert (2006) directly model each individual learner’s prior probabilities and subsequent inferences. Steyvers et al. (2003) model learning given participant-chosen interventions on an individual basis. Tenenbaum and Griffiths (2001) do not directly model individuals, but do obtain probability judgments that can be compared to the predicted posterior probability distribution (assumed to be the same for all individuals). Whether these analyses have provided more support for the Bayesian models is, we believe, open to question. The concern about the methodology of the more typical methods discussed here remains, independently.

  7. Recall that ‘Bayesian model’ refers (in the cognitive science community) to a model of Bayesian belief updating. Throughout this section, we share this focus, and so will ignore the many arguments for synchronic Bayesianism: the theory that degrees of belief are (or should be) given by a coherent subjective probability distribution. Our issue here is only with the correct way to change one’s beliefs over time, not whether synchronic Bayesianism is the correct way to understand degrees of belief at some moment in time.

  8. One natural assumption is that learners should only invoke computable functions, but this constraint is sometimes incompatible with the requirement (on synchronic Dutch book grounds) that a Bayesian reasoner know all (relevant) logical and mathematical implications of the various hypotheses that she entertains (Gaifman and Snir 1982). Moreover, there are learning problems that can be solved in the long run by computable falsificationist methods (e.g., Popper’s method of “assert the hypothesis until it is refuted”) that cannot be solved by a computable Bayesian reasoner (Juhl 1993; Kelly and Schulte 1995; Osherson et al. 1988). In these circumstances the Bayesian reasoner only converges to the truth (whenever the truth can be learned) if she can sometimes “compute” uncomputable functions.

  9. If the true hypothesis H is empirically distinguishable from other hypotheses and P(H) ≠ 0, then for all ε, P(Bayesian reasoner has degree of belief greater than 1–ε in the truth) → 1 as the number of datapoints goes to infinity; see Savage (1972) for a canonical expression of this result.

  10. More precisely, Bayesian updating (for any non-dogmatic prior probability distribution) provably satisfies the condition that there is no method that gets to the truth faster than Bayesian updating in every “world” (i.e., regardless of which hypothesis is true, and the order of the randomly sampled evidence). There may be alternative non-Bayesian methods that get to the truth faster in particular worlds, but none outperforms Bayesian updating in every world (Schulte 1999). There are, however, methods that are similarly non-dominated by Bayesian updating, so any argument along these lines does not identify Bayesian inference as uniquely rational.

  11. Virtually no experiment that uses Bayesian models has a degenerate posterior distribution. If the Bayesian model is supposed to provide a normatively correct description of the belief update of an individual, then it follows that the experiment is explicitly considering circumstances in which the distribution over the hypotheses has not converged. Moreover, much of the appeal of Bayesian models (in contrast to logic-based models) results from the ability to describe shifts in degree of belief that are not complete, i.e. where uncertainty over the true hypothesis remains.

  12. The Luce choice axiom states:

    1. (1)

      If options a and b are in a choice set S and a is never chosen over b in the binary choice situation, then a can be removed from S without affecting any choice probabilities; and

    2. (2)

      If R is a subset of S, then the choice probabilities for the choice set R are identical to the choice probabilities for S conditional on R having been chosen (i.e. P R (a) = P S (a | R) for all a in R).

    Luce (1959) shows how the axiom implies that PS(x) = v(x)/Σy in S v(y), where v(.) is a measure of value or weight over the options y in the choice set S.

  13. This question is particularly pressing, since rational analyses must ultimately provide a developmental story (in phylogenetic time, ontogenetic time, or both) that exploits some optimality property of the model. Such a story seems much less plausible if people are probability matching.

  14. An alternative suggestion is made in Fiorina (1971), who argues that many experiments use non-random event probabilities that may appear non-constant to the participant. If event probabilities fluctuate, then choosing the hypothesis with maximal posterior probability is no longer optimal. This response is insufficient, though, since fluctuating probabilities do not automatically imply probability matching behavior is optimal. Moreover, probability matching occurs even in experiments in which participants appear to consider the event-probabilities stable.

  15. We have omitted many technical details, including some additional necessary assumptions. Interested readers should consult any standard book on optimal foraging theory (e.g., Stephens and Krebs 1986) or see the appendix for a simple concrete example. We argue that this response is unsuccessful, and so these additional constraints on its scope are irrelevant for our present purposes.

  16. Josh Tenenbaum (personal communication) has suggested that PAC-Bayesian approaches provide a potential solution to the dilemma we point to and so we address the proposal here. A thorough description of PAC-Bayesian learning would go far beyond the scope of this article. We only aim here to indicate the reasons why we do not think this is a fruitful approach.

  17. We note that any rejection of proposition 3 that exclusively focuses on the prior faces a significant formal challenge. Suppose N participants have priors P 1(H), …, P N (H). Assume also that all individuals have the same likelihoods (P(D|h) for all h in H), and that they choose optimally given their beliefs. The “population prior” (the initial aggregate response distribution) is the distribution over H of the number of participants for which h maximizes their prior, i.e. P pop (h) = # N argmax H P i (h)/N. If all individuals use Bayesian updating, then the “population posterior” (the final aggregate response distribution) is P pop (H|D) = # N argmax H P i (h|D)/N. The puzzle can be solved by appeal to variation in the participant priors only if the population posterior is distributionally equivalent to Bayesian updating on the population prior: P pop (H|D) must be distributionally equivalent to P(D|H)P pop (H)/P(D). Mathematically, this holds when: # N argmax H P(D|h)P i (h) ≈ P(D|H) # N argmax H P i (h). For arbitrary likelihoods, this condition is not satisfied for standard prior distributions (e.g., flat or Gaussian), although it may be satisfied for such priors given particular likelihoods P(D|H). (We know of no such analyses.) But satisfaction in special cases would only provide further support that the appearance of probability matching is largely accidental.

  18. If e(.) is i.i.d. with a Weibull distribution for each hypothesis in the space (and each participant), then the distribution of responses is given by \( \exp (U(H_{i} ))/\sum\nolimits_{j} {\exp (U(H_{j} ))} \). The Weibull distribution is sufficient; weaker constraints on the distribution of e(.) can be given.

  19. For example, Schulz et al. (2007), which we discussed in Section “Confirmation of Bayesian Models”, claim in their conclusion: “The Bayesian model presented in the Appendix [in their paper] … show[s] that the judgments of the 4- and 5-year-olds in our experiments are close to the probabilities entertained by an ideal Bayesian learner using a particular causal theory.”

  20. If competitor C1 selects location X with probability p and competitor C2 selects location Y with probability q, then C1 receives 0.75*(1 − q) for location X and 0.25 q for location Y. At equilibrium these have to be equal, hence q = 0.75. The argument is the same for p since the payoff structure is symmetric.

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Acknowledgments

Numerous conversations with Josh Tenenbaum, Tom Griffiths, Noah Goodman, and Chris Lucas helped shape the arguments and ideas in this paper, though we doubt that they would endorse many (or any) of our conclusions. We also received valuable feedback from several anonymous reviewers. The first author was partially supported by a grant from the James S. McDonnell Foundation Causal Learning Collaborative. The second author was partially supported by a James S. McDonnell Foundation Scholar Award.

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Appendix

Appendix

Recent books on Bayesian models include: Oaksford and Chater (2007) with Open Peer Commentary (Oaksford and Chater 2009); Doya et al. (2007); and Chater and Oaksford (2008). Bayesian models were also the focus of a 2006 special issue of Trends in Cognitive Science (Chater et al. 2006; Courville et al. 2006; Körding and Wolpert 2006; Steyvers et al. 2006; Tenenbaum et al. 2006; Yuille and Kersten 2006). A very incomplete sample of phenomena for which Bayesian models have been proposed includes:

  • Category learning and inference (Heit 1998; Kemp and Tenenbaum 2003; Kemp et al. 2007; Tenenbaum and Griffiths 2001)

  • Causal learning and reasoning (Bonawitz et al. 2006; Griffiths and Tenenbaum 2005; Schulz et al. 2007; Sobel and Kushnir 2006; Sobel et al. 2004; Steyvers et al. 2003)

  • Inference about conditionals (Oaksford and Chater 2007; Oaksford et al. 2000)

  • Covariation assessment (McKenzie and Mikkelsen 2007)

  • Imitation (Rao et al. 2004)

  • Information selection (Oaksford et al. 1997)

  • Framing effects (McKenzie 2004)

  • Memory effects (Schooler et al. 2001; Shiffrin and Steyvers 1997; Steyvers and Griffiths 2008)

  • Object perception (Kersten and Yuille 2003; Kersten et al. 2004)

  • Repetition effects and priming (Mozer et al. 2002, 2003)

  • Word learning (Xu and Tenenbaum 2005, 2007)

Example of Optimality of Probability Matching Strategy in Competitive Circumstances

Suppose that a resource occurs with probability 0.75 at location X and with probability 0.25 at locations Y. Further, suppose that for two competitors the pay-off is structured such that a competitor only obtains resources if (1) resources are present at the chosen location and (2) the other competitor did not select the same location. The pay-off matrices for the two possible locations of resources are shown below:

 

Resource is at location X (75% of cases)

Competitor 2 selected location X

Competitor 2 selected location Y

Competitor 1 selected location X

0/0

1/0

Competitor 1 selected location Y

0/1

0/0

 

Resource is at location Y (25% of cases)

Competitor 2 selected location X

Competitor 2 selected location Y

Competitor 1 selected location X

0/0

0/1

Competitor 1 selected location Y

1/0

0/0

In foraging theory the optimal strategy is generally taken to be the one that maximizes the average pay-off over several foraging trials (Stephens and Krebs 1986). We can thus combine the two possible payoff structures by weighting them according to the rate of occurrence of the resource at each location:

 

Combined pay-off structure

Competitor 2 selected location X

Competitor 2 selected location Y

Competitor 1 selected location X

0/0

0.75/0.25

Competitor 1 selected location Y

0.25/0.75

0/0

If both competitors select location X with probability 0.75 and location Y with probability 0.25 then no unilateral change of strategy by a competitor would improve that competitor’s payoff, that is, these two strategies that “match the probabilities” of the occurrence of the resource at the two locations specify a Nash equilibrium.Footnote 20 Note, however, that the optimality of the probability matching strategy crucially depends on the combination of the resource distribution and the specific resource payoff structure. In particular, if competitors can split resources when they both select the same location, then probability matching no longer constitutes a Nash equilibrium.

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Eberhardt, F., Danks, D. Confirmation in the Cognitive Sciences: The Problematic Case of Bayesian Models. Minds & Machines 21, 389–410 (2011). https://doi.org/10.1007/s11023-011-9241-3

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