Implicit and explicit Bayesian inference

Description

A major current revolution in cognitive science concerns the rapid ascendance of Bayesian modeling of probabilistic (or uncertain) reasoning (Chater, Tenenbaum, & Yuille, 2006). In the last couple of years, Bayesian modeling has surpassed both symbolic and connectionist methods in frequency of conference presentations and publications. An entire special issue of Trends in Cognitive Sciences (July 2006) was devoted to Probabilistic Models of Cognition. Bayesian modeling is being successfully applied to a wide range of problems in cognitive science including sensorimotor control (Körding & Wolpert, 2006), vision (Yuille & Kersten, 2006), conditioning (Courville, Daw, & Touretzky, 2006), induction and inference (Tenenbaum, Griffiths, & Kemp, 2006), and language (Chater & Manning, 2006). In each of these areas, Bayesian models are accounting for (and in some cases even predicting) subtle data patterns in a wide variety of psychological experiments.

A general conclusion emerging from this work is that people (and other animals) optimize their performance by conforming to Bayes’ rule that specifies how posterior conditional probabilities (of a hypothesis being true, given a pattern of data) are computed from the product of the likelihood of those data given the truth of the hypothesis and the prior probability of the hypothesis. This likelihood x prior product is divided by the sum of similar products for all relevant hypotheses (a sum known as the marginalized probability of the data). Part of the appeal of this approach is that, unlike the emphasis on representational structure in symbolic models and soft statistical constraints in connectionist models (Shultz, 2003), Bayesian approaches emphasize both structure and statistics – effectively, by computing statistics over structures, in a way that describes how knowledge is modified by new evidence.

This resurgence of interest in Bayesian methods is somewhat surprising given the Nobel-Prize-winning work showing that people are rather poor Bayesians, subject to, e.g., the base-rate fallacy and the representativeness heuristic (Kahneman, Slovic, & Tversky, 1982; Kahneman & Tversky, 1996; Tversky & Kahneman, 1974, 1981). There is also evidence that people confuse the direction of conditional probabilities, e.g., the probability of a symptom given a disease vs. the probability of a disease given a symptom (Eddy, 1982). Even experienced medical professionals deviate from Bayes in these ways, creating medical inefficiencies and sometimes disastrous outcomes.

Clearly, the new evidence that people are Bayesian optimizers needs to be reconciled with this older work suggesting that people routinely ignore prior probabilities and confuse conditional probabilities. An hypothesis currently being floated to explain this discrepancy is that people are implicit, but not explicit, Bayesians (Chater et al., 2006). This seems a bit implausible given that the distinction between implicit vs. explicit distinction is rather vague and may not make much of a psychological difference in any case. It also seems to conflict with experiments showing that prior probabilities are more likely to be used if made more explicit (Bar-Hillel & Fischoff, 1981; Fischoff, Slovic, & Lichtenstein, 1979; Gigerenzer, Hell, & Blank, 1988). Interestingly, however, the implicit-explicit hypothesis does seem to be testable. One could, for example, design psychological experiments that a) take a modern experiment showing implicit conformity to Bayes’ rule and add a condition in which the numeric features are explicit, or b) take an older experiment showing explicit deviations from Bayesian rationality and add a condition in which the numeric features are implicit. According to the implicit-explicit hypothesis, such new conditions should reverse the usual trends.

Another, perhaps related hypothesis worth considering is that what upsets explicit Bayesian reasoning is the use of probabilities in problem descriptions. There is evidence that people do much better on otherwise explicit uncertainty problems if the numerical information is presented in terms of frequencies, rather than probabilities, although subjects still don’t come very close to Bayesian norms (Chase, Hertwig, & Gigerenzer, 1998; Gigerenzer & Hoffrage, 1995; Gigerenzer & Todd, 1999).

Qualifications

Ability to design, set up, and run a psychology experiment. Ability to use statistical packages such as SPSS for ANOVA.

References

Bar-Hillel, M., & Fischoff, B. (1981). When do base rates affect predictions? Journal of Personality and Social Psychology, 41, 671-680.

Chase, V. M., Hertwig, R., & Gigerenzer, G. (1998). Visions of rationality. Trends in Cognitive Sciences, 2, 206-214.

Chater, N., & Manning, C. D. (2006). Probabilistic models of language processing and acquisition. Trends in Cognitive Sciences, 10, 335-344.

Chater, N., Tenenbaum, J. B., & Yuille, A. (2006). Probabilistic models of cognition: Conceptual foundations. Trends in Cognitive Sciences, 10, 287-291.

Courville, A. C., Daw, N. D., & Touretzky, D. S. (2006). Bayesian theories of conditioning in a changing world. Trends in Cognitive Sciences, 10, 294-300.

Eddy, D. M. (1982). Probabilistic reasoning in clinical medicine: Problems and opportunities. In D. Kahneman, P. Slovic & A. Tversky (Eds.), Judgment under uncertainty: Heuristics and biases (pp. 249-267). Cambridge: Cambridge University Press.

Fischoff, B., Slovic, P., & Lichtenstein, S. (1979). Subjective sensitivity analysis. Organizational behavior and human decision processes, 23, 339-359.

Gigerenzer, G., Hell, W., & Blank, H. (1988). Presentation and content: The use of base rates as a continuous variable. Journal of Experimental Psychology: General, 14, 513-525.

Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102, 684-704.

Gigerenzer, G., & Todd, P. M. (1999). Simple heuristics that make us smart. New York: Oxford University Press.

Kahneman, D., Slovic, P., & Tversky, A. (1982). Judgment under uncertainty: Heuristics and biases. Cambridge: Cambridge University Press.

Kahneman, D., & Tversky, A. (1996). On the reality of cognitive illusions. Psychological Review, 103, 582-591.

Körding, K. P., & Wolpert, D. M. (2006). Bayesian decision theory in sensorimotor control. Trends in Cognitive Sciences, 10, 319-326.

Shultz, T. R. (2003). Computational developmental psychology. Cambridge, MA: MIT Press.

Tenenbaum, J. B., Griffiths, T. L., & Kemp, C. (2006). Theory-based Bayesian models of inductive learning and reasoning. Trends in Cognitive Sciences, 10, 309-318.

Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185, 1124-1131.

Tversky, A., & Kahneman, D. (1981). The framing of decisions and the psychology of choice. Science, 211, 453-458.

Yuille, A., & Kersten, D. (2006). Vision as Bayesian inference: analysis by synthesis? Trends in Cognitive Sciences, 10, 301-308.