[This post summarizes “Confirmation in the cognitive sciences: The problematic case of Bayesian models”, by Eberhardt and Danks (2011)]
Eberhardt and Danks begin this paper with a nice summary of the experimental set-up of a lot of Bayesian cognitive science papers. First, you try to induce uniform priors in all the subjects, using some cover story. Then, you present some evidence; then, you somehow make subjects identify what they now take to be the most likely hypothesis. Possible ways of doing this include free response (just asking them), and forced choice (making them bet on one hypothesis). Bayesians model this change over time in to indicate that subjects are learning in light of evidence, which the Bayesian models as Bayesian updating: updating a prior probability to derive a posterior probability, in accordance with Bayes Rule.
Now, it’s hard (with a forced-choice task at least) to get an individual-level read on what exactly people’s posterior probability distributions are. Instead, what we get is an aggregate distribution from the entire group of people. What do I mean by this? Let’s suppose that each subject has been forced to pick either hypothesis A or hypothesis B as more likely. If you pick the right one, you get $1. And let’s suppose that the rational posterior would be to have a 0.8 credence in A, and 0.2 in B.
Now, if people are epistemically rational, they should all converge on this same posterior–0.8 to A and 0.2 to B. And if they instrumentally rational, in the forced choice task they should all pick A. Why? For all of them, picking A maximizes expected reward. So, the chart of responses for subjects should just show 100% of responses favoring A. But this is not what we see. On the group level, we see 80% of people choosing A, and 20% of people choosing B.
Why would this be? As Danks and Eberhardt put it: “These analyses thus imply the puzzling conclusion that the population as a whole acts as a rational Bayesian learner, but the individual learners do not.”
Eberhardt and Danks argue that we have four jointly inconsistent claims of the Bayesian modeling paradigm.
1. Individuals are Bayesian [explicit claim]
2. People choose the option that maximizes expected utility given their beliefs [requirement of rationality]
3. Experiments successfully constrain participants’ priors and utilities [methodological assumption]
4. The distribution of participant responses matches the model posterior [empirical data].
They consider two ways of rejecting claim 2, i.e. two ways of characterizing circumstances in which the behavior that we see–probability matching–could be instrumentally rational. One reply is that probability matching can be rational in cases of competition and resource constraints where you can’t communicate (apparently fish probability match according to food distribution). But as Eberhardt and Danks point out, these situations are not present in the lab. Another is to appeal to PAC-Bayesian theory, but they say that this framework should also predict that probability matching is sub-optimal in these particular tasks.
Next, they consider rejecting claim 3.
Strong rejection: people in these experiments actually do not have uniform priors and utilities (401). But as they point out, this is just to undermine the whole methodology of these experiments (402).
Weak rejection: variation of priors and utilities in the population is small, but sufficient to explain the empirical results. That is, individual variation in utilities will account for the pattern we see. This is an empirical question: “It is unknown whether there is a plausible distribution for individual deviations that implies identity between the model posterior and the response distribution.” But I don’t think it is likely.
Finally, you can also just reject that the artificial settings of the laboratory are sufficient to focus people on the task. In the real world, they would care more, and behave more rationality. This is always a possibility in psychological experiments. But then to be a Bayesian, you would have to ditch the psychological evidence, and maintain (without any evidence) that people are Bayesian in everyday life. This reply undercuts the whole Bayesian modeling program.
Eberhardt and Danks conclude by asking why we want Bayesian models in the first place. Sure, you could always justify Bayesianism as a compact and unifying way of summarizing a bunch of psychological data. But to really justify Bayesian models (over and above a mere paradigmatic justification), you either want to say that Bayesian models unify all these domains either because:
-there is some common mechanism across these domains that explains why it reoccurs
-it is optimal in all these domains
In short: “A Bayesian model must be rational for it to play a role in any substantive explanation at all.” But Eberhardt and Danks are not convinced that the data really do fit into some rational pattern.
Experience Machines 