On Bayesian integration in sensorimotor learning: Another look at Kording and Wolpert (2004)

Kording and Wolpert (2004), hereafter referred to as KW, describe an experiment where subjects engaged in a repeated task entailing movements of their finger. Subjects strove for accuracy in the stochastic environment and, on some trials, received mid-trial and post-trial feedback. KW claims that subjects learned the underlying stochastic distribution from the post-trial feedback of previous trials. KW also claims that subjects regarded mid-trial feedback that had a smaller visual size as more precise and they were therefore more sensitive to such mid-trial feedback. KW concludes that the observations are consistent with optimal Bayesian learning. Indeed, under mild assumptions, it is well-known that Bayesian learners will have posterior beliefs that converge to the true distribution. We note that the KW analysis is based on data that had been averaged across important trial-specific details and averaged across trials. Averaging data disregards possibly valuable information and its dangers have been known for some time. Notably, the KW analysis does not exclude non-Bayesian explanations. When we analyze the trial-level KW data, we find that subjects were less--not more--sensitive to mid-trial feedback when it had a smaller visual size. Our trial-level analysis also suggests a recency bias, rather than evidence that the subjects learned the stochastic distribution. In other words, we do not find that the observations are consistent with optimal Bayesian learning. In the KW dataset, it seems that evidence for optimal Bayesian learning is a statistical artifact of analyzing averaged data.