Complex internal representations in sensorimotor decision making: a Bayesian investigation
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The past twenty years have seen a successful formalization of the idea that perception is a form of probabilistic inference. Bayesian Decision Theory (BDT) provides a neat mathematical framework for describing how an ideal observer and actor should interpret incoming sensory stimuli and act in the face of uncertainty. The predictions of BDT, however, crucially depend on the observer’s internal models, represented in the Bayesian framework by priors, likelihoods, and the loss function. Arguably, only in the simplest scenarios (e.g., with a few Gaussian variables) we can expect a real observer’s internal representations to perfectly match the true statistics of the task at hand, and to conform to exact Bayesian computations, but how humans systematically deviate from BDT in more complex cases is yet to be understood. In this thesis we theoretically and experimentally investigate how people represent and perform probabilistic inference with complex (beyond Gaussian) one-dimensional distributions of stimuli in the context of sensorimotor decision making. The goal is to reconstruct the observers’ internal representations and details of their decision-making process from the behavioural data – by employing Bayesian inference to uncover properties of a system, the ideal observer, that is believed to perform Bayesian inference itself. This “inverse problem” is not unique: in principle, distinct Bayesian observer models can produce very similar behaviours. We circumvented this issue by means of experimental constraints and independent validation of the results. To understand how people represent complex distributions of stimuli in the specific domain of time perception, we conducted a series of psychophysical experiments where participants were asked to reproduce the time interval between a mouse click and a flash, drawn from a session-dependent distribution of intervals. We found that participants could learn smooth approximations of the non-Gaussian experimental distributions, but seemed to have trouble with learning some complex statistical features such as bimodality. To investigate whether this difficulty arose from learning complex distributions or computing with them, we conducted a target estimation experiment in which “priors” where explicitly displayed on screen and therefore did not need to be learnt. Lack of difference in performance between the Gaussian and bimodal conditions in this task suggests that acquiring a bimodal prior, rather than computing with it, is the major difficulty. Model comparison on a large number of Bayesian observer models, representing different assumptions about the noise sources and details of the decision process, revealed a further source of variability in decision making that was modelled as a “stochastic posterior”. Finally, prompted by a secondary finding of the previous experiment, we tested the effect of decision uncertainty on the capacity of the participants to correct for added perturbations in the visual feedback in a centre of mass estimation task. Participants almost completely compensated for the injected error in low uncertainty trials, but only partially so in the high uncertainty ones, even when allowed sufficient time to adjust their response. Surprisingly, though, their overall performance was not significantly affected. This finding is consistent with the behaviour of a Bayesian observer with an additional term in the loss function that represents “effort” – a component of optimal control usually thought to be negligible in sensorimotor estimation tasks. Together, these studies provide new insight into the capacity and limitations people have in learning and performing probabilistic inference with distributions beyond Gaussian. This work also introduces several tools and techniques that can help in the systematic exploration of suboptimal behaviour. Developing a language to describe suboptimality, mismatching representations and approximate inference, as opposed to optimality and exact inference, is a fundamental step to link behavioural studies to actual neural computations.