6. Explicit representation of the posterior and it’s evolution in time in a human estimation task

A. Aldo Faisal aaf23@cam.ac.uk Daniel M. Wolpert wolpert@eng.cam.ac.uk

CBL, Dept. of Engineering, University of Cambridge, Cambridge, UK

When driving in traffic, our perception provides snapshots of different parts of the surrounding world at different instances in time (e.g. traffic ahead and behind). This sensory information has to be combined requiring us to estimate how the surroundings may have changed in absence of new sensory information. It is an open question what internal representations and computations are used (and which should be used) to achieve this, which we address here computationally and experimentally. Here, we show that humans estimating the position of an occluded moving object, retain second-order information about the estimate’s posterior probability distribution and evolve their estimate’s uncertainty over time in a way that is consistent with an explicit representation of the posterior. We carried out a psychophysics experiment to address this question (6 subjects, 1200 trials each). Subjects observed a particle moving away in a random direction on a linear trajectory (at one of two different velocities). After 200 ms the object vanished and continues moving for T=200-900 ms. After this time the radial position of particle is shown (to eliminate bias of the position estimation task by uncertainty about the time passed) and subjects had to indicate the angular position of the particle and their perceived uncertainty (”error bars”) about their estimate. Subjects gained reward if the particle was within error bars or lost points reward if not. Reward increased as error bars were made smaller and vice versa. At the end of each trial subjects received feedback about the particle position and the score achieved. We found that, subjects estimate’s of the position was related to a subject specific level of uncertainty. Moreover, as the time in which the object moved invisibly increased subject’s uncertainty scaled as a square root power-law of time (σ = T0.540.03). This scaling is consistent with an explicit representation of the estimate in terms of a probability distribution updated in time by a stochastic partial differential equation of the Fokker-Planck type. However, subject’s actual sensory estimation error (thus their performance and not their belief about their performance) scaled linearly with time (σ = T0.950.08, yet subjects did not correct the way they estimated how their uncertainty evolved over time (N=1200 trials). This suggesting that human’s may use a fairly generic model of how the unobserved world changes: in the form of random additive noise. This is appealing from a computational perspective as it allows to retain the Bayesian framework in a straightforward manner by updating the relevant probability distributions to the same instant in time using stochastic calculus.