A Bayesian framework for modeling intuitive dynamics

A Bayesian Framework for Modeling Intuitive Dynamics Adam N. Sanborn (asanborn@gatsby.ucl.ac.uk) Gatsby Computational Neuroscience Unit, University College London London, UK Vikash K. Mansinghka (vkm@mit.edu) Brain and Cognitive Sciences Department, Massachusetts Institute of Technology Boston, MA USA Thomas L. Griffiths (tom griffiths@berkeley.edu) Department of Psychology, University of California, Berkeley Berkeley, CA USA Abstract were moving independently. Studies in this area have col- lected data on how changes in physical variables relate to the judgments of causality (e.g., Chaput & Cohen, 2001; Spelke, 1994; Schlottmann & Anderson, 1993). People have strong intuitions about the masses of objects and the causal forces that they exert upon one another. These in- tuitions have been explored through a variety of tasks, in par- ticular judging the relative masses of objects involved in col- lisions and evaluating whether one object caused another to move. We present a single framework for explaining two types of judgments that people make about the dynamics of objects, based on Bayesian inference. In this framework, we define a particular model of dynamics – essentially Newtonian physics plus Gaussian noise – which makes predictions about the tra- jectories of objects following collisions based on their masses. By applying Bayesian inference, it becomes possible to reason from trajectories back to masses, and to reason about whether one object caused another to move. We use this framework to predict human causality judgments using data collected from a mass judgment task. Keywords: collisions; Bayesian modeling; perception; causality; mass judgments Following the ground-breaking work of Michotte (1963), the perception of collisions has been studied as a way of de- termining how people infer unobservable variables from ob- servable variables. In visual perception of collisions, the ob- server can see the movements of the objects, but needs to infer the hidden properties of the objects (Chaput & Cohen, 2001; Cohen & Ross, in press; Gilden & Proffitt, 1989; Runeson, 1977; Runeson, Juslin, & Olsson, 2000; Schlottmann & An- derson, 1993; Todd & Warren, 1982). There are many pos- sible questions that could be asked of observers. Two of the most commonly used tasks are judging which object is heav- ier and judging whether a collision occurred. Judgments of the ratio of masses of colliding objects have motivated arguments about the perceptual invariants. In a mass judgment task, subjects are presented with two objects colliding on a screen and are asked to choose which object has greater mass. People make characteristic patterns of errors, which have led researchers to propose that human mass ra- tio judgments are based on heuristics (e.g., Cohen & Ross, in press; Gilden & Proffitt, 1989; Todd & Warren, 1982), though other researchers argue that the correct mass ratios are com- puted for experienced observers (Runeson, 1977; Runeson et al., 2000). Michotte (1963) has directly motivated a second area of research. In this similar task, subjects had to deter- mine whether two objects had collided or whether the objects Despite the similar tasks used in mass judgments and judg- ments of causality, these two lines of research have pro- ceeded almost always independently (but see Kaiser & Prof- fitt, 1984). Our goal is to present a unifying explanation for these two types of judgments. We use a simple probabilistic model of physical dynamics that makes the same assumptions about the probability with which different physical properties of objects are likely to be observed, but differs in the hypothe- ses to be evaluated and the data that are used. Unlike previous models of mass judgments, we assume that people are not us- ing heuristics, but instead are making their decisions based on the underlying physical laws. Using these laws, we assume subjects are making the optimal decision based on noisy ob- servations of the velocities of the objects. The plan of this paper is as follows. First we describe the mass and causality judgment tasks, along with the physical laws that dictate the optimal decisions in these tasks. Next we generalize the optimal decisions to a more realistic situa- tion in which the observed velocities are perceived with noise. Our generative probabilistic model applies to both mass and causality tasks. Next we replicate two venerable experiments: one for mass judgments and one for causality judgments. We fit our model to the data from the mass judgment task, and show that we can accurately mimic human performance in this task. In addition, using the data from the mass judg- ment experiment, we successfully predict how subjects will respond in the causality judgment task. Finally, we discuss how our model relates to other work in these two areas. Task Descriptions and Physical Laws The mass judgment and causality judgment tasks are similar experimental paradigms, with a few key differences. In this section we describe the variables in the two tasks and what a classical model of mechanics would predict.