A representation-level algorithm for detecting spatial coincidences Jennifer Lee (jenn.laura.lee@gmail.com) New York University, 4 Washington Pl New York, NY 10003, USA Wei Ji Ma (weijima@nyu.edu) New York University, 4 Washington Pl New York, NY 10003, USA Abstract Spatial coincidences can lead to causal discoveries. We might expect to find a few ants on the sidewalk, but an unusually large cluster tips us off about the presence of a nearby food source. The leading cognitive explanation for this class of reasoning is Bayesian, but Bayesian mod- els remain notoriously agnostic about the inner work- ings of the cognitive “black box.” In this cluster detec- tion paradigm, we ask what algorithms the brain might actually implement to detect spatial coincidences in an “approximately Bayesian” way. We find evidence that the brain represents two variables of the generative model: 1) the location of a hypothesized causal source and 2) the location of the points to which it gave rise. However, we propose that the brain is limited to representing probabil- ity distributions over one but not both of these variables, resulting in strong deviations from Bayes-optimality. We find, counterintuitively, that subjects become more prone to false alarms as the amount of information increases, and our proposed cognitive algorithm accounts for this pattern. Our representation-level algorithm elucidates the cognitive processes underlying coincidence detec- tion, and helps explain our tendency to perceive causal patterns where none exist. Keywords: Causal inference; Coincidence; Probabilistic rea- soning; Perceptual grouping; Bayesian models Introduction Griffiths and Tenenbaum have proposed that a sense of co- incidence can lead to causal discoveries (Griffiths & Tenen- baum, 2007)– for instance, in the “London bombing problem,” an individual might look at a map of bombings to determine whether they are indiscriminate or targeted. Their normative Bayesian framework provides a highly unifying account of our sense of coincidence in a variety of contexts. But while the framework provides an approximate “as if” description of hu- man behaviour, it falls short of making any commitments about the mental representations and algorithms carried out by the brain during this assessment. Indeed, if we were to translate the Bayesian model for spatial coincidence detection into a representation-level model of the inner workings of the brain, we expect that the number of computations required to solve a simple spatial coincidence detection task would quickly ex- ceed a number which might plausibly be implemented by the brain. In a decision-making task, a nuisance parameter is a vari- able that does not bear directly on the decision, but that must be accounted for in order to arrive at the variable of interest. Previous studies on category learning (Fleming, Maloney, & Daw, 2013) (Murphy, Chen, & Ross, 2012) and perceptual decision-making suggest that subjects might use simplified point-estimates of intermediate nuisance parameters instead of marginalizing over their full probability distributions (though see (Shen & Ma, 2016)), resulting in particular patterns of sub- optimal behaviour. In the current study, we employ a spatial coincidence task inspired by the “London bombing” problem. Our version of the task uses the spatial distribution of pigeons in a park, affected by a pigeon feeder whose location is not directly observable. Pigeons cluster around the pigeon feeder, if she is present. The subject’s goal is to infer the presence or absence of the feeder. The generative model of the task entails two abstract parameters: 1) the location of the causal object (feeder) and 2) which of all observations “are affiliated with” (i.e., “result from”) the causal object. We ask whether these two parame- ters are represented by the brain at all, and if so, whether they are represented in full probabilistic form, or as collapsed point estimates. Our analyses rely on two important assumptions: Firstly, we assume that practice trials and explicit verbal and graphical instruction are sufficient for subjects to learn the correct generative model. This involves, for instance, the assumption that subjects learn the general statistics of where and how often the bird feeder appears. Secondly, we assume that the variables entailed by the gen- erative model (e.g., the location of an unobserved object) must be represented by the brain as either a single-point estimate (e.g., the number ‘5’), or as a full probability distribution (e.g., a ‘heatmap’ of probability over multiple locations), or else not represented at all. We then test three hypotheses about probabilistic represen- tation in the brain during a spatial coincidence detection task. On the Strong Bayesian representation hypothesis, the brain actually represents all of the abstract parameters of the gener- ative model, including their full probability distributions. On the Non-Probabilistic representation hypothesis, the abstract vari- ables of the generative model are not mentally represented at all: instead, subjects assess spatial coincidences using some heuristic metric like the mean distance between points. Lastly, the Weak Bayesian representation hypothesis holds that the 966 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0