Introduction

Many forms of perception or cognition require the inference of high-level categorical variables from a multitude of stimuli. For example, the spatial distribution of visual items allows the perceptual decision-making system to infer the presence or absence of latent causes in the world (a high-level categorical variable). The Bayesian framework for perceptual decision-making takes a “generative models” approach [1], positing perception as inference over a latent state of the world based on noisy data. A generative model specifies how a stimulus may be generated from the presence or absence of combinations of latent causes (or objects) in a scene. The Bayes-optimal observer knows this generative model and uses it to perform inference based on observed sensory data. The Bayesian approach is successful at capturing human decision-making data for many cases of perceptual multisensory cue integration (e.g. [2, 3]) and sensorimotor learning (e.g. [4]), and has also been successful in providing a computational account of various perceptual grouping phenomena [5].

The generative model approach has been useful in tasks that require the grouping or organization of ambiguous image fragments into latent causes. The problem of perceptual grouping can be conceptualized as an instance of the broader problem of detecting a latent cause (e.g., a partially occluded object), or delimiting its effects, based on directly observable stimuli (e.g., image elements). Bayesian hierarchical grouping [6] is one such example of a framework for perceptual grouping based on generative mixture models. Image elements are assumed to be generated by a mixture of distinct objects, with each object generating image elements according to certain generative assumptions, offering an explanation for dot clustering, contour integration, and part decomposition in visual perception. Similarly, in the contour integration literature, researchers have re-interpreted the “association field” (how strongly the visual system associates two line segments with a particular configuration of positions and orientations as belonging to one contour) [7] as the conditional link probability between two oriented line segments based on a generative model [8]. Contour integration– the ability to detect curvilinearly aligned edge configurations despite randomly oriented distractors– was found to be optimal in humans [9]. These findings suggests that the generative model approach may be a useful starting point for constructing observer models that describe how the brain performs these tasks.

Similar processes may underlie our ability to make high-level inferences about hidden states of the world based on graphically abstracted visual data. For example, in the “London bomb problem,” an individual looks at a map of bombings to determine whether they are all indiscriminate or whether some bombs are in fact aimed at a common target. In this problem, the ideal observer must consider all possible target locations, and the possibility that each bomb was generated by a targeted vs. random process. The normative Bayesian framework proposed by Griffiths and Tenenbaum [10] provides a unifying account of our sense of coincidence in a variety of contexts— including, for instance, coincidences over space (e.g., many bombs landing in a cluster on the map), time (e.g., many birthdays falling close together in date), and category (e.g., landing many heads in a row in a series of coin flips)– providing an account of how we may decide certain states are mere coincidences while others lead us to infer a common cause. Empirically, they show that a Bayesian model roughly tracks trends in human ratings of coincidence on a variety of causal inference tasks.

However, in each of these cases of perceptual grouping or causal inference, the Bayesian framework provides an approximate “as if” description of human behaviour, and falls short of making any commitments about the mental representations and algorithms actually carried out by the brain during these judgements. The success of Bayes-optimal models does not necessarily entail that humans perform full Bayesian probabilistic computations at the algorithmic or mechanistic level [11, 12].

Moreover, if one were to interpret the computational Bayesian model for spatial coincidence detection as a representation-level model of the inner workings of the brain, the number of computations required to solve a relatively simple spatial coincidence detection task would quickly exceed a number which the visual system might plausibly implement. This is especially problematic for natural visual scenes with a large number of elementary visual components. How might the brain then be implementing something that approximates the Bayesian computation in the case of perceptual grouping and spatial causal inference tasks?

In a Bayesian model, one of the most computationally taxing steps involves “marginalizing” an often high-dimensional probability distribution over a large number of generative model variables (the joint distribution) in order to arrive at the probability distribution for the specific task-relevant variable of interest (the marginal distribution). This step is often intractable in natural settings with many objects and visual features [13]. For example, in the London bombing problem described above, for a map with N visual items, the Bayesian calculation amounts to considering 2^N distinct combinations of “targeted” vs. “random” items.

In a decision-making task, these “nuisance” generative model variables are variables of the generative model that do not bear directly on the decision of the observer, but that must be accounted for (as in “marginalized out”) in order to arrive at the variable of interest in Bayesian inference. For example, in the London bombing problem, the variable of interest is the binary top-level category (whether or not the bombings are targeted), while an example of a nuisance parameter is the location of the target, if it exists. Taking into account possible identities of the nuisance parameter (e.g. hypothetical target location) is necessary on the Bayesian model to arrive at an optimal estimate about the variable of interest (e.g. presence or absence of a target). The generative model employed by an ideal Bayesian observer is often assumed to correspond to the true underlying process that generates the observations, meaning those nuisance variables and variables of interest are those determined by the true underlying process itself. Previous studies in perceptual decision-making [14, 15] suggest that subjects might use simplified point-estimates of certain nuisance variables instead of marginalizing over their full probability distributions, resulting in particular patterns of suboptimal behaviour (but see [16]). One way to potentially evade the combinatorial explosion of unique representations entailed by the Bayesian computation during spatial coincidence detection is to represent only some task parameters as full probability distributions, while committing to others as single-point estimates. This family of approximately Bayesian models may be considered “partially committal”, in that they commit to at least one of the nuisance variables of the generative model as a point estimate, while still framing the problem as one involving inference about generative model variables (rather than taking on an entirely heuristic strategy that requires no knowledge of the generative model at all).

In the current study, we employ a spatial coincidence task inspired by the “London bombing” problem to test whether spatial causal inference is optimal or may be better accounted for by “point-estimating” observer models. 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 parameters are represented by the brain at all, and if so, whether they are represented in full probabilistic form, or as collapsed point estimates.

We then test three hypotheses about probabilistic representation in the brain during a spatial coincidence detection task. On the Strong Bayesian representation hypothesis, the brain represents all of the abstract parameters of the generative model, including their full probability distributions. On the Non-Probabilistic representation hypothesis, the abstract variables 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 Point Estimate representation hypothesis holds that the variables entailed by the generative model are indeed represented by the brain, but that not all such parameters can be represented as full probability distributions: at least some are represented as single-point estimates.

Task

10 subjects were given a cluster detection task (Fig 1), in which they were asked whether a set of dots was drawn from a random uniform distribution, or from a mixture of a uniform distribution and a Gaussian. In task context, dots denoted the location of pigeons in a park. A “causal object” was introduced as an invisible “pigeon feeder” whose location was not directly observable. Subjects were instructed as follows: “On days when the pigeon feeder is present, pigeons tend to cluster around her location. But even when the feeder is present, there’s only a 50% chance that a given pigeon will be affiliated with her.” Pigeons which were not affiliated with the feeder were drawn randomly. The location of the feeder herself was drawn from a Gaussian distribution centered at the center of the screen. Subjects indicated whether a feeder was present by pressing a button. Their decision was based on the spatial distribution of pigeons on the screen. Subjects completed 20 practice trials with full feedback (“correct”/ “incorrect,” including an image showing the true underlying partition of pigeons and actual location of the feeder if there was one), followed by the main task, comprised of 2000 trials with partial feedback (“correct”/ “incorrect”).

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Fig 1. Task design and example stimuli.

Example of “feeder present” (top) and “feeder absent” stimuli (bottom) for N = 10. The stimulus is shown for 400 ms, after which subjects are prompted to respond about the presence or absence of the feeder, and then asked to rate their confidence level from 1 (least confident) to 4 (most confident).

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Observer models

Despite its apparent simplicity, the pigeon task poses a strong computational challenge to any inference system. This is because of the problem of nuisance parameters, variables that an optimal observer would marginalize over to compute a posterior distribution over the variable of interest. In our task, the generative model entails two nuisance parameters: μ, the location of the bird feeder (if present), and the “partition variable” z, a binary vector that denotes which of the N pigeons are affiliated or unaffiliated with the feeder. The ideal Bayesian observer would represent both μ and z as probability distributions. Here, we test variants of the Bayesian observer model, fully heuristic models, and models that commit to (instead of marginalizing over) at least one of these two nuisance parameters.

Given N observations x = {x₁, …, x_N}, the observer is tasked to infer category C. The log posterior ratio is (1) Since p(C = 0) = p(C = 1) = 0.5, we have (2) where we have introduced L₀ ≡ p(x|C = 0) and L₁ ≡ p(x|C = 1).

We test five families of observer models, each with a different assumption about what it is that cognitive systems represent during the task (Fig 2). In Family A (the Strong Bayesian model), the brain represents full probability distributions over both variables of the generative model. L₁ is calculated through a double marginalization: by integrating over all possible feeder locations μ and by summing over all possible z: Here, L(z, μ; x) ≡ p(x|z, μ) represents the likelihood of a particular feeder location and partition given a set of pigeon observations. The prior probabilities of each feeder location and partition are independent of one another and multiply this likelihood.

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Fig 2. Taxonomy of model families.

Family H models do not posit the representation of either location or partition.

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In Family B, the brain commits to and represents only a single feeder location (), but represents (and sums over) the full probability distribution of partitions (z): For instance, in one model, the brain simply chooses the center of mass of all of the pigeons as its committed μ. For a given pigeon observation, since no marginalization of μ is needed, and summing over different partitions involves consideration of its “affiliated” and “unaffiliated” likelihoods, can be re-expressed as a product over observed pigeons, where can be broken down into a given pigeon’s “affiliated” and “unaffiliated” terms: , where each factor is given by Eq (1).

Conversely, in Family C, the brain commits to and represents only a single partition . That is, it only represents one of all possible combinations of affiliated pigeons, while still representing the feeder location μ as a probability distribution over all possible locations: The particular set of pigeons to represent as “affiliated”, , is determined by the specifics of each of the models within Family C. For example, in model C8, the chosen is the partition that maximizes the marginal likelihood, so that . In model C9, the observer commits to through “agglomerative clustering”– an algorithm that begins by calculating the centroid of the pigeons as a reference point, then iteratively considers increasingly distant pigeons for inclusion into a “feeder affiliated” cluster.

In Family D, the brain represents both feeder location and partition as point estimates rather than probability distributions. The general form of models in this family is: For instance, one model commits to both feeder location and partition by maximizing their joint posterior: .

For all of Families A through D, once the log posterior ratio d is calculated, observer responses are modelled allowing for decision noise (Gaussian noise on the log posterior ratio with variance σ_d), and a lapse rate λ, which represents the probability of randomly choosing either category with probability 0.5. The ideal Bayesian observer would respond C = 1 when d > k where k = 0. However, we also allow for the possibility that observers have unequal utilities, costs, or incorrect prior assumptions when reporting C = 0 and C = 1, by fitting the decision criterion k as a free parameter over all trials or as four separate parameters for each of N = 6, 9, 12, 15.

Putting everything together, the probability of reporting C = 1 becomes (3) where for Family A and Φ denotes the cumulative normal distribution.

Lastly, in Family H, the brain does not represent feeder location or partition at all. Instead, observers solve the task by representing some other abstract variable outside of the generative model, like “pigeon density.” For instance, the observer might simply represent the density of the points on screen and respond “feeder present” if that number exceeds some threshold. Each heuristic model has the form where the function f represents the model’s heuristic.

Family A represents the Strong Bayesian representation hypothesis, Families B, C, and D the Point-Estimate representation hypothesis, and family H the non-probabilistic representation hypothesis. See S1 Fig for the full list of observer models and their parameters by family, and see Methods for a more detailed description of each model.

Results

Previous studies suggest that observers may combine their observations of ambiguous visual elements with knowledge about a generative model in order to perform inferences about latent causes. The aim of the present study is to investigate how observers represent and compute over the intermediate “nuisance” variables entailed by these generative models, especially when those computations become intractable. To achieve this, we designed a task paradigm in which observers are asked to detect the presence of a latent cause (a “pigeon feeder”) based on a large number of image fragments (the distribution of “pigeons” in a park). We then fit four different families of models, each of which represents the generative model’s nuisance variables as either a point-estimate or full probability distribution.

On “feeder absent” trials, the pigeons are drawn from a uniform distribution over the park. On “feeder present” trials, the pigeons are drawn from a mixture of a uniform and a Gaussian distribution, where the Gaussian is centred at the location of the hidden pigeon feeder. When the feeder is present, every pigeon has a 50% chance of affiliation with it (i.e. a 50% chance of having been drawn from the Gaussians).

We first characterize behavioural responses as a function of basic stimulus properties like “pigeon density.” As one might expect, subjects are more likely to respond “feeder present” the more densely packed the pigeons are on screen. Intuitively again, when the distance between the two closest pigeons is very small (i.e., when the two closest pigeons are very close together), subjects are also more likely to respond “feeder present.” (Fig 3) Nevertheless, subjects are surprisingly adept at telling apart trials when the feeder truly is or isn’t present even when statistics like the “mean distance” between points are controlled for across trials. This fact is evidenced by the vertical offset of the red and blue curves in Fig 3. Therefore, from the outset, we see that if a summary statistic like density or minimum distance is represented, they cannot possibly be the only thing that subjects represent in order to make accurate judgements about spatial coincidence in this task.

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Fig 3. Proportion of “feeder present” responses as a function of each distance-based heuristic.

“Feeder absent” trials in blue, “feeder present” trials in red, with colour saturation indicating the number of pigeons on a given trial (N = 6, 9, 12, 15). For each trial, the following quantities were computed: the mean pairwise distance of all pigeons (A), the maximum local density, where local density is computed by convolving a gaussian of σ = 1.4cm over the circular arena (B), the nearest neighbour distance, calculated as the mean distance between all pigeons and their nearest neighbour (C), and the distance between the two nearest pigeons (D). Points are binned in equal quantiles along the abscissa quantity such that an equal number of trials are represented in each point. For stimulus histogram distributions along each abscissa quantity, see S2 Fig.

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We next examine the trend in responses, broken down by “feeder present” and “feeder absent” trials, as the number of pigeons (N) varies. One unexpected finding is that the proportion of “feeder present” responses increases as a simple function of the number of pigeons (N) (Fig 4), even for “feeder absent” trials. In other words, as the number of uniformly drawn dots increases, subjects are more likely to sound a false alarm and report “Feeder present”. Examining model predictions, we see that this unexpected qualitative trend is at odds with the Strong Bayesian hypothesis (Fig 4)– intuitively, if we were able to reason optimally, observing more dots drawn from a uniform distribution should make us more certain that the distribution is in fact uniform. Instead, we see that the number of “feeder present” responses increases with increasing N. Henceforth, we refer to the slope of this curve as the “effect of N on false alarms.”

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Fig 4. Model fits of proportion of “feeder present” responses as a function of number of pigeons (N) denoted by shaded area; subject data denoted by solid lines.

The Bayesian model incorrectly predicts a decreasing effect of N on false alarms.

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We next ask whether the strong Bayesian model can be rescued by fitting a different prior over category (or decision criterion) for each N. We tested this Bayes variant (see Model A2, S1 Fig), which allowed for flexible, N-dependent decision criteria, to see whether the increasing effect of N on false alarms seen in the data might be accounted for by a Bayesian model with suboptimal decision criteria (see S8 and S9 Figs for the fitted decision criteria for each model). On this modified Bayesian model, subjects are calculating the decision variable (log likelihood ratio) in a perfectly Bayesian way, but may have a different decision criterion for each N, such that they are generally more permissive of “Feeder present” reports as the total number of pigeons increases. Together, S7 and S8 Figs show that the flexible criterion Bayesian model (A2), among many other models, may correctly capture the effect of N on false alarms by adopting a more liberal criterion for declaring “feeder present” as N grows larger.

We moreover tested another permissive variant of the Bayesian model which allowed for subjects to have incorrectly internalized the spread of pigeons around the feeder (σ), and the spread of the feeder around the center of the park (σ_s). We turned both of these variables into free parameters in model A3 (producing a σ- and σ_s- flexible variant of the Bayesian model), but this flexible variant performed similarly poorly to the ideal Bayes model (See Fig 6, model A3).

To further visualize model performance, we next examined “Feeder present” trials in greater detail. Recall that during “feeder present” trials, the generative model was such that any one pigeon had a 50% chance of being “affiliated” with the feeder (i.e., drawn from the feeder-centered Gaussian distribution). As a result, some “feeder present” trials have more affiliated pigeons than others (following a binomial distribution over “feeder present” trials). We would expect trials with a greater number of affiliated pigeons to be more easily assessed as “Feeder present” trials. As expected, for “feeder present” trials, subjects are more likely to correctly respond “feeder present” as the proportion of affiliated pigeons increases, and are more likely to incorrectly respond “feeder absent” as the proportion of unaffiliated pigeons increases (Fig 5). We show that the Strong Bayesian model manages to capture these general trends, but that it provides a rather crude fit to the data. The modified Bayesian model with N-dependent decision criteria better captures these data. (S12 Fig)

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Fig 5. Subject data (solid) and model fits (shaded), with standard error, showing proportion of “feeder present” responses during “feeder present” trials, as a function of the number of affiliated or unaffiliated pigeons.

Model fits shown for strong Bayes (A1) and Agglomerative Clustering (C8).

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In sum, a Bayesian model with flexible, N-dependent criterion (A2) captures subject data well on a number of dimensions. Nevertheless, it still provides an implausible process-level account of the cognitive process, as it requires marginalizing over 2^N distinct representations. We next asked whether other models which do not require this taxing marginalization over both variables of the generative model (Families B, C, and D) are able to fit subject data as well as model A2.

To summarize performance over the full range of models, we show two of the most informative aspects of model performance: the model’s predicted effect of N on false alarms, and the overall predicted subject accuracy (% correct) in Fig 6. Notably, a number of partially committal models (those of Families B, C, and D) perform just as well as the modified Bayesian model A2 on these dimensions. While the ideal Bayesian model A1 predicts subject accuracy reasonably well, it is one of the only that predicts the wrong direction of the effect of N on false alarms. Family H models may predict the correct direction of the effect of N on false alarms, but are much farther afield in terms of predictions of overall accuracy.

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Fig 6. Two informative dimensions of model predictions.

Overall predicted subject accuracy, and effect of N on false alarms (the slope of the blue shaded line in S7 Fig.

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Formal model comparison using Akaike Information Criterion (Fig 7) shows that each of the three “Weak Bayesian” families (B, C, and D) present at least one strong contender for the model that best fits subject data: B6 (in which μ is committed to via estimate of the posterior mean), C7 (in which z is committed to via maximizing the marginal likelihood), C8 (in which z is committed to via agglomerative clustering), and D9 (in which both μ and z are jointly committed to via maximizing the joint posterior). These results hold for model comparison via Bayesian Information Criterion (see S3 Fig).

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Fig 7. Model comparison (AIC).

MML denotes “Maximum Marginal Likelihood,” MMP denotes “Maximum Marginal Posterior,” MJP denotes “Maximum Joint Posterior.” See NN denotes “Nearest Neighbour.” See “Model Descriptions” under “Methods” for definitions of each model. AIC (Akaike information criterion) is an estimated score for the quality of each model, and the difference in AIC scores between models (Δ AIC) plotted provides an estimate of the quality of each model relative to the fully Bayesian model (A1) in panel A, or the best-fitting model (D9) in panel B. The four best models (highlighted) are indistinguishable via AIC and BIC (see S3 Fig).

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Lastly, it’s possible that subjects have not properly learned that the probability of affiliation of a given pigeon is 0.5, and instead hold the false belief that probability of affiliation is N-dependent. We tested family A, B, C, and D models again but allowed for probability of affiliation to vary by N (See Fig 8 for model comparison, S1 Fig for model parameters, and S11 Fig for fitted p_aff values for each N for each model). For instance, subjects may have a different internalized probability of affiliation for each N, such that they generally believe the feeder to have a higher probability of producing affiliated pigeons when the number of pigeons decreases. This might similarly work to account for the effect of N on false alarms: it would mean that subjects require higher clustering to declare “Feeder present” on low-N trials, because “Feeder present” as the total number of pigeons increases. We see that with this variation, the Bayesian model can be rescued (see Fig 8 model A1 vs. A2 and S12 Fig model A1P vs. A2P). A number of family B and D models also become indistinguishably well-fitting with the additional N-dependent free p_aff parameters. On this assumption of false belief, models B6P, D9P, 10DP, and 11DP marginally outperform A2P, C7P, and C8P.

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Fig 8. Model comparison (AIC) with probability of affiliation fitted as a separate free parameter for each of the four N conditions (N = 6, 9, 12, 15) are denoted with the addition of “P” to the model name.

MML denotes “Maximum Marginal Likelihood,” MMP denotes “Maximum Marginal Posterior,” MJP denotes “Maximum Joint Posterior.” NN denotes “Nearest Neighbour.” See “Model Descriptions” under “Methods” for definitions of each model. AIC (Akaike information criterion) is an estimated score for the quality of each model, and the difference in AIC scores between models (Δ AIC) plotted provides an estimate of the quality of each model relative to the fully Bayesian model (A1) in panel A, or the best-fitting model (B6P) in panel B. The five best models (highlighted) are indistinguishable via AIC and BIC (see S4 Fig) Adding a flexible N-dependent criterion can help rescue the Bayesian model (see model A2 compared to A1), and adding 4 flexible N-dependent probability of affiliation parameters increases goodness of fit for all models. See S1 Fig for more information about the parameters included in each model.

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Among the basic (non-false-belief) set of winning models, Agglomerative clustering (C8) is the only one of the four winning models that evades the combinatorially explosive marginalization step over z while also providing a potential process-level model of mental representation in the brain, and may be able to account for choice, reaction time, and confidence rating data simultaneously. Fig 9 and S5 Fig show that the approximate log likelihood ratio given by the agglomerative clustering algorithm tracks both reaction times and confidence ratings in a similar manner to that of the Bayes-optimal log likelihood ratio, and Fig 9 suggests that the number of iterations performed by the agglomerative clustering algorithm is comparable to the reaction times of participants for trials binned by agglomerative clustering approximated LLRs. Importantly, Figs 5 and 4 demonstrate that the agglomerative clustering model is able to provide a superior fit to response data compared to the Bayesian model (A1) and an indistinguishably good fit to the other three basic winning models (Fig 7).

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Fig 9.

(A) Subject reaction times and (B) Predicted reaction times (based on the number of iterations of the agglomerative clustering algorithm), according to the log likelihood ratios (LLR) of the Bayesian model, binned into 10 LLR quantiles (equal number of trials in each plot point).

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In Agglomerative Clustering, the observer commits to a particular z rather than representing a full probability distribution. We choose this z by picking a single point as the cluster “seed.” The cluster is hypothesized to belong to the causal source. One iteratively adds the next-nearest point to the cluster, each time evaluating the log likelihood ratio which results from that particular z. We continue adding points until the log likelihood ratio no longer increases, resulting in some spatially contiguous set of points hypothesized to belong to the source if such a source exists, represented by the committed z. The z-dependent log likelihood ratio is then calculated as a decision variable with noisy threshold. This algorithm is loosely based on work by Heller and Gharamani on agglomerative hierarchical clustering [17]. Our algorithm is sequential rather than hierarchical, in that a single cluster is defined and single observations are merged to the existing cluster via Bayesian hypothesis testing (i.e., the evaluation of the log likelihood ratio). For a histogram comparing the Bayesian LLR to the Agglomerative Clustering approximation of the LLR, see Fig 10.

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Fig 10. Histogram of decision variables across trials, pooled across all subjects (2000 trials x 10 subjects, plotted in 35 equal bins from -10 to 25).

The decision variable is the log likelihood ratio for Bayes (A1), and an estimated log likelihood ratio for agglomerative clustering (C8), calculated based on the stimulus on each trial. Note that as the number of pigeons increases during feeder-absent trials, the distribution of the decision variable for agglomerative clustering shifts to the right (light blue to dark blue lines), whereas the Bayesian decision variable exhibits the opposite trend.

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S8 Fig shows fitted decision criteria for each model across subjects. There are a number of factors that contribute to the fitted decision criterion across models– for instance, the goodness of the committed variables, and the subject’s fitted p_aff beliefs. Some models commit to a particular μ or z that is particularly good, thereby biasing the log posterior ratio estimates to higher values. Overall positive decision criteria might therefore be expected if the committed variables are quite good– for example, Model 9 evaluates the LLR at only the μ and z that maximizes the joint posterior, meaning the estimated LLR for a given trial will shift upwards (see S10 Fig, panel A). Similarly, models with p_aff > 0.5 may cause the decision variable distributions (and therefore the fitted decision criteria) to shift downwards. If the probability of affiliation for the feeder present (C = 1) hypothesis is higher than 0.5, then it should take a greater amount of dot “clustering” to support the feeder present hypothesis, and the observer’s decision variable on a given trial will shift lower, in the direction of the feeder absent (C = 0) hypothesis (see S10 Fig, panel B). Note that the decision variable’s zero-point is only meaningful for the perfect Bayesian model (since a Bayesian observer should set their decision criterion to zero), whereas observers likely set a different decision threshold when the log posterior ratio is estimated (as in all other models), in compensation for the fact that these estimates produce shifts in both C = 0 and C = 1 distributions.

Decision variables (ex, the Bayesian log likelihood ratio, or suboptimal estimates of it) largely tracked confidence ratings and reaction times in expected ways. Confidence ratings reliably tracked Bayesian log likelihood ratios in a monotonic manner (S5 Fig). High-confidence “Feeder present” responses corresponded to high positive log likelihood ratios, while high-confidence “Feeder absent” responses tracked low negative log likelihood ratios. Reports of higher confidence were also related to shorter reaction times (S6 Fig). Trials which fell into the highest positive LLR quantile (highly clustered pigeon configurations which greatly favour “Feeder present” responses) were also related to shorter reaction times.

Notably, Agglomerative Clustering, being the only model with a structured iterative “time course”, is able to predict trends in reaction time data (Fig 9). When binning stimuli based on their estimated LLRs, trends in the number of iterations required of the agglomerative clustering algorithm mirror trends in subject reaction times, as seen in (Fig 9. For most subjects (8 out of 10), there is a very weak but significant (p < 0.001) ranked correlation between the reaction times and the number of iterations of the agglomerative clustering algorithm across trials (across subjects, mean Spearman ρ across subjects = 0.12, SEM = 0.02).

Discussion

In this paper, we asked how people infer the presence or absence of a latent cause based on a spatial configuration of items, especially when optimal inference would require the integration of a computationally intractable number of world states. We first asked whether observer models that make inferences over aspects of the true generative model (i.e., Bayesian models and point-estimating variants) accounted better for human behaviour than models based on simple visual heuristics, such as mean density or minimal distance. We found that this was the case, with the heuristic models deviating greatly from the data. We fitted three families of partially committal or point-estimating models, in each of which either or both of the nuisance variables of the generative model were represented as point estimates rather than probability distributions. We found that the Bayesian model (which represents both generative model variables as full probability distributions) exhibits qualitative deviations from subject behaviour. Notably, the Bayesian model fails to capture the trend that observers are more likely to falsely detect a latent cause as the number of visual elements increased. Several point estimate observer models—at least one from each family—can account for these deviations, as well as for fine-grained subject behaviour in its entirety. Therefore, point estimate representation of certain generative model variables may be how the brain performs causal inference, especially when full marginalization over these variables requires accounting for a large number of world states.

In context of this complex latent cause detection task, the effect of the number of observations on false alarm rates was strong. Recent work tested the ability of subjects to infer a hidden low-level variable (the proportion of red vs. blue team supporters leaving a given airplane) based on both observations of differently sized samples, and an inferred higher-level contextual variable (the general red or blue-team bias for a set of planes). [18] show that inferred context is integrated with observations to inform confidence estimate, and notably that the sample size of previous observations guides context reliability. The sample size of observations may therefore be a factor that subjects take into consideration when performing hierarchical latent variable inference in the real world, affecting both choice and confidence judgements. In the case of our study, however, subjects suboptimally set a more liberal detection criterion as sample size increased, perhaps reflecting certain real-world prior beliefs that a greater number of observations provides evidence for the presence of a latent spatial cause, regardless of the distribution of the observations.

Among the basic set of models (that do not make false belief assumptions about the probability of feeder affiliation), the four best-performing models (B6, C7, C8, D9) were functionally indistinguishable according to AIC and BIC. However, taking seriously the computations required of some of these winning models at the algorithmic level would still mean granting the brain upwards of 2^N evaluations in the case of marginalization over z. The agglomerative clustering account (C8) stands out because the number of operations required to commit to a plausible partition is less than , making it a more attractive candidate as a biologically plausible algorithm compared to any of the other three statistically indistinguishable partially Bayesian models. The algorithm further coheres with intuitions that the least plausible possibilities of z may not be represented or considered at all during the decision-making process. The iterative nature of the agglomerative clustering algorithm moreover allows it to predict response times, whereas the Bayesian model and variants thereof do not provide an immediate mechanism to do so.

However, the present study only reflects a small sampling of the possible models in each tested family. This means that we have to be cautious when making any inferences about which generative model variables may be committed to as a point estimate. The study is also limited in its ability to provide an explanation for why the brain may represent certain generative model variables as point estimates while representing others as full probability distributions. Nevertheless, the models tested represent a new and more or less principled category of approximately Bayesian algorithms, distinct from sampling and variational algorithms.

These results advance an understanding of perceptual grouping and causal inference based on spatially-related image elements, suggesting that, while observers likely represent the structure of the generative model, they probably do not represent full probability distributions and marginalize over each nuisance variable. Committing to the representation of some nuisance variables as point estimates may be one strategy employed by the brain to avoid computationally costly marginalizations in real-world visual environments, if these marginalizations are in fact shown to be costly to the brain. Though the present study cannot shed light on why some variables may be represented as full probability distributions but not others, testing algorithms that implement these partially committal models may be a fruitful inductive modelling approach for other perceptual decision-making tasks, as they may help reformulate intractable “as if” Bayesian computations in terms of a set of tractable process-level algorithms potentially used by the brain.

Methods

Model derivations

Below, we introduce the notation used for all models and formalize the true generative model that determines how observations are in fact generated based on stimulus category. Then, we formalize each of the observer models within the five observer model families, each of which perform full (family A) or partial (families B, C, D) generative model inferences about stimulus category based on observations, or which simply employ a heuristic decision model (family H).

Notation

All location variables have two dimensions, but we do not denote them as vectors (with boldface). We reserve boldface notation for vectors corresponding to N observations.

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Supporting information