On the Origins of Suboptimality in Human Probabilistic Inference

Humans have been shown to combine noisy sensory information with previous experience (priors), in qualitative and sometimes quantitative agreement with the statistically-optimal predictions of Bayesian integration. However, when the prior distribution becomes more complex than a simple Gaussian, such as skewed or bimodal, training takes much longer and performance appears suboptimal. It is unclear whether such suboptimality arises from an imprecise internal representation of the complex prior, or from additional constraints in performing probabilistic computations on complex distributions, even when accurately represented. Here we probe the sources of suboptimality in probabilistic inference using a novel estimation task in which subjects are exposed to an explicitly provided distribution, thereby removing the need to remember the prior. Subjects had to estimate the location of a target given a noisy cue and a visual representation of the prior probability density over locations, which changed on each trial. Different classes of priors were examined (Gaussian, unimodal, bimodal). Subjects' performance was in qualitative agreement with the predictions of Bayesian Decision Theory although generally suboptimal. The degree of suboptimality was modulated by statistical features of the priors but was largely independent of the class of the prior and level of noise in the cue, suggesting that suboptimality in dealing with complex statistical features, such as bimodality, may be due to a problem of acquiring the priors rather than computing with them. We performed a factorial model comparison across a large set of Bayesian observer models to identify additional sources of noise and suboptimality. Our analysis rejects several models of stochastic behavior, including probability matching and sample-averaging strategies. Instead we show that subjects' response variability was mainly driven by a combination of a noisy estimation of the parameters of the priors, and by variability in the decision process, which we represent as a noisy or stochastic posterior.


Methods
Ten subjects (3 male and 7 female; age range 21-33 years) that had taken part in the main experiment also participated in the control experiment.
The experimental setup had the same layout as the main experiment (see Methods and Figure 1 in the paper), with the following differences: (a) no discrete distribution of targets was shown on screen, only a horizontal target line; (b) in all trials the target was drawn randomly from a uniform distribution whose range covered the width of the active screen window; (c) as usual, half of the trials featured short-distance cues and the other half long-distance cues, but both types of cues had no added noise. In each trial the target was always perfectly above the shown cue, with x ≡ x cue .
Subjects performed a short practice session (64 trials) followed by a test session (288 trials). Full performance feedback was provided during both practice and test. Feedback consisted in a visual display of the true position of the target and an integer-valued score that was maximal (10 points) for a perfect 'hit' and decreased rapidly away from the target, according to the following equation: where r is the response in the trial, x is the target position, σ score is one-tenth of the cursor diameter (8.3 · 10 −3 screen units or 2.5 mm) and x denotes the floor function.
All subjects' datasets for the sensorimotor estimation session are available online in Dataset S1.

Results
Results of the sensorimotor estimation session for all subjects are plotted in Figure 1. The root-meansquared error (RMSE) of the response with respect to the true target position was on average (9.3 ± 0.8) · 10 −3 screen units for long-distance cues and (5.2 ± 0.3) · 10 −3 screen units for short-distance cues (mean ± SE across subjects). In general, the RMSE can be divided in a constant bias term and a variance term, but the bias term was overall small, on average (0.6 ± 0.5) · 10 −3 screen units, and not significantly different than zero (p = 0.26), which means that the error arose almost entirely from the subject's response variability.  Since subjects knew that the cues were fully informative about the target position, all variability in their responses originated from two sources: sensory noise (error in projecting the cue position on the target line) and motor noise. We assumed that sensory and motor noise were independent and normally distributed, and that sensory variability was proportional to the distance of the cue from the target line (Weber's law). Under these assumptions, variance of subjects' responses was described by the following formula: σ 2 response = σ 2 motor + w 2 sensory d 2 cue (S1) where w sensory is Weber's fraction and d cue is the distance of the cue from the target line. Using Eq. S1 we were able to estimate participants' sensorimotor parameters; results are reported in Table 1.
The estimated parameters in Table 1 allowed us to assess the typical impact of realistic values of sensorimotor noise on subjects' performance. First, we computed the performance of the optimal ideal observer model with added realistic noise. In order to do so, we generated 1000 subjects by sampling from the distribution of estimated sensorimotor parameters and we then simulated their behavior on our subjects' datasets according to the optimal observer model. We found an average optimality index of 0.997 ± 0.001 which is empirically indistinguishable from one. The difference in performance induced by the sensorimotor noise was analogously negligible for the simulations of other ideal observer models, such as the 'prior-only' or 'cue-only' models (see Figure 5 in the paper). These results show that motor and sensory noise had a very limited impact on subjects' performance.

Informative priors for the model comparison
The pooled estimated parameters summarized in Table 1 were used to construct informative priors for the motor and sensory parameters that were applied in our model comparison (see paper and Text S1). Bootstrapped parameters were fit with log-normal distributions with log-scale µ and shape parameter σ (which correspond to mean and SD in log space; see Figure 2). The resulting parameters of the priors were µ = log 3.4 · 10 −3 screen units, σ = 0.38 for σ motor ; and µ = log 7.7 · 10 −3 screen units, σ = 0.32 for Σ high . The prior on σ motor was used in all observer models, whereas the prior on Σ high was used only in the observer models with sensory noise (model factor S).
Using an independent experiment to construct informative priors can be thought of as a 'soft' generalization of the typical procedure that consists in directly applying independently estimated parameters to an observer model [1]. In that case, the constructed priors are delta functions on point estimates of the subjects' parameters. Here, instead, pooled measured parameters were used to compute distributions that represent realistic values for the model parameters in our task (that is, informative priors).  Figure 2. Priors over sensorimotor parameters. The experimental estimates of individual parameters for motor noise (brown dots) and sensory noise (purple dots) are used to construct informative log-normal priors for σ motor (brown line) and Σ high (purple line) in the main experiment. Error bars are 95% confidence intervals, computed via bootstrap.