Bayesian inference underlies the contraction bias in delayed comparison tasks

Abstract

Delayed comparison tasks are widely used in the study of working memory and perception in psychology and neuroscience. It has long been known, however, that decisions in these tasks are biased. When the two stimuli in a delayed comparison trial are small in magnitude, subjects tend to report that the first stimulus is larger than the second stimulus. In contrast, subjects tend to report that the second stimulus is larger than the first when the stimuli are relatively large. Here we study the computational principles underlying this bias, also known as the contraction bias. We propose that the contraction bias results from a Bayesian computation in which a noisy representation of a magnitude is combined with a-priori information about the distribution of magnitudes to optimize performance. We test our hypothesis on choice behavior in a visual delayed comparison experiment by studying the effect of (i) changing the prior distribution and (ii) changing the uncertainty in the memorized stimulus. We show that choice behavior in both manipulations is consistent with the performance of an observer who uses a Bayesian inference in order to improve performance. Moreover, our results suggest that the contraction bias arises during memory retrieval/decision making and not during memory encoding. These results support the notion that the contraction bias illusion can be understood as resulting from optimality considerations.

Figure 1. The delayed comparison task and subjects' performance.

A, The standard task. Subjects viewed a horizontal bar (L ₁) on a computer screen for 1 sec and memorized its length. After a delay period of 1 sec, during which the screen remained blank, the subjects viewed a second bar (L ₂) and were instructed to report which of the two bars was longer. The second bar, L ₂ remained visible until subjects made a response. The difference in length between L₁ and L₂ varied between −30% and +30%. Unbeknownst to the subjects, on roughly 50% of the trials, the lengths of the first and second bars were equal (L ₁ = L ₂). B, The average psychometric curve of 9 subjects. The abscissa corresponds to the difference between the two bar lengths, and the ordinate corresponds to the fraction of trials in which subjects chose L ₁ as longer than L ₂. Error bars depict standard error of the mean (SEM). Line is a least-square fit of an error function: where and . C, Average response curve of 9 subjects. Fraction of times in which subjects reported ‘L ₁>L ₂’ on the impossible trials are plotted as a function of bar length. Subjects overestimated the magnitude of the memorized L ₁ bar when it was relatively small and underestimated L ₁ when it was relatively long, consistent with the contraction bias. Each data point corresponds to 21 impossible trials per subject. Error bars depict SEM.

Figure 2. Bayesian inference and the contraction bias.

A, Response curve of a Bayesian model with infinite noise in the representation of L ₁ (no memory of L ₁) and no noise in representation of L ₂. The model reports ‘L ₁>L ₂’ in trials where L ₂ is smaller than the median of the prior distribution, and ‘L ₁<L ₂’ in trials in which L ₂ is larger than the median. This behavior arises because the posterior distribution of L ₁ is the same as the prior distribution of the bar lengths, whereas the posterior distribution of L ₂ is not influenced by the prior at all. B, Response curve of the Bayesian model with equal noise in the representations of L ₁ and of L ₂. The contribution of the prior to the posteriors of L ₁ and L ₂ is identical because the levels of noise in the two representations are equal. Thus, in trials where L ₁ = L ₂, the model reports ‘L ₁>L ₂’ at chance level independently of the length of the bars. C, Response curve of the Bayesian model assuming that there is more noise in the representation of L ₁ than in representation of L ₂. Response curve is a combination of A and B.

Figure 3. Effect of the prior on the response curve.

Assuming that noise is independent of bar length, the model predicts that the shape of the response curve is independent of the physical range of the stimuli. Thus, a lateral shift in the prior would result in a lateral shift in the response curve. Two groups of subjects completed the task in Figure 1A for two overlapping uniform priors, 50 to 200 pixels (open circles) and 150 to 600 pixels (filled circles). The response curve in the impossible trials was not significantly different between the two groups. Each data point corresponds to 14 impossible trials per subject. Error bars depict SEM. Lines are the best fit of the Bayesian model, see Materials and Methods.

Figure 4. Effect of noise on the response curve.

A, Subjects performed a modified experiment where a secondary task had to be performed between the presentations of the two bars on randomly selected 50% of the trials. Top row depicts sequence of events in trials with interference: a sequence of 4 colors was presented on the screen 500 msec after the presentation of L ₁. Each color was presented for 400 msec and subjects were instructed to memorize the sequence. 400 msec after the disappearance of the last color, a number from 1 to 4 appeared on the screen. Subjects were instructed to recall the color that corresponded to the number. B, Percentage correct in bar length comparison in the standard (black) and modified (red) trials. The ability to memorize the length of L ₁ was impaired in the modified trials compared to the standard unperturbed trials, in both the easy (±30%, left), intermediate (±15%, center) and hard (±7.5%, right) trials. These results suggest that the secondary task increased uncertainty in the memory of the length of L ₁. C, Response curve in the standard (filled circles) and modified trials (open circles). The larger slope of the response curve on the modified trials compared to the standard trials suggests that the secondary task caused an enhancement of the contraction bias. Each data point corresponds to 6 impossible trials per subject. Error bars depict SEM. Lines are the best fit of the Bayesian model, see Materials and Methods.

Figure 5. Ideal decision maker solution to the task in Figure 1A .

A, The likelihood of a representation R_i given a particular length (here L_i = 0.85, σ_i = 0.24) assuming . B, The prior distribution of bar lengths. C, The posterior distribution of L_i given a particular measurement (here R_i = 0.85), calculated using Bayes' rule. D, The probability that L ₁>L ₂ for different values of R ₁ and R ₂, computed using the posteriors. The black line corresponds to the values of R ₁ and R ₂ such that Pr(L ₁>L ₂|R ₁,R ₂) = 0.5 (here, and ). E, Response curve of the model on the impossible trials in which L ₁ = L ₂.