**Abstract**:

**Abstract**
**The two-alternative forced-choice (2AFC) task is the workhorse of psychophysics and is used to measure the just-noticeable difference, generally assumed to accurately quantify sensory precision. However, this assumption is not true for all mechanisms of decision making. Here we derive the behavioral predictions for two popular mechanisms, sampling and maximum a posteriori, and examine how they affect the outcome of the 2AFC task. These predictions are used in a combined visual 2AFC and estimation experiment. Our results strongly suggest that subjects use a maximum a posteriori mechanism. Further, our derivations and experimental paradigm establish the already standard 2AFC task as a behavioral tool for measuring how humans make decisions under uncertainty.**

*maximum a posteriori*(MAP) choice (Green, 1966). An alternative theory proposes that the brain cannot compute the MAP but can instead get samples from these believed waiting-time distributions. By using many such samples, the brain can estimate the expected wait times for each line and make a choice. This is called the sampling hypothesis (Ackley, Hinton, & Sejnowski, 1985; Hoyer & Hyvärinen, 2003). Both MAP and sampling can be viewed as normative models of decision making (Sakai & Fukai, 2008b; Vul, Goodman, Griffiths, & Tenenbaum, 2009; Wozny et al., 2010; Duda, Hart, & Stork, 2012), making either an informative candidate for how the brain makes decisions.

*s*

_{1}and

*s*

_{2}. We denote the subject's response as

*z*(

*z*= 0 if false,

*z*= 1 if true). By asking the subject to perform many trials while systematically manipulating the disparity

*δ*=

*s*

_{2}–

*s*

_{1}, we can measure a psychometric curve, characterizing the probability of a subject's decision as a function of the disparity. As is standard, we fit this curve with the cumulative normal distribution: where erf denotes the error function and PSE is the probability of subjective equality: the subjective bias in stimulus disparity. One of the central measures obtained from fitting the psychometric curve is the JND, which quantifies how different the stimuli must be for subjects to reliably discriminate between them. Though the definition of JND varies slightly across studies, here we define it to be the best fit of

*σ*

_{JND}to behavior. This JND is of central importance across much of psychophysics.

*s*

_{1}and

*s*

_{2}and instead have to rely on noisy sensations of the cues,

*c*

_{1}and

*c*

_{2}. Formally, our sensory information induces the likelihood of all possible values of the stimulus given the cues,

*P*(

*c*

_{1}|

*s*

_{1}) and

*P*(

*c*

_{2}|

*s*

_{2}). Subjects' percepts can be thought of as a belief distribution over stimuli values given the cue, which we can describe as a posterior distribution,

*P*(

*s*|

*c*). Applying Bayes' formula, we have where

*P*(

*s*) is the subject's prior expectation of stimulus values. Now we can formally interpret the 2AFC task as a decision,

*z*, based on beliefs about the stimulus values,

*P*(

*s*

_{1}|

*c*

_{1}) and

*P*(

*s*

_{2}|

*c*

_{2}). This allows us to predict different distributions,

*P*(

*z*|

*δ*), for different candidate decision-making mechanisms.

*P*(

*c*

_{1}|

*s*

_{1}) =

*s*

_{1},

*P*(

*c*

_{2}|

*s*

_{2}) =

*s*

_{2},

*P*(

*s*) =

*μ*,

*δ*is the difference between the expected posterior beliefs in the two stimuli: Therefore, a subject's response is determined through the random variable,

_{MAP}*δ*, which is defined by the two random cues,

_{MAP}*c*

_{1}, and

*c*

_{2}. Next, using

*P*(

*c*

_{1}|

*s*

_{1}) and

*P*(

*c*

_{2}|

*s*

_{2}), we can integrate out

*c*

_{1}and

*c*

_{2}to obtain the probability distribution for

*δ*in terms of the two experimental variables,

_{MAP}*s*

_{1},

*s*

_{2}: Finally, we compute the probability of a subject's response:

*s*

_{1}and

*s*

_{2}, and cannot be rewritten in terms of their difference,

*s*

_{2}–

*s*

_{1}; that is, the psychometric curve is actually a surface. Also note that the point of subjective equality (PSE) is found when

*σ*

_{1}=

*σ*

_{2}=

*σ*, then the psychometric curve collapses to the more familiar form defined by the difference of the two stimuli,

*δ*: To summarize, if subjects choose according to the MAP hypothesis, then Equation 9 accurately models their behavior, whereas Equation 1 is used to fit their behavior. By comparing terms with Equation 1, we can define the experimentally derived JND in terms of the precision of a subject's likelihood: This result is implicitly used by most studies that employ the 2AFC task to measure sensory uncertainty. If these assumptions are true, then the prior indeed has no influence on 2AFC behavior or the JND, making the approach particularly attractive (Figure 1B). However, as we demonstrate later, alternative decision-making mechanisms predict distinct results.

*k*samples from the posterior distributions

*P*(

*s*

_{1}|

*c*

_{1}) and

*P*(

*s*

_{2}|

*c*

_{2}), respectively; that is,

*s*

_{1}

*∼*

_{i}*P*(

*s*

_{1}|

*c*

_{1}). Here again, if we assume Gaussian distributions for the likelihoods and prior we can rewrite the choice as follows: where

*δ*=

_{sample}*c*

_{1}, and

*c*

_{2}; and

*P*(

*c*

_{2}) is defined similarly. Just as with the MAP decision, we can integrate out

*c*

_{1}and

*c*

_{2}using the likelihoods

*P*(

*c*

_{1}|

*s*

_{1}) and

*P*(

*c*

_{2}|

*s*

_{2}), respectively, to obtain the probability distribution for

*δ*in terms of the two experimental variables,

_{sample}*s*

_{1},

*s*

_{2}: where the mean and variance are

*s*

_{1}and

*s*

_{2}. The PSE is the same as in the MAP case (Equation 7). The JND is different, however, and given by Only when the two likelihoods have the same variance,

*σ*, does the psychometric curve collapse to a form defined by the difference of the two stimuli,

*δ*: We note several features of this result, first of which is the appearance of the term from the prior,

*σ*. By matching terms with Equation 1, the experimentally derived JND is not merely a subject's sensory accuracy but rather a combination of both sensory and prior uncertainties: This result is in stark contrast with the traditional interpretation of the 2AFC task. We see that when a subject's prior is certain (relatively small

_{s}*σ*), the JND increases (Figure 1B). In the limit of an infinite number of samples, the JND under the sampling hypothesis is equivalent to the MAP prediction. In the case of a uniform prior (or, more precisely, in the limit where the prior's variance tends to infinity), the JND is

_{s}*k*= 1, which is the so-called matching hypothesis (Vulkan, 2000; Wozny et al., 2010). Matching is equivalent to the scenario where subjects choose between their choices with a probability that is proportional to the probability of being correct; that is,

*z*= 1, with probability

*P*(

*s*

_{2}>

*s*

_{1}|

*c*

_{2},

*c*

_{1}) and a JND of

*z*. We examined this latter hypothesis (see Supplemental Appendix), and while there is no closed-form expression for the resulting distribution analogous to Equation 19, we found that this model's influence on the JND was nearly identical to that of our model. Additionally, subjects could have a prior belief that is different from the correct Gaussian distribution in our experiment (again, see Experimental protocol). To this end we examined the influence of a uniform prior over stimuli locations. Here too we found nearly indistinguishable predictions from our Gaussian model (see Supplemental Appendix). Thus, we suggest that the results we present here for sampling may be typical for alternative interpretations.

*s*

_{2}–

*s*

_{1}that results in

*P*(

*z*= 1|

*s*

_{1},

*s*

_{2}) = 0.5 is given by Equation 7. From this equation, we can obtain a relationship between either stimulus value and the PSE. For example, we can isolate the value of

*s*

_{2}that corresponds to the PSE as and then substitute this value to rewrite the PSE in terms of

*s*

_{1}as follows: We see how this linear relationship changes with the variance of the prior. We can exploit this relationship as a valuable control to verify that subjects use their subjective prior during our 2AFC experiments.

*1*for the first coin,

*2*for the second coin). The data collected during the 2AFC trials were used to construct psychometric curves. The curves allowed us to measure the subjects' JNDs and to verify that the subjects used a consistent prior across the estimation and 2AFC trials.

*ŝ*is the estimated coin location,

*c*is the cue,

*μ*is the mean of the prior, and

*r*

_{reliance}is the relative reliance on the likelihood, defined as

*σ*

_{sensory}≪

*σ*

_{prior}), then

*r*

_{reliance}≈ 1 and the expected stimulus position would be at the center of the sensory feedback/cue location. Similarly, if the prior is very important relative to sensory feedback (i.e.,

*σ*

_{sensory}

_{≫}

*σ*

_{prior}), then

*r*

_{reliance}≈ 0 and the estimated stimulus position would be at the prior's mean. The reliance on the likelihood is an indirect measurement of the variance of a subject's prior (Kording & Wolpert, 2004; Berniker et al., 2010; Vilares et al., 2012).

*c*, and the respective subjects' estimation of the stimulus position,

*s*. By using ordinary least squares estimation, it is possible to estimate

*r*

_{reliance}without bias (Hastie, 2009). Similarly, we can manipulate Equation 23 to obtain the mean of the prior.

*δ*is the discrepancy between stimuli and

*z*is the decision of the subject. We find the values of PSE and

*σ*

_{JND}by a maximum-likelihood estimation algorithm.

*λ*probability. The psychometric curve can then be modified to accommodate this change as follows:

*t*tests were two-sided and significance level was set to 0.05.

*k*sample values drawn from each belief distribution (see Materials and methods). If the brain uses this mechanism, the JND does not measure sensory precision. Rather, the JND is influenced by sensory noise and prior beliefs (see Table 1 and Figure 1B, C). We also note the special case where

*k*= 1 (Table 1), which is the matching hypothesis (Estes, 1950; Myers, 1976; Vulkan, 2000; Wozny et al., 2010).

*σ*

_{prior}), the JND increases (see Figure 1C). Intuitively, we can interpret this as follows: As the prior becomes more and more certain, sensory information becomes less relevant, and distinguishing a difference between the two stimuli requires increasingly large differences. By investigating multiple variations of these decision-making hypotheses, we can derive a corresponding interpretation of the experimental JND. In the next section we present results from an experiment that exploits these differing interpretations.

*t*(135) = 1.04,

*p*= 0.3 (see Figure 3B for example subject). Pooling each subject's data across days, we found differences across subjects,

*F*(6, 18) = 3.26,

*p*= 0.02, but not conditions,

*F*(3, 18) = 0.06,

*p*= 0.97 (two-way analysis of variance, or ANOVA). This suggested that overall the subjects learned the correct, condition-independent, experimentally defined mean.

*F*(1, 19) = 108.1,

*p*< 0.01, and likelihoods,

*F*(1, 19) = 21.74,

*p*< 0.01, but not subjects,

*F*(1, 19) = 0.874,

*p*= 0.53. This suggested that all subjects reacted to the four conditions early on, and did so similarly. Examining the reliance on the likelihood across 250-trial bins for the first day, we found that the distance from the optimal slope significantly diminished for the NP-WL condition,

*t*(26) = −2.44,

*p*

_{one-sided}= 0.01, but not for the other conditions; NP-WL:

*t*(26) = 2.51,

*p*

_{one-sided}= 0.99; WP-NL:

*t*(26) = −1.37,

*p*

_{one-sided}= 0.09; WP-WL:

*t*(26) = −0.78,

*p*

_{one-sided}= 0.22. Overall, these results suggest that subjects' behavior converged quickly and that no significant learning was observable even during the first day.

*F*(1, 123) = 324.8,

*p*< 0.01, likelihoods,

*F*(1, 123) = 91.1,

*p*< 0.01, and subjects,

*F*(6, 123) = 4.42,

*p*< 0.01, but not across days,

*F*(4, 123) = 1.43,

*p*= 0.22. Therefore, subjects' overall responses did differ, but these differences were dominated by the changes across prior and likelihood conditions. Pooling the data across subjects and days, the overall reliance on likelihood was as follows [mean ± SE (optimal),

*n*= 34]: NP-NL: 0.76 ± 0.014 (0.91); NP-WL: 0.47 ± 0.016 (0.39); WP-NL: 0.96 ± 0.007 (0.99); and WP-WL: 0.92 ± 0.006 (0.94). We note here that on the whole, both within and across subjects, these numbers are all statistically distinct from the theoretical optimum. Regardless, we are able to precisely characterize each subject's prior. Furthermore, our analysis demonstrated that across conditions, the subjects had stable priors across days.

*t*(67) = 1.71,

*p*= 0.091, averaged across subjects, days, and conditions. By performing a three-way ANOVA, we found that there were no significant differences in PSE across priors,

*F*(1, 56) = 1.74,

*p*= 0.19, days,

*F*(4, 56) = 0.45,

*p*= 0.73, or subjects,

*F*(6, 56) = 2.25,

*p*= 0.05. Overall, these results suggest that subjects did not have strong biases in their decisions.

*F*(6, 56) = 1.38,

*p*= 0.23, or days,

*F*(4, 56) = 1.27,

*p*= 0.28. There were significant differences across priors,

*F*(1, 56) = 34.3,

*p*< 0.001, and the measured JNDs were larger in the wide prior condition (WP-WL; the data from the WP-NL trials are used in the analysis in the subsequent section). This suggests that subjects were self-consistent across days, but the prior did have some influence.

*t*test

*t*(6) = −2.35,

*p*= 0.028 (Figure 5B). Thus, subjects used the prior during the 2AFC task.

*t*(8) = 3.19,

*p*

_{one-sided}= 0.99; S2:

*t*(8) = 6.2,

*p*

_{one-sided}= 0.99; S3:

*t*(8) = 3.5,

*p*

_{one-sided}= 0.99; S4:

*t*(8) = 1.41,

*p*

_{one-sided}= 0.90; S5:

*t*(8) = 1.23,

*p*

_{one-sided}= 0.87; S6:

*t*(8) = 1.63,

*p*

_{one-sided}= 0.92; S7:

*t*(6) = 1.6,

*p*

_{one-sided}= 0.92 (Figure 6A). We performed the same analysis across subjects after pooling data across days. Again, we found that the data did not follow a negative trend, and there was no significant decrease in JNDs,

*t*(12) = 1.497,

*p*

_{one-sided}= 0.919 (Figure 6B). Collectively, these results provide no evidence of the JND dependence on priors predicted by sampling.

*k*, the sample size. Both model parameters were fit using the subject averages (Figure 6B, dotted and dashed lines). The fitted sensory noises for the MAP and sampling models were 35% and 31% larger than the true experimental standard deviation (0.05 screen units), respectively. This should be expected since the inherent noise in the visual system can only increase the uncertainty in the experimental stimuli. For the sampling model, the best fit for

*k*was 24; 95% confidence interval [5, 222], bootstrapped across subjects and conditions. The sampling model's large

*k*effectively approximated the MAP model. A quantitative comparison of the two models accounting for the difference in free parameters strongly favored the MAP model over the sampling model (Bayesian Information Criterion [BIC] = −62.6 and −47.4, respectively). As a further analysis, we also fit the two models to each individual subject. Here again the results favored the MAP over sampling when comparing BIC,

*t*

_{paired}(6) = −18.61,

*p*< 0.001.

*t*(6) = −8.3,

*p*< 0.001, which means that the precision of their decisions was better than that possible with matching. That is, even if subjects had perfect visual acuity (i.e., their likelihood was equivalent to that experimentally imposed), these data rule out the matching mechanism. Similarly, we find sampling with

*k*= 2 unlikely,

*t*(6) = −3.9,

*p*= 0.006. Matching as well as sampling with small

*k*seems incompatible with the data from this condition.

*k*will always appear like MAP. Our analysis shows that if subjects were sampling, they did so with a good number of samples. Limiting the stimulus exposure may limit the number of samples subjects can acquire. Sampling time was limited by the presentation of a visual mask that followed the stimulus after just 25 ms. It would thus seem that if subjects sample, they do so extremely rapidly. Future studies can address alternative techniques for limiting the potential number of samples in an effort to find further evidence for sampling. Regardless of the techniques employed, if the brain is making decisions with a large number of samples, distinguishing between MAP and sampling at the behavioral level will always remain a fundamental challenge.

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