Decision making with stochastic posterior distributions.

<p><b>a–c</b>: Each panel shows an example of how different models of stochasticity in the representation of the posterior distribution, and therefore in the computation of the expected loss, may affect decision making in a trial. In all cases, the observer chooses the subjectively optimal target (blue arrow) that minimizes the expected loss (purple line; see Eq. 4) given his or her current representation of the posterior (black lines or bars). The original posterior distribution is showed in panels b–f for comparison (shaded line). <b>a</b>: Original posterior distribution. <b>b</b>: Noisy posterior: the original posterior is corrupted by random multiplicative or Poisson-like noise (in this example, the noise has caused the observer to aim for the wrong peak). <b>c</b>: Sample-based posterior: a discrete approximation of the posterior is built by drawing samples from the original posterior (grey bars; samples are binned for visualization purposes). <b>d–f</b>: Each panel shows how stochasticity in the posterior affects the distribution of target choices (blue line). <b>d</b>: Without noise, the target choice distribution is a delta function peaked on the minimum of the expected loss, as per standard BDT. <b>e</b>: On each trial, the posterior is corrupted by different instances of noise, inducing a distribution of possible target choices (blue line). In our task, this distribution of target choices is very well approximated by a power function of the posterior distribution, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003661#pcbi.1003661.e185" target="_blank">Eq. 7</a> (red dashed line); see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003661#pcbi.1003661.s003" target="_blank">Text S2</a> for details. <b>f</b>: Similarly, the target choice distribution induced by sampling (blue line) is fit very well by a power function of the posterior (red dashed line). Note the extremely close resemblance of panels e and f (the exponent of the power function is the same).