Effect of the integration of the prior with the sensory evidence

Let’s supposse that we have a normal prior to respond right, for example, and that the sensory evidence is also normal.

\[X_{prior} \sim N(\mu_{prior}, \sigma^2_{prior})\] \[X_{s} \sim N(\mu_{s}, \sigma^2_{s})\] Then the posterior is distributed

\[X_{p} \sim N(\mu_{p}, \sigma^2_{p})\] with \[\frac{1}{\sigma^2_{p}} = \frac{1}{\sigma^2_{prior}} + \frac{1}{\sigma^2_{s}}\] \[\mu_p = \frac{\sigma_p^2}{\sigma^2_{prior}} \mu_{prior} + \frac{\sigma_p^2}{\sigma^2_{s}} \mu_{s}\] Assuming linear transduction of the stimulus \(s\) (the point should hold also for other transductions)

\[\mu_s = a s + b\]

If we assume a standard signal detection theory model (for example, here) and a non-biased decision rule, the proportion correct is

\[ P_{correct}(s) = \int_{0}^{\infty}N(\frac{\sigma_p^2}{\sigma^2_{prior}} \mu_{prior} + \frac{\sigma_p^2}{\sigma^2_{s}} (a s + b), \sigma_p^2) dx \] This is psychometric function with the cumulative normal shape. The location parameter is shifted towards the prior and the prior makes the slope stepper.

The point here is that this model of integration does not produce lapses which could be introduce ad hoc

\[\Phi(s) = \gamma + (1- \gamma - \lambda) P_{correct}(s)\]