Bayesian models

Basic formulation

A classical way to model perception is as an observer trying to infer the state of the world $s$ from noisy sensory measurements $x$. From Bayes rules this is: $$ p(s|x)=\frac{p(s).p(x|s)}{p(x)} $$ The distribution $p(x|s)$ is called the likelihood function. It describes the probability to observer $x$ given the state $s$. Typically it is assumed to be a noisy measurement of $s$, corrupted by normally distributed noise of variance $\sigma^2$: $$ p(x|s)=\mathcal{N}(s,\sigma) $$ The distribution $p(s)$ is called the prior. It indicates the a priori belief of the observer that $s$ has different values. Note that the prior may not match the true distribution of $s$. In other words the observer might be misguided about what the values of $s$ can be, however it is classical to assume that observers are well calibrated.