# 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.

## Assumptions

## Estimators