Bayesian Filter

Definition

Estimates the hidden signal from the measured signal containing noise, using Bayes theorem.

Algorithm

• $x$: hidden signal
• $y$: observed signal
• $p(x)$: prior distribution
• $p(x|y)$: likelihood probability
• $p(y|x)$: posterior distribution

prior and posterior depends on sampling step $n$ $p_n(x),p_n(x|y)$

Compute posterior distribution $p_n(x|y)\propto p(y|x)p_n(x)$

Calculate MAP estimator for the given observation ${\hat{x}}_n=\underset{x}{\mathrm{argmax}}{p}_n(x|y=y_n)$

Update prior distribution $p_{n+1}(x)\leftarrow p_n(x|y)$

Wiener filter

Wiener filter add a Brownian motion to the model before computing the posterior distribution It makes the prior distribution smoother $p_n(x)=p_n(x)+\alpha \frac{\partial p_n(x)}{\partial x^2}$