Bayes’ rule:

\[\tag{1.1} p(A|B)=\frac{p(B|A)p(A)}{p(B)}\]

Prior :

\(p(A)\) the probability as the belief in the hypothesis before seeing the new evidence.

Posterior :

\(p(A|B)\) the probability of the hypothesis being true, if the evidence is present.

Likelihood :

\(p(B|A)\) the probability of the evidence being present, given the hypothesis is true.

Marginal :

\(p(B)\) the probability of the evidence being present, whether the hypothesis is true or false.

Bayes Filter:

From bayes’ rule we have:
\(p(Hypothesis|Evidence) = p(Hypothesis)* \frac{p(Evidence|Hypothesis)}{p(Evidence)}\)

\[posterior \propto prior * likelihood\]

We can obtain a recursive state estimation algorithm:

1. Prediction
   \(\underbrace{p(x_t|z_{1:t-1})}_{\text{prediction}}=\int \underbrace{p(x_t|x_{t-1})}_{\text{motion model}}\underbrace{p(x_{t-1}|z_{1:t-1})}_{\text{prev. posterior}}d_{x_{t-1}}\)

2. Correction
   \(\underbrace{p(x_t|z_{1:t})}_{\text{posterior}}\propto \underbrace{p(z_t|x_t)}_{\text{observation}}\underbrace{p(x_t|z_{1:t-1})}_{\text{prediction}}\)