Kalman Filter:

Prediction

Motion prediction
\(\bar{x}_{k|k-1}=F_{k-1}\bar{x}_{k-1|k-1}\)

Increase uncertainity by motion
\(P_{k|k-1}=F_{k-1}P_{k-1|k-1}F^{T}_{k-1}+Q_{k-1}\)

Update

Measurement prediction
\(\bar{z}=H_{k}\bar{x}_{k|k-1}\)

Innovation
\(\varepsilon_{k}=z_{k}-\bar{z}_{k}\)

Innovation covariance
\(S_k=H_kP_{k|k-1}H^T_k+R_k\)

Kalman gain
\(K_k=P_{k|k-1}H^T_kS^{-1}_k\)

Weighted average
\(\bar{x}_{k|k}=\bar{x}_{k|k-1}+K_k\varepsilon_k\)

Uncertainty reduction
\(P_{k|k}=P_{k|k-1}-K_kH_kP_{k|k-1}\)

Likelihood

\[p(z_k|z_{1:k-1})=\mathcal{N}(z_k;\bar{z}_k, S_k)\]