What is Kalman filter, innovation, Kalman gain?

The Kalman filter (Kalman, 1960) computes recursively the conditional law $\alpha_t \mid y_1, \dots, y_{t-1} \sim \mathcal N(a_t, P_t)$. Given $(a_t, P_t)$, the innovation is $v_t = y_t - Z_ta_t$, with variance $F_t = Z_tP_tZ_t^\top + H$; the Kalman gain $K_t = P_tZ_t^\top F_t^{-1}$ updates the state, $a_{t|t} = a_t + K_tv_t$ and $P_{t|t} = P_t - K_tZ_tP_t$, and the prediction step gives $a_{t+1} = Ta_{t|t}$ and $P_{t+1} = TP_{t|t}T^\top + Q$.