What is Kalman smoother?

The Kalman smoother computes $\E[\alpha_t \mid y_1, \dots, y_n]$ and its variance from the filter’s output by a backward pass (Rauch, Tung and Striebel, 1965): $\hat\alpha_t = a_{t|t} + J_t(\hat\alpha_{t+1} - a_{t+1})$ with $J_t = P_{t|t}T^\top P_{t+1}^{-1}$.