# Smoothing process of Kalman filter

I have a question about the smoothing (backward) process of Kalman filter. Is it correct to say

$E[x_{t|T}] = x_{t|t}$

where $x_{t|t}$ is the estimated result from forward process?

I am struggling to understand the smoothing process of Kalman filter. Any help will be appreciated.

• Do you want the complete derivation of the Kalman Smoother or doesn't understand something about it? – Royi Aug 10 '15 at 5:53
• I would like to have the complete derivation of the Kalman Smoother. I can find some by Google. But it is appreciated if this bit can be explained. Thank you – Ben Aug 10 '15 at 7:01
• Why don't you add a reference, copy its Latex put it as a community answer and we'll add it with explanations? – Royi Mar 26 '16 at 13:19

The filtering distribution $p(x_k|y_1,...,y_k)$ is computed via the Kalman filter and given a linear-time invariant system is parametrized by the mean $E[x_k|y_1,...,y_k]$ and covariance matrix.
When doing smoothing we want to compute $p(x_k|y_1,...,y_N)$ with $N \ge k$, in this case the mean we compute is $E[x_k|y_1,...,y_N]$.