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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.

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  • $\begingroup$ Do you want the complete derivation of the Kalman Smoother or doesn't understand something about it? $\endgroup$
    – Royi
    Aug 10, 2015 at 5:53
  • $\begingroup$ 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 $\endgroup$
    – Ben
    Aug 10, 2015 at 7:01
  • $\begingroup$ Why don't you add a reference, copy its Latex put it as a community answer and we'll add it with explanations? $\endgroup$
    – Royi
    Mar 26, 2016 at 13:19

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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 $E[x_k|y_1,...,y_N]$.

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