Smoothing process of Kalman filter
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 $E[x_k|y_1,...,y_N]$.