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My question is, do the smoothed values affect the estimates obtained from the Maximization Step? If not then we could eliminate the smoothing. Once the estimates are obtained, they do not change apart from substituting the smoothed values $x_n^s, P_n^s$. So, why do smoothing?

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The result of the $M$-step is expressed in terms of the matrices $\mathbf{P}_n^2$ and the vectors $\mathbf{x}_n^s$. These quantities are defined as expectations. Smoothing is used to compute these expectations, because in practice expectations are usually approximated by time averages (hence 'smoothing'). If you have another method for computing these expectations, the results of the $M$-step will of course be different, but the expressions for the results of the $M$-step remain valid, you just plug in different values for the respective matrices and vectors.

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  • $\begingroup$ Just to clarify if I understood your point: Once, the analytical expressions for the estimates in M steps have been derived, containing the $P_n^s$ and $X_n^s$ terms, they remain fixed. Then, based on the smoothing technique for Kalman smoother the values from the smoothing steps will be plugged into these $P_n^s$ and $X_n^s$ terms. If nonlinear estimation is performed, then using the results from the $M step, the smoothed expressions for extended Kalman and Unscented Kalman are plugged. Is my understanding correct? $\endgroup$ – Ria George Mar 18 '15 at 18:20
  • $\begingroup$ @RiaGeorge: Yes, I think you understood what I mean: the formulas from the $M$-step remain valid. You just use different values for the matrices and vectors in the expression. $\endgroup$ – Matt L. Mar 18 '15 at 21:14

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