Beginner level: Help in learning Kalman Smoother (Part 1) Parameter estimation of Linear Dynamical system is a tutorial which explains Kalman Filter, Smoothing, and Expectation Maximization. I have followed the derivation for Kalman Filter. But cannot understand the smoothing. I have read textbooks and many other resources including the references cited in this document but failed to understand how the smoothing equations are derived. Particularly in this case document, Section 6 mentions the Smoother. In the E Step, the smoother is calculated and the result of the Smoothing is included in the estimates obtained in the Maximization step. For the model, 
$h(t) = \mathbf{A^T} h(t-1) + \eta^h(t)$
$v(t) = \mathbf{B^T}h(t) + \eta^v(t)$
$\eta^h(t) = N(0,Q), \eta^v(t) =N(0,R)$
The log likelihood is $Q= - \sum_{t=1}^{} \big(\frac{1}{2}[v(t) - Bh(t))'R^{-1}[v(t)-Bh(t)] \big) - \frac{T}{2} \log |R| - \sum_{t=2}^T \big( \frac{1}{2} [h(t)' - Ah(t-1)]'Q^{-1}[h(t) - Ah(t-1)]\big) - \frac{T-2}{2} \log |Q| -\frac{1}{2} {[h_1 - \pi_1]}' V_1^{-1}[h_1 - \pi_1] - \frac{1}{2} \log |V_1| - \frac{T(p+2) \log 2 \pi}{2}$
where $\pi_1, V$ is the mean and variance of the initial condition of $h$.
Section 6: I cannot understand how the expressions 2--6 in Pg7 have come and what is the technique of including this in the M step. It shall be really helpful if the derivation of any one expression and the plugging in of the smoothed estimate into the M step is illustrated.
 A: Let me take a few steps back. EM algorithm is not required in a Kalman Filter if the design matrices(A, B, Q, R, etc) are known. They are known only if you know which physical system you are modelling from the beginning. If not then you will have to estimate these matrices. Filtering and smoothing operations are performed assuming that these matrices are already known. 
EM utilizes the filtering or smoothing equations by starting with some random initial values of design matrices and then running the filtering equations (expectation step) and then lowering the prediction error (maximization step). It is possible to replace filtering equations with smoothing equations in the above procedure. You get to choose either filtering or smoothing (To be very precise filtering is the first step of smoothing. Hence smoothing is like an add-on). The difference is filtering only uses past values, smoothing on the other hand takes also future values into consideration. 
EM Derivation for Kalman Filter is probably the most complete derivation of the EM procedure for Kalman Filter/Smoother.
To sum up, when design matrices are known, you run either filtering or smoothing equations to execute the filter. If matrices are not known you execute the filter by either filtering or smoothing equations then modify the matrices so that the results of the previous operation are improved. You repeat this procedure until the matrices don't change much. And this is called the EM procedure. The good thing is Kalman EM has neat solutions for the derivatives in the maximization procedure therefore you don't need numerical techniques for maximization.
