I do not understand the properties of EM algorithm of Kalman.

Assume one has a state-space model: $$ z_{t} = B x_{t} + v_{k}\\ x_{t+1} = A x_{t} + w_{k}, $$ where noise terms are i.i.d. such that $v_{k} \sim N(0, Q)$ and $w_{k} \sim N(0, R)$. Assume that the matrices $A$ and $B$ are known.

The question: does EM algorithm provide a consistent estimation of $Q$ and $R$, the observation and transition covariances?

My discovery:

According to https://stackoverflow.com/questions/46198246/em-algorithm-with-pykalman/58560992#58560992

and some simple simulations, the estimate of the covariance matrix is far from the true values.

From the discussion here: https://dsp.stackexchange.com/questions/24417/em-algorithm-and-kalman-filter

However, EM algorithm has a drawback. It only gives suboptimal solution. In other words, the parameters estimated by EM algorithm are only local minimum/maximum, rather than global minimum/maximum. Therefore, further turning may be needed.

Therefore, my conclusion is that in general Kalman EM does not provide a consistent estimator of the variance.


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