Unknown parameter - augmenting state equation (Kalman filter) First, we have a state space model with mean reversion and $\mu$ is unknown
$y(t )= F* x_t  +e_t$
$x_t- \mu = G* (x_{t-1}-\mu) +n_t$
There is a option to add unknown parameters to the state vector and pass them on with zero error to estimate them.
The state equation can be writen as
$$\pmatrix{x_t \\ \mu_t} = \pmatrix{G & I-G \\ 0 & I}\pmatrix{x_{t-1} \\\mu_{t-1}}  + \pmatrix{n_t \\ 0}.$$
After that I can run the Kalman filter and get an estimate for $\mu$. The question is, which estimate is the one that would be equivalent to a maximum likelihood estimate in advance.
Is it the estimate that would result in the last time step?
Or to put it another way: $\mu_t$ changes over time. Which $\mu_t$ then represents the estimate, so to speak, if, for example, the estimate of $\mu$ is asked for? Is it the estimate from the last time step in the Kalman filter?
Would be very grateful for help.
 A: Suppose that the model does not involve any parameter other than
$\boldsymbol{\mu}$. You can regard $\mu$ as a random quantity in a Bayesian
approach. Then, if the prior on $\boldsymbol{\mu}$ is Gaussian, the posterior is
also Gaussian and is given by the last step of the Kalman Filter. The
filtered mean $\widehat{\boldsymbol{\mu}}_{T\vert T}$ for the last time $T$ gives the posterior mean
which also is the posterior mode, and the posterior variance is the
filtered variance. Moreover, you can use a diffuse (improper) prior
for $\boldsymbol{\mu}$; provided that $\boldsymbol{\mu}$ can be identified, the posterior will be
proper, and the mode will be the value of $\mu$ that maximises the
likelihood. Note that since $\boldsymbol{\mu}$ does not depend on $t$, a better
notation would be something like $\widehat{\boldsymbol{\mu}}_{\bullet \vert T}$.
Things are more complex in the case where the model involves more
parameters forming a vector $\boldsymbol{\theta}$.

*

*The Bayesian approach would then lead to regard
$\boldsymbol{\theta}$ as random as well. With mild conditions the full
conditional $p(\boldsymbol{\mu} \vert \mathcal{Y}, \, \boldsymbol{\theta})$ will
be Gaussian where $\mathcal{Y}$ is the whole series. This can be of
some help.


*In a frequentist approach, you can maximise the profile-likelihood
that depends only of $\boldsymbol{\theta}$. Indeed, the value of $\boldsymbol{\mu}$
which maximises the likelihood for a given $\boldsymbol{\theta}$ is
given by the Kalman filter as above. Remind however that you have to
use the diffuse prior.
In the second case, the estimate of $\boldsymbol{\mu}$ is provided by the Kalman
filter at a very small cost.
Note that depending on the dimensions of the system, it can
be better to use an alternative state
vector $\mathbf{x}_t^\star := \mathbf{x}_t - \boldsymbol{\mu}$ along with
$\boldsymbol{\nu} := \mathbf{F} \boldsymbol{\mu}$ and the state space model
\begin{align*}
   \mathbf{x}_t^\star &= \mathbf{G} \mathbf{x}^\star_{t-1} + \mathbf{n}_t\\
   \mathbf{y}_t  &= \mathbf{F} \mathbf{x}^\star_ t + \boldsymbol{\nu} +
   \mathbf{e}_t.
\end{align*}
The augmented state form would then take $\boldsymbol{\nu}$ as
augmented part with a block diagonal transition matrix. The motivation is that $\boldsymbol{\nu}$ can be of smaller length
than $\boldsymbol{\mu}$, especially when the observation is a scalar
$y_t$. Note also that as in in this question linked in the comment by
@Cam.Davidson.Pilon, we can cope similarly
with the case where $\boldsymbol{\nu}$ is replaced by
$\mathbf{z}_t^\top \boldsymbol{\beta}$ for a vector $\mathbf{z}_t$ of
known covariates. Then we would put the vector $\boldsymbol{\beta}$ of
unknown regression coefficients in the augmented state.
