How to treat non-identifiable states in Kalman filtering/dynamic linear models?

Question

Let $x_t = G_tx_{t-1}+\omega_t$ with $\omega_t \sim \mathrm{N}(\mathbf{0}, \mathbf{W}_t)$ be a state equation and $y_t = F_tx_t+\nu_t$ with $\nu_t \sim \mathrm{N}(\mathbf{0}, \mathbf{V}_t)$ be a measurement equation in a DLM. In my model, $F_t$ is an $m \times n$ matrix for which $n \gg m$, so that for an observed $y_t$, there may be many $x_t$ which satisfy the measurement equation. Then, the particular $x_t$ at time $t$ which produced $y_t$ is non-identifiable. In static contexts this situation is tackled through regularization (e.g., Tikhonov, Lasso...), but how is identifiability problems treated within the context of Kalman filtering/DLM?

Answer 1

The concept of observability answers this question. In a nutshell, observability tells you if you can reconstruct the state $x_{t}$ from the measurement $y_{t}$. Also, a system with $n > m$ can be observable, and a system with $n < m$ can be unobservable.

There is a simple test for observability for linear time-invariant systems. However, for linear time-varying systems, you must look at the Gramian.

If you system does turn out to be unobservable, you must then construct a minimal realization of the system, such that the system is observable.