Deriving a filter like a Kalman filter from a non-Gaussian state space model

Question

Assume we specify a state space model as

$$Y_t = a X_t + W_t$$

and

$$X_{t+1} = b X_t + V_t$$

where $b,a \in R$, $E[W_t] = E[V_t] = 0 \quad \forall{t }$ and $W_t $ and $V_t$ are indipendent for all t and both have finite second moment.

Notice that $W_t$ and $V_t$ are not assumed to be Gaussian.

I think I once saw somewhere a derivation of a filter à la Kalman filter with these assumptions only, has this been tried? Does somebody have any references?

EDIT: just to be clear, I am looking for a reference to the derivation of a filter with only these hypothesis, I have seen derivations of the Kalman filter but the ones I have seen all rely on the Gaussian assumption.

Answer 1

I am skeptical that such a general kalman filter exists. To derive the classical kalman filter relies heavily on the (closed) formulas for the posterior of the gaussian distribution and the fact that the normal is a conjugate prior for itself. Gamma also has this property so perhaps you can derive a kalman filter for gamma errors, too.

That doesn't mean you can't implement a non-gaussian kalman filter. You can do it using MCMC software such as Stan. But without closed formulas you won't be able to integrate out the latent states so you'll need to explicitly estimate many more parameters ($X_t$ in your notation). It will be many orders of magnitude (computationally-- not necessarily statistically) less efficient.