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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.

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    $\begingroup$ It’s still the optimal linear filter if that’s what you’re after. $\endgroup$ – hejseb Sep 30 '18 at 19:56
  • $\begingroup$ @hejseb thanks, It might well be! Do you have a reference to a derivation? $\endgroup$ – Monolite Sep 30 '18 at 20:06
  • $\begingroup$ As @hejseb hinted at, the Kalman filter is the answer to more than one question. Are you looking for a least-squares linear filter (which the Kalman filter is, even without normality), or are you looking to compute the sequence of filtering distributions $p(X_t|Y_1,..Y_t)$ in general (which the Kalman filter does but only by assuming normality)? $\endgroup$ – Chris Haug Oct 2 '18 at 19:54
  • $\begingroup$ @ChrisHaug a least squares linear filter I guess. The only ways I have seen the derivations of the Kalman filter is with the normality assumption. $\endgroup$ – Monolite Oct 2 '18 at 20:09
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    $\begingroup$ @Monolite Just like OLS is the minimum mean squared error estimator (optimal linear estimator) even in the presence of non-normality, so is KF the optimal linear filter. That doesn't mean that it's a good filter, because the class of linear filters could be very restrictive.I just checked Hamilton's Time Series Analysis (1994) and it too doesn't make use of normality and that derivation is more explicit than Wikipedia's. $\endgroup$ – hejseb Oct 5 '18 at 11:26
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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.

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  • $\begingroup$ Thanks for the answer, So the optima linear filter cited by @hejseb is not an appropriate filter in this case? $\endgroup$ – Monolite Oct 1 '18 at 10:48

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