I'm looking for a Bayesian filter where observations are generated according to $s_t = \gamma s_{t-1} + w_p$ and $w_p \sim Normal(0, \sigma_p^2)$. Both $\gamma$ and the variance of the process noise $\sigma_p$ are known. However, different from the classical Kalman filter, the observational noise is not assumed to be Gaussian. Is there a specific name for this kind of filter?
I'm particularly interested in comparing the Kalman filter's estimates to the optimal estimates for a problem where the observation noise is only approximately Gaussian. Hence, I tried to recursively compute the posterior (numerically) using simple grid approximation, but I'm struggling with the marginalisation part:
$$p\left({s}_{t} \mid {m}_{1: t-1}\right)=\int p\left({s}_{t} \mid {s}_{t-1}\right) p\left({s}_{t-1} \mid {m}_{1: t-1}\right) {d} x_{t-1}$$
How would something like this look with a simple grid approximation? Is it feasible?
Best Answer
I don't recall any specific for the kind of filters you mentioned. Suppose that your likelihood model at hand is student-t, then just search student-t filters. There is no need to come up with a generic term.
Solving filtering problems with non-Gaussian measurement noises is very common. The most well-known solution is particle filters. Another solution is to approximate your measurement model with a Gaussian one by moment matching.