# Approximating a 1-d Kalman Filter with non-Gaussian Observation Noise

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?

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Mar 17, 2022 at 17:09