#state-space-models

A one-dimensional linear-Gaussian state-space model has $h_s = A h_{s-1} + B \xi_s$ and $v_s = C h_s + D \eta_s$ with independent standard normal $\xi_s, \eta_s$. The filtering distribution at time $s - 1$ is $p(h_{s-1} \mid v_{1:s-1}) = \mathcal{N}(0, 1)$. With $A = 1$, $B^2 = 1$, $C = 1$, $D^2 = 2$ you observe $v_s = 3$. What is the mean $\mu_s$ of $p(h_s \mid v_{1:s})$?