Exponential decay rate of the filter’s dependence on the initial distribution

We review how tools from multiplicative ergodic theory and the theory of positive operators are used in the analysis of exponential stability of the optimal nonlinear filter. Particularly, in the case of finite state, we relate the filter sensitivity to perturbations in its initial data to the Lyapunov spectral gap associated with the filtering equation, and, in a general setting, use Hilbert’s metric and Birkhoff’s contraction coefficient to estimate the decay rate of the error.