The nonlinear filtering equation describes the conditional law of a Markov process $\{x_t\}$ at time $t$ conditioned on the path $y_s \int_0^s g(x_u)du + w_s$ for $s\in[0,t]$. Here $g$ is a fixed function and $\{w_t\}$ is a Brownian motion independent of $\{x_t\}$. We study the exponential decay rate of the variation distance between solutions that correspond to different initial conditions. The question is related to Lyapunov exponents, Birkhoff's contraction and large deviations.