Abstract: The Filtering Problem consists in estimating the state of a system from a record of noisy measurements. More precisely, given the random process $ X_n $ describing the state of the system, and the random process $ Y_n $ representing a sequence of noisy observations of the system of $ X_n $, the filtering problem amounts to computing the conditional probability $ \pi_n $ of $ X_n $ given the measures $ Y_1,Y_2,\ldots,Y_n $. The sequence of conditional probabilities $ \pi_n $ is called the filter process. The theory of the stochastic filter has applications to many different areas (e.g., control problems, wireless communications, mathematical finance, signal processing). An important problem is to determine the asymptotic stability of the filter process depending on the choice of the initial condition $ \pi_0 $. The filter $ \pi_n $ is asymptotically stable if it forgets its initial condition. In this talk, I will discuss joint work with J.~Broecker on the asymptotic stability of the filter $ p_n $ when the process $ X_n $ is generated by random maps expanding on average.