This paper proposes a probabilistic approach to the problem of intrinsic filtering of a system on a matrix Lie group with invariance properties. The problem of an invariant continuous-time model with discrete-time measurements is cast into a rigorous stochastic and geometric framework. Building upon the theory of continuous-time invariant observers, we introduce a class of simple filters and study their properties (without addressing the optimal filtering problem). We show that, akin to the Kalman filter for linear systems, the error equation is a Markov chain that does not depend on the state estimate. Thus, when the filter's gains are held fixed, the noisy error's distribution is proved to converge to a stationary distribution, under some convergence properties of the filter with noise turned off. We also introduce two novel tools of engineering interest: the discrete-time invariant extended Kalman filter, for which the trusted covariance matrix is shown to converge, and the invariant ensemble Kalman filter. The methods are applied to attitude estimation, allowing to derive novel theoretical results in this field, and illustrated through simulations on synthetic data.