The asymptotic entropy h(m) of a probability measure m with finite Shannon entropy on a countable group G encodes information about the asymptotic behavior of the m-random walk on G. In many classes of groups, the asymptotic entropy can be computed through the action of G on a Polish space X, which is often a geometric boundary of G. In such cases, the space X can be endowed with a m-stationary probability measure m such that L-infinity (X, l) is isomorphic to the space of bounded m-harmonic functions on G (i.e., (X, l) is the Poisson boundary of (G, m)). I will explain how the uniqueness of the stationary measure on X leads to the continuity of the function from m to h(m). This result gives a new proof of continuity in the case of hyperbolic groups, which was already known, and extends it to new classes of groups, such as SLd(Z), for d at least 3.