On the Euclidean sphere, by defining suitable analogs of the mean and variance in terms of the first circular moment, it can be shown that the von Mises distribution is the maximum entropy distribution among all distributions with a given (complex) value of the circular moment.
This approach can't be immediately translated to hyperbolic space via the analogy of the hyperboloid in Minkowski space to the sphere in Euclidean space.
Is there another approach (with a suitable definition of "mean" and "variance") that yields results about maximum entropy distributions (with given "mean" and "variance") in hyperbolic space?
