 The "natural" gradient of formula_144, complying with the Fisher information metric (an informational distance measure between probability distributions and the curvature of the relative entropy), now reads

where the Fisher information matrix formula_154 is the expectation of the Hessian of and renders the expression independent of the chosen parameterization.Combining the previous equalities we get

A Monte Carlo approximation of the latter expectation takes the average over samples from 

where the notation formula_64 from above is used and therefore formula_77 are monotonically decreasing in formula_12.Ollivier "et al."