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Derivatives of Probability Measures
Simulation of Derivatives
Keywords Derivatives - Stochastic optimization
For stochastic optimization problems of the form
where H(x, v) is a cost function, μ x a family of probability measures indexed by x and F(x) the objective value function (OVF), the necessary condition ∇ x F(x) = 0 must be expressed in terms of the derivatives of H(x, ·) and μ x w.r.t. x. In particular, concepts of differentiability of probability measures are needed.
Suppose that the family (μ x ) is dominated , i. e. there is a Borel measure ν such that the densities
exist for all x. Then the differentiability of the measures may be defined by the differentiability of the densities.