Gradient and Hessian of functions with non-independent variables

Abstract

Mathematical models are sometime given as functions of independent input variables and equations or inequations connecting the input variables. A probabilistic characterization of such models results in treating them as functions with non-independent variables. Using the distribution function or copula of such variables that comply with such equations or inequations, we derive two types of partial derivatives of functions with non-independent variables (i.e., actual and dependent derivatives) and argue in favor of the latter. The dependent partial derivatives of functions with non-independent variables rely on the dependent Jacobian matrix of dependent variables, which is also used to define a tensor metric. The differential geometric framework allows for deriving the gradient, Hessian and Taylor-type expansion of functions with non-independent variables.