Relationship between the Hessian and Covariance Matrix for Gaussian Random Variables

Relationship between the Hessian and Covariance Matrix for Gaussian Random Variables 

Consider a Gaussian random vector 0 with mean 0 and covariance matrix Z so its joint probability density function (PDF) is given by  

For Gaussian random variables, the second derivatives of the objective function are constant for all 0 because the objective function is a quadratic function of 0. Therefore, the Hessian matrix can be computed without obtaining the mean vector 0.  

The elements in the Hessian matrix carry the conditional information of the random vector because they are obtained by fixing all other parameters. The diagonal elements are the curvature of the objective function in the corresponding direction. The reciprocals of these diagonal 

Bayesian Methods for Structural Dynamics and Civil Engineering Ka-Veng Yuen 2010 John Wiley Sons (Asia) Pte Ltd 

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