Unfamiliar hessian matrix expression

[Monamandala]

I am familiar with the hessian matrix having the square in the numerator and a product of partial derivatives in the denominator:

$Hessian = \frac{\partial^2 f}{\partial x_i \partial x_j}$

However, I have come across a different expression, source: https://users.ugent.be/~yrosseel/lavaan/lavaan2.pdf (slide 40)

$nCov(\hat\theta) =A^{-1}=[-Hessian]^{-1} = [-\partial F(\hat\theta)/(\partial\hat\theta\partial\hat\theta')]^{-1}$

A - represents a hessian matrix.

I am curious are the one attached and usual hessian matrix interchangeable somehow? Why is there no square and a derivative appears in the denominator in the attached example?

 

[mmwave]

Hello there,

Thank you for sharing your question with me. I am a scientist and I would be happy to provide some clarification on the hessian matrix and the expression you have come across.

The hessian matrix, as you mentioned, is typically represented by the square in the numerator and the product of partial derivatives in the denominator. This matrix is used in multivariate calculus to measure the curvature of a function at a given point. It is also commonly used in optimization algorithms to find the minimum or maximum of a function.

The expression you have come across is also related to the hessian matrix, but it is a slightly different representation. This expression is known as the negative inverse hessian, and it is denoted by $-Hessian^{-1}$ or $[-\partial F(\hat\theta)/(\partial\hat\theta\partial\hat\theta')]^{-1}$. This expression is used in the field of statistics, specifically in the context of structural equation modeling.

In this case, the hessian matrix is being used to estimate the covariance matrix of the parameter estimates, denoted by $nCov(\hat\theta)$. This is why the expression is written as $nCov(\hat\theta) = A^{-1} = [-Hessian]^{-1} = [-\partial F(\hat\theta)/(\partial\hat\theta\partial\hat\theta')]^{-1}$. The negative inverse hessian is used because it allows for a more efficient and accurate estimation of the covariance matrix.

To answer your question, the two expressions are not interchangeable because they serve different purposes. The usual hessian matrix is used for curvature and optimization, while the negative inverse hessian is used for estimating covariance in statistics. I hope this helps clarify the difference between the two expressions.