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(I posted a similar question on math.se.)

In information geometry, the determinant of the Fisher information matrix is a natural volume form on a statistical manifold, so it has a nice geometrical interpretation. The fact that it appears in the definition of a Jeffreys prior, for example, is linked to its invariance under reparametrizations, which is (imho) a geometrical property.

But what is that determinant in statistics? Does it measure anything meaningful? (For example, I would say that if it is zero, then the parameters are not independent. Does this go any further?)

Also, is there any closed form to compute it, at least in some "easy" cases?

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In many examples, the inverse of the fisher information matrix is the covariance matrix of the parameter estimates $\hat{\beta}$, exactly or approximately. Often it gives that covariance matrix asymptotically. The determinant of a covariance matrix is often called a generalized variance.

So the determinant of the Fisher information matrix is the inverse of that generalized variance. This can be used in experimental design to find optimal experiments (for parameter estimation). In that context, this is called D-optimality, which has a huge literature. so google for "D-optimal experimental design". In practice, it is often easier to maximize the determinant of the inverse covariance matrix, but that is obviously the same thing as minimizing the determinant of its inverse.

There are also many posts on this site, but few has good answers. Here is one: Experimental (factorial) design not exploiting the variance

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