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- TL;DR Summary
- A silly question on off-diagonal elements of the Fisher-Rao metric

Let ## \mathcal{S} ## be a family of probability distributions ## \mathcal{P} ## of random variable ## \beta ## which is smoothly parametrized by a finite number of real parameters, i.e.,

## \mathcal{S}=\left\{\mathcal{P}_{\theta}=w(\beta;\theta);\theta \in \mathbb{R}^{n}, \theta=(\theta^{i})\right\} ## . The statistical model ## \mathcal{S} ## carries the structure of smooth Riemannian manifold ## \mathcal{M} ## , with respect to which ## \theta=(\theta^{i}) ## play the role of coordinates of a point ## \mathcal{P}_{\theta}\in \mathcal{S} ## , and whose metric is defined by the Fisher's information matrix ## \mbox{H}=(g_{ij}(\theta)) ## , where the coefficients of this matrix, which yields a positive definite metric, are calculated as the expectation of a product involving partial derivatives of the logarithm of the probability density function's (PDF)

## g_{ij}(\theta)=\int^{+\infty}_{-\infty} \displaystyle \frac{\partial^{2}ln \left( w(\beta;\theta)\right)}{\partial \theta^{i} \partial \theta^{j}}w(\beta;\theta)d\beta ## .

How do we neglect the off-diagonal terms ## g_{12}= g_{21} ## ?

In other words, is there a mathematical argument, wherein it is possible to consider ## g_{12}=g_{21}=0 ## ?

## \mathcal{S}=\left\{\mathcal{P}_{\theta}=w(\beta;\theta);\theta \in \mathbb{R}^{n}, \theta=(\theta^{i})\right\} ## . The statistical model ## \mathcal{S} ## carries the structure of smooth Riemannian manifold ## \mathcal{M} ## , with respect to which ## \theta=(\theta^{i}) ## play the role of coordinates of a point ## \mathcal{P}_{\theta}\in \mathcal{S} ## , and whose metric is defined by the Fisher's information matrix ## \mbox{H}=(g_{ij}(\theta)) ## , where the coefficients of this matrix, which yields a positive definite metric, are calculated as the expectation of a product involving partial derivatives of the logarithm of the probability density function's (PDF)

## g_{ij}(\theta)=\int^{+\infty}_{-\infty} \displaystyle \frac{\partial^{2}ln \left( w(\beta;\theta)\right)}{\partial \theta^{i} \partial \theta^{j}}w(\beta;\theta)d\beta ## .

How do we neglect the off-diagonal terms ## g_{12}= g_{21} ## ?

In other words, is there a mathematical argument, wherein it is possible to consider ## g_{12}=g_{21}=0 ## ?