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Suppose I fit a Gaussian process to data such that the posterior distribution over any output is also a Gaussian process, $\mathcal{G}\mathcal{P}(\mu(x),\sigma^2(x))$ where $x$ is some valid input. Now I define an estimator in terms of the posterior mean and posterior variance, $\theta(\mu(x), \sigma^2(x))$. I want to compute the variance of the estimator.

In frequentist statistics, the standard device to compute the variance of the estimator is using delta-method which needs the Fisher information matrix of the parameters.

However, to compute the Fisher information matrix of the posterior mean and posterior variance for the Gaussian process one needs functional derivative. It should also be a process.

Can someone please point out if computing information matrix in this way is possible and suggest any references.

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