I would like to calculate the Fisher information for a purturbed Gaussian distribution: Suppose we have a varible $x$ distributed as $f(x|\alpha)=\frac{1}{(2\pi \delta^2(\alpha))^{1/2}} Exp\{-\frac{(x-\mu(\alpha))^2}{2\delta^2(\alpha)}\}$ using ${\rm FisherInformation}[\{\mu, \delta(\alpha)\}, f]$, where Fisher information is: $F(\alpha)=\int_{-\infty}^{\infty}dxf(x|\alpha)[\partial_{\alpha}\log f(x|\alpha)]^2$ .
Suppose the distribution of $x$ is perturbed and becomes:
$g(x|\alpha)=f(x|\alpha)[1+\varepsilon f(x)]/k(\alpha)$
where $\varepsilon$ is very small and $k(\alpha)$ is a normalization constant. How do we calculate the Fisher information? What will be the condition for $f(x)$ to have a valid probability for $g(x|\alpha)$? If $f(x)=x^n$.
dist = NormalDistribution[μ, σ]; f = PDF[dist, x]; h = D[Log[f], {{μ, σ}, 2}]; (fim = Expectation[-h, x \[Distributed] dist]) // MatrixForm
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