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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 $f(x)=x^n$.

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  • $\begingroup$ Ask it at another site. See a Wiki article to this end. $\endgroup$
    – user64494
    Oct 23, 2020 at 4:44
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    $\begingroup$ dist = NormalDistribution[μ, σ]; f = PDF[dist, x]; h = D[Log[f], {{μ, σ}, 2}]; (fim = Expectation[-h, x \[Distributed] dist]) // MatrixForm $\endgroup$
    – flinty
    Oct 23, 2020 at 9:23
  • $\begingroup$ @flinty, I would like to evaluate the above expression: $\endgroup$
    – user0322
    Oct 24, 2020 at 16:18
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    $\begingroup$ @user0322 not sure what you mean, just evaluate it - do you have Mathematica? That's the Fisher matrix for your distribution which has a normal PDF. It's a matrix: $$\left( \begin{array}{cc} \frac{1}{\sigma ^2} & 0 \\ 0 & \frac{2}{\sigma ^2} \\ \end{array} \right)$$ If you want just $F(\alpha)$ in terms of unknown $\mu,\delta$ then I think it works out to be $$\frac{2 \delta '(\alpha )^2+\mu '(\alpha )^2}{\delta (\alpha )^2}$$ which you get running this alteration of my code pastebin.com/HzzLELAn . If you use the square instead of a 2nd derivative, then just flip the sign. $\endgroup$
    – flinty
    Oct 24, 2020 at 16:47

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A direct calculation confirms @flinty's result:

f[x_, α_] = E^(-((x-μ[α])^2/(2*δ[α]^2)))/(δ[α]*Sqrt[2π]);
Integrate[f[x,α]*D[Log[f[x,α]],α]^2, {x, -∞, ∞}, Assumptions -> δ[α] > 0]

(*    (2*δ'[α]^2+μ'[α]^2)/δ[α]^2    *)
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