The Fisher information metric is defined as:
$$g_{jk}(\theta)= \int_X \frac{\partial \log p(x,\theta)}{\partial \theta_j} \frac{\partial \log p(x,\theta)}{\partial \theta_k} p(x,\theta) \, dx.$$
Say $p(x,\theta):= \exp \frac{\theta}{\log x},$ for $x\in(0,1).$
Then $\theta \in (0,\infty)$ gives us a coordinate on a $1-$manifold equivalent to the positive real line $(0,\infty).$
I calculated (confirming with wolfram alpha) the integral to be a Bessel function: $$\frac{2K_1(2\sqrt{\theta})}{\sqrt{\theta}} $$
But this calculation was done without the normalization constant. Also I'm not sure how to write my result in the proper form i.e. as a metric.
Am I on the right track here? What is the correct calculation and proper way to write the metric?
I think I basically got the answer but looking for verification of the solution and additional feedback on how to write the metric.
Edit:
With the normalization constant included, I found the metric to be:
$$ g(\theta)=\frac{K_0(2\sqrt{\theta})^3}{\theta K_1(2\sqrt{\theta})^2}-2K_0(2\sqrt{\theta})^2+\frac{1}{\theta}. $$
I think this is probably the correct form of the metric.
Still unsure of the $jk$ subscripts and where they come into play in writing the metric properly.