I asked the following questionquestion in MSE for which I couldn't get any answer yet. I thought this would be a better place for that question.
In statistical maniolds $S=\{p_\theta\}$,$\theta=(\theta_1,\dots,\theta_n)$, the Riemaanian metric usually defined is the Fisher information metric $$g_{ij}(\partial_i,\partial_j)=\int \partial_i(\log p_\theta) \partial_i(\log p_\theta)~p_\theta~dx$$
The associated connection coefficients are defined by $$\Gamma_{ij}^k=\int \partial_i\partial_j(\log p_\theta)\partial_k (\log p_\theta)~ p_{\theta}~dx$$
where $\partial_i=\frac{\partial}{\partial\theta_i}$.
My question is, what is the intuition behind defining these? Is there a way to prove using the above metric and connection that the linear family of probability distributions $$L=\{p:\int f_i(x)p(x)~dx=m_i, i=1,\dots,k\}$$ intersects "orthogonally" the associated exponential family $$\mathcal{E}=\{p:p(x)=c(\theta)q(x)\exp(-\sum_{i=1}^k\theta_i f_i(x))\}$$ in the sense that $L\cap\mathcal{E}=\{p^*\}$ where $p^*$ satisfies $$D(p\|q)=D(p\|p^*)+D(p^*\|q)$$ for every $p\in L, q\in \mathcal{E}$.
I recently came to know about the connection between Fisher information metric and the relative entropy: $$D( p(\cdot , a+da) \| p(.,a) )\approx\frac{1}{2} g_{i,j} da^{i} da^{j}$$ Would this be a backbone in establishing the above result?
