I asked the following question 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?


Very short "answer": we've discussed in your other question the roots of the idea of geometrization of statistical models by Jeffreys, how they become Riemannian manifolds with metric $g_{ij}$, etc. Much later, classical statisticians such as Efron, Amari, etc, got interested in the idea of the curvature of these manifolds, because there is a link with the concept of second order efficiency of classical estimators. It turns out that the Jeffreys metric (aka Fisher-Rao Information Metric) is kind of boring from the curvature standpoint. For example, for the normal model, if you consider the connection compatible with $g_{ij}$, compute the corresponding Christoffel symbols $\Gamma^i_{jk}$, and the Riemann tensor $R^i_{jkl}$, you will figure out that this is a space of constant scalar curvature $R=g^{ij} R^k_{ijk}$ (this is a good exercise: do it!). To "enrich" the geometry, Amari considered a family of $\alpha$-connections, which are not, in general, compatible with $g_{ij}$, and studied the curvature induced by this family of connections. Amari succeeded in establishing links with second order efficiency, and did a lot of interesting things with his $\alpha$-connections. You can find all this and much more in his book.


  • $\begingroup$ Thank you. I will try to get Amari's book and look at these concepts very carefully. $\endgroup$ – Kumara Mar 5 '13 at 5:12
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    $\begingroup$ One more question: I came to know about the result mentioned in my question from Csiszar's Information Theory And Statistics: A Tutorial where the result is proven for the finite sample space using basic calculus and linear algebra. Do you think the fundamental idea might change when we try to prove the same result using Riemannian geometry? $\endgroup$ – Kumara Mar 5 '13 at 5:19
  • $\begingroup$ I would need to take a look at Csiszar's book. $\endgroup$ – Zen Mar 5 '13 at 20:32
  • $\begingroup$ You may need to look at only page no. 24-26 (Theorem 3.2/Cor.3.1). $\endgroup$ – Kumara Mar 6 '13 at 6:41

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