The Fisher-Rao length is the Riemannian length induced by Fisher information. It turns a statistical model into a metric space when the geodesic distance is finite and nondegenerate.
Length of statistical curves
A smooth curve of distributions has length obtained by integrating Fisher speed.
Definition. For a smooth curve \(t\mapsto\theta(t)\), \[ L(\theta)=\int_0^1\sqrt{g_{ij}(\theta(t))\dot\theta^i(t)\dot\theta^j(t)}\,\mathrm{d}t. \] The Fisher-Rao distance is the infimum of this length over smooth curves joining two model points.
Proposition. The length is invariant under smooth reparametrization of the statistical model.
Proof. Under coordinate change, tangent components and the Fisher tensor transform by inverse rules. The scalar \(g_{ij}\dot\theta^i\dot\theta^j\) is therefore unchanged. The integral over curve parameter is also unchanged under monotone reparametrization of \(t\).
Square-root embedding
For the full positive simplex, Fisher geometry becomes spherical geometry after taking square roots.
Theorem. On the probability simplex, the map \(p\mapsto 2\sqrt p\) embeds the simplex into the positive orthant of the sphere of radius \(2\), and the Fisher metric is the pullback of the Euclidean metric.
Proof. For a tangent vector \(u\) with \(\sum_i u_i=0\), the differential is \(u_i/\sqrt{p_i}\). Its Euclidean squared norm is \(\sum_i u_i^2/p_i\), which is the Fisher metric on the simplex. The image has norm squared \(4\sum_i p_i=4\).
Distance on the simplex
The spherical embedding gives an explicit Fisher-Rao distance.
Corollary. For strictly positive probability vectors \(p,q\), \[ d_{\mathrm{FR}}(p,q)=2\arccos\left(\sum_i\sqrt{p_iq_i}\right). \]
Proof. The Euclidean sphere of radius \(2\) has geodesic distance \(2\arccos(\langle x,y\rangle/4)\). With \(x=2\sqrt p\) and \(y=2\sqrt q\), \(\langle x,y\rangle/4=\sum_i\sqrt{p_iq_i}\).
Distance versus divergence
Fisher-Rao distance is intrinsic but harder to compute than divergences.
Remark. KL divergence is directed and not a metric; Fisher-Rao distance is symmetric and geodesic. Locally they agree to second order through the Fisher tensor, but globally they encode different geometry.
Length, energy, and geodesic distance
For an absolutely continuous statistical curve \(\gamma:[0,1]\to\mathcal M\), define \[ L_g(\gamma) =\int_0^1\sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,\mathrm{d}t, \qquad E_g(\gamma) =\frac12\int_0^1g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))\,\mathrm{d}t. \] Length is invariant under increasing reparametrization. By Cauchy--Schwarz, \(L_g(\gamma)^2\le2E_g(\gamma)\), with equality for constant-speed curves. The Fisher--Rao distance is the infimum of length over curves joining two laws in the model.
The simplex as a spherical orthant
Theorem. On the interior of the probability simplex \(\Delta_{m-1}^{\circ}\), the map \[ \Phi(p)=2(\sqrt{p_1},\ldots,\sqrt{p_m}) \] is an isometric embedding into the positive orthant of the sphere \(S^{m-1}(2)\). Consequently, \[ d_{\mathrm{FR}}(p,q) =2\arccos\!\left(\sum_{i=1}^m\sqrt{p_iq_i}\right). \]
Proof. A tangent vector \(u\) satisfies \(\sum_i u_i=0\). Since \(\mathrm D\Phi_p[u]=(u_i/\sqrt{p_i})_{i=1}^m\), \[ \langle\mathrm D\Phi_p[u],\mathrm D\Phi_p[v]\rangle =\sum_i\frac{u_iv_i}{p_i}=g_p(u,v). \] Great-circle distance on a radius-two sphere equals twice the angle between its radius vectors. Their normalized inner product is \(\sum_i\sqrt{p_iq_i}\), proving the formula.
Example (explicit geodesic). Put \(\alpha=\arccos\sum_i\sqrt{p_iq_i}\). For \(0<\alpha<\pi\), the constant-speed Fisher geodesic is obtained by squaring the coordinates of \[ s(t) =\frac{\sin\{(1-t)\alpha\}}{\sin\alpha}\sqrt p +\frac{\sin(t\alpha)}{\sin\alpha}\sqrt q, \qquad p_i(t)=s_i(t)^2. \] It is generally neither the ordinary mixture segment nor the exponential segment.
Normal location-scale geometry
For \(N(\mu,\sigma^2)\), \(\sigma>0\), \[ g=\frac{\mathrm{d}\mu^2+2\,\mathrm{d}\sigma^2}{\sigma^2}. \] With \(x=\mu/\sqrt2\) and \(y=\sigma\), this is \(2(\mathrm{d}x^2+\mathrm{d}y^2)/y^2\), twice the Poincaré upper-half-plane metric. Hence the Gaussian location-scale manifold has constant sectional curvature \(-1/2\), and \[ \cosh\!\left(\frac{d_{\mathrm{FR}}}{\sqrt2}\right) =1+\frac{(\mu_1-\mu_2)^2+2(\sigma_1-\sigma_2)^2} {4\sigma_1\sigma_2}. \] This example shows that Fisher geometry need not be flat even when the parameter space is an open subset of Euclidean space.
Intrinsic distance versus statistical divergence
A divergence \(D(P,Q)\) is directed and need not satisfy a triangle inequality; \(d_{\mathrm{FR}}\) is symmetric and metric whenever points can be joined within the regular component. Locally, \[ D_{\mathrm{KL}}(P_\theta\|P_{\theta+\mathrm{d}\theta}) =\frac12\,\mathrm{d}s_{\mathrm{FR}}^2+o(\|\mathrm{d}\theta\|^2), \] but no universal global identity relates KL divergence to Fisher distance. Restricting the ambient simplex to a curved submodel can also enlarge distance: the shortest spherical arc may leave the model.