Concrete models show how Fisher geometry encodes scale, curvature, and parameter interactions. The same formulas can look very different in different coordinates.
Bernoulli family
The one-dimensional Bernoulli model has an explicit Fisher line element.
Example. For \(P_\theta(Y=1)=\theta\), \(0<\theta<1\), \[ g(\theta)=\frac{1}{\theta(1-\theta)}. \]
Proof. The score is \(Y/\theta-(1-Y)/(1-\theta)\). Its variance under Bernoulli\((\theta)\) is \(1/\theta+1/(1-\theta)=1/\{\theta(1-\theta)\}\).
Normal location-scale family
The normal family is a basic curved surface with orthogonal location and scale directions.
Proposition. For \(N(\mu,\sigma^2)\), \[ g_{\mu\mu}=\frac{1}{\sigma^2},\qquad g_{\mu\sigma}=0,\qquad g_{\sigma\sigma}=\frac{2}{\sigma^2}. \]
Proof. The scores are \(s_\mu=(X-\mu)/\sigma^2\) and \(s_\sigma=-1/\sigma+(X-\mu)^2/\sigma^3\). Using central moments \(\mathbb E(X-\mu)^2=\sigma^2\), \(\mathbb E(X-\mu)^3=0\), and \(\mathbb E(X-\mu)^4=3\sigma^4\) gives the entries.
Categorical family
The categorical family is the probability simplex with Fisher metric.
Example. For \(p=(p_1,\ldots,p_n)\), tangent vectors satisfy \(\sum_i u_i=0\) and the Fisher metric is \(g_p(u,u)=\sum_i u_i^2/p_i\).
Remark. This example is the local form behind multinomial asymptotics and chi-square approximations.
Poisson family
The Poisson mean has information inverse to the mean.
Proposition. For \(X\sim\operatorname{Poisson}(\lambda)\), \[ g(\lambda)=\frac1\lambda. \]
Proof. The score is \(X/\lambda-1\). Its variance is \(\operatorname{Var}(X)/\lambda^2=\lambda/\lambda^2=1/\lambda\).
One-parameter exponential families
For \(p_\theta(x)=\exp\{\theta T(x)-\psi(\theta)\}h(x)\), \[ I(\theta)=\psi''(\theta)=\operatorname{Var}_\theta T(X). \] Every one-dimensional metric is locally flattened by \(s(\theta)=\int^\theta\sqrt{I(t)}\,\mathrm{d}t\), but boundary distances still encode statistical behavior.
Example (Poisson). For \(X\sim\operatorname{Poisson}(\lambda)\), \(I(\lambda)=1/\lambda\). The coordinate \(s=2\sqrt\lambda\) yields \[ d_{\mathrm{FR}}(\lambda_1,\lambda_2) =2|\sqrt{\lambda_1}-\sqrt{\lambda_2}|. \]
Example (exponential rate). For density \(p_\lambda(x)=\lambda e^{-\lambda x}\mathbf1_{x\ge0}\), \(I(\lambda)=1/\lambda^2\). Thus \(s=\log\lambda\) and \(d_{\mathrm{FR}}(\lambda_1,\lambda_2)=|\log(\lambda_1/\lambda_2)|\). Both \(0\) and \(\infty\) are infinitely far away.
Normal location and scale
For \(N(\mu,\sigma^2)\), \[ \ell_\mu=\frac{X-\mu}{\sigma^2}, \qquad \ell_\sigma=-\frac1\sigma+\frac{(X-\mu)^2}{\sigma^3}, \qquad I(\mu,\sigma)= \begin{pmatrix}\sigma^{-2}&0\\0&2\sigma^{-2}\end{pmatrix}. \] Location and scale are orthogonal, but their common factor creates negative curvature. With fixed \(\sigma\), distance is \(|\mu_1-\mu_2|/\sigma\); allowing scale to vary can shorten the path by moving through larger variances.
Multivariate Gaussian covariance geometry
Proposition. On zero-mean Gaussian laws \(N(0,\Sigma)\), \(\Sigma\in\mathbb S_{++}^d\), \[ g_\Sigma(U,V)=\frac12\operatorname{tr} (\Sigma^{-1}U\Sigma^{-1}V), \] and \[ d_{\mathrm{FR}}(\Sigma_0,\Sigma_1) =\frac1{\sqrt2}\left\|\log\!\left( \Sigma_0^{-1/2}\Sigma_1\Sigma_0^{-1/2} \right)\right\|_{\mathrm F}. \]
Proof. Along \(\Sigma_t=\Sigma+tU\), \[ \ell_U(x) =-\frac12\operatorname{tr}(\Sigma^{-1}U) +\frac12x^\top\Sigma^{-1}U\Sigma^{-1}x. \] The Gaussian quadratic-form covariance identity yields the metric. Congruence invariance reduces distance to \(\Sigma_0=I\); diagonalizing \(\Sigma_1\) leaves a product of scalar log-scale geodesics and gives the formula.
Location families beyond Gaussianity
For \(p_\mu(x)=f(x-\mu)\), assuming finite information, \[ I_\mu=\int\{\nabla\log f(z)\}\{\nabla\log f(z)\}^\top f(z)\,\mathrm{d}z \] is constant in \(\mu\). The location manifold is therefore flat, although different noise shapes induce different constant matrices. Gaussianity is special globally through entropy and projection properties, not through this flatness alone.