Likelihood ratios encode an experiment relative to a base measure. Hellinger geometry works with their square roots, placing probability measures on the positive part of the unit sphere in \(L_2\). This geometry is stable under products and gives quantitative bounds for testing, contiguity, and quadratic likelihood approximation.
Likelihood-ratio coordinates
Definition. If \(Q\ll P\), the likelihood ratio is \[ L=\frac{\mathrm{d}Q}{\mathrm{d}P}. \] It is nonnegative and satisfies \(E_PL=1\). For an experiment dominated by \(P_{\theta_0}\), the likelihood-ratio process is \[ L_\theta = \frac{\mathrm{d}P_\theta}{\mathrm{d}P_{\theta_0}}, \qquad\theta\in\Theta. \]
Proposition. For a finite parameter set \(\{\theta_0,\theta_1,\ldots,\theta_k\}\), the law under \(P_{\theta_0}\) of \[ (L_{\theta_1},\ldots,L_{\theta_k}) \] determines the experiment up to Blackwell equivalence.
Proof. Map the observation to its likelihood-ratio vector. Its distribution under \(\theta_0\) is \(\mu\), and under \(\theta_j\) it has density \(z_j\) with respect to \(\mu\), because \[ E_{\theta_j}f(L) =E_{\theta_0}\{L_{\theta_j}f(L)\} =\int z_jf(z)\mu(\mathrm{d}z). \] The vector is sufficient by factorization. Hence the original experiment is equivalent to this canonical likelihood-ratio experiment. \(\square\)
Hellinger affinity and distance
Definition. For probability measures \(P,Q\), dominated by \(\mu\) with densities \(p,q\), define the Hellinger affinity and distance by \[ \rho(P,Q)=\int\sqrt{pq}\,\mathrm{d}\mu, \] \[ H^2(P,Q) = \int(\sqrt p-\sqrt q)^2\,\mathrm{d}\mu =2\{1-\rho(P,Q)\}. \]
The definition is independent of the dominating measure. Since \(\|\sqrt p\|_2=\|\sqrt q\|_2=1\), \(H(P,Q)\) is the chordal distance between two unit vectors in \(L_2(\mu)\).
Proposition. \[ \frac12H^2(P,Q) \leq\|P-Q\|_{\mathrm{TV}} \leq H(P,Q). \]
Proof. Since \[ |\sqrt p-\sqrt q|^2 \leq|\sqrt p-\sqrt q|(\sqrt p+\sqrt q) =|p-q|, \] integration and \(\|P-Q\|_{\mathrm{TV}}=\frac12\int|p-q|\) give the first inequality. For the second, Cauchy–Schwarz yields \[ \begin{aligned} 2\|P-Q\|_{\mathrm{TV}} &=\int|\sqrt p-\sqrt q|(\sqrt p+\sqrt q)\,\mathrm{d}\mu\\ &\leq H(P,Q) \left\{\int(\sqrt p+\sqrt q)^2\,\mathrm{d}\mu\right\}^{1/2}\\ &\leq2H(P,Q). \end{aligned} \] \(\square\)
Testing affinity
Proposition. For every test \(0\leq\varphi\leq1\), \[ P\varphi+Q(1-\varphi) \geq1-\|P-Q\|_{\mathrm{TV}} \geq1-H(P,Q). \] Thus two measures close in Hellinger distance cannot simultaneously have small type-I and type-II errors.
The first lower bound is exact after minimizing over tests. Hellinger distance is often easier to calculate, especially for independent products.
Product geometry
Theorem. For product measures, \[ \rho\!\left(\bigotimes_{i=1}^nP_i, \bigotimes_{i=1}^nQ_i\right) = \prod_{i=1}^n\rho(P_i,Q_i). \] Hence \[ H^2\!\left(\bigotimes_iP_i,\bigotimes_iQ_i\right) = 2\left[ 1-\prod_{i=1}^n \left\{1-\frac12H^2(P_i,Q_i)\right\} \right] \leq\sum_{i=1}^nH^2(P_i,Q_i). \]
Proof. Fubini's theorem factors the integral of the square root of the product densities. The inequality follows from \(1-\prod_i(1-a_i)\leq\sum_i a_i\) for \(0\leq a_i\leq1\). \(\square\)
If every component distance is small, then \[ H^2\!\left(\bigotimes_iP_i,\bigotimes_iQ_i\right) \sim\sum_iH^2(P_i,Q_i). \] This near-additivity is a principal reason Hellinger expansions are effective for independent non-identically distributed experiments.
Likelihood ratios in Hellinger form
If \(Q\ll P\) with likelihood ratio \(L\), then \[ \rho(P,Q)=E_P\sqrt L, \qquad H^2(P,Q)=E_P(\sqrt L-1)^2. \] Thus Hellinger closeness is an \(L_2(P)\) statement about the square-root likelihood ratio. Quadratic mean differentiability takes exactly this form: \[ \sqrt{p_{\theta+t}} =\sqrt{p_\theta} +\frac12t^\top\dot\ell_\theta\sqrt{p_\theta} +o(\|t\|) \quad\text{in }L_2. \]
Kullback–Leibler and Hellinger comparison
Proposition. If \(P\ll Q\), then \[ H^2(P,Q)\leq D_{\mathrm{KL}}(P\|Q). \]
Proof. With \(u=p/q\), the scalar inequality \[ (\sqrt u-1)^2\leq u\log u-u+1 \] and integration against \(Q\) give \[ H^2(P,Q) \leq\int(p\log(p/q)-p+q)\,\mathrm{d}\mu =D_{\mathrm{KL}}(P\|Q). \] \(\square\)
KL divergence is directional and additive for products; Hellinger distance is symmetric and bounded. Le Cam theory uses whichever geometry matches the needed operation, often deriving total-variation control from one of them.