Cencov's theorem explains why the Fisher metric is canonical: under invariance and monotonicity requirements, it is essentially the unique Riemannian metric on finite probability simplices.
Statement on finite simplices
The theorem is a uniqueness result, not merely another formula for Fisher information.
Theorem. On finite positive probability simplices, any smooth Riemannian metric that is invariant under congruent Markov embeddings is a constant multiple of the Fisher metric \[ g_p(u,v)=c\sum_i\frac{u_iv_i}{p_i}. \]
Proof. The full theorem is proved by reducing the metric to its behavior under finite category refinements. Permutation invariance first removes preferred labels. Congruent embeddings then compare a point \(p\) with a refinement in which one mass \(p_i\) is split into \(ap_i\) and \((1-a)p_i\), while tangent vectors split in the same proportions. These comparisons force the metric to be diagonal in the coordinate tangent directions and force each diagonal coefficient to solve the splitting functional equation \(a(x)=c/x\). Summing over coordinates gives the displayed Fisher form, with a single constant \(c\) because permutations identify all coordinates.
Remark. Different formulations use invariance under sufficient statistics, congruent embeddings, or monotonicity under Markov morphisms. The conclusion is the same canonical Fisher tensor up to scale.
Idea of the proof
Permutation symmetry and splitting maps force the metric coefficients to have Fisher form.
Proposition. Permutation invariance makes the metric diagonal with coefficients depending only on the probability coordinate, and invariance under splitting a category forces that coefficient to be proportional to \(1/p_i\).
Proof. Permutations exchange categories, so cross terms and diagonal terms must obey strong symmetry. A congruent embedding that splits one category into two subcategories preserves statistical distinguishability. Comparing the length of a tangent perturbation before and after such a split yields a functional equation for the coefficient \(a(p)\), whose solution is \(a(p)=c/p\).
Why uniqueness matters
The theorem gives a structural reason that Fisher geometry appears in statistics, not only an algebraic accident.
Remark. Cencov's theorem is the information-geometric analogue of a representation theorem: if a metric respects stochastic transformations of experiments, it must be Fisher.
Beyond the classical theorem
Infinite-dimensional, quantum, and noncommutative settings have related but more delicate uniqueness statements.
Remark. In quantum information geometry, monotone metrics form a family indexed by operator-monotone functions, so the classical uniqueness of Fisher becomes a richer classification problem.
Finite statistical models and congruent maps
Let \(\Delta_{n-1}^{\circ}=\{p\in(0,1)^n:\sum_i p_i=1\}\). A tangent vector is \(u\in\mathbb R^n\) with \(\sum_i u_i=0\). For a stochastic matrix \(K\), the induced maps are \(p\mapsto pK\) and \(u\mapsto uK\). A Markov map is congruent when it has a stochastic left inverse on the image; it amounts to splitting each outcome into parameter-independent sub-outcomes.
Theorem (Cencov). Suppose that for every \(n\ge2\), \(g^{(n)}\) is a continuous Riemannian metric on \(\Delta_{n-1}^{\circ}\), and every congruent Markov embedding is an isometry. Then a constant \(c>0\), independent of \(n\), satisfies \[ g_p^{(n)}(u,v)=c\sum_{i=1}^n\frac{u_iv_i}{p_i}. \] Thus the Fisher metric is unique up to one common positive scale.
The rational-point reduction
Lemma. At the uniform distribution \(\bar p_n=(1/n,\ldots,1/n)\), permutation invariance forces \[ g_{\bar p_n}(u,v)=a_n\sum_{i=1}^n u_iv_i \quad\text{on }\mathbf1^\top u=\mathbf1^\top v=0 \] for some \(a_n>0\).
Proof. The metric matrix at \(\bar p_n\) commutes with every permutation matrix. Such matrices have the form \(\alpha I+\beta\mathbf1\mathbf1^\top\). The second term vanishes on the tangent hyperplane, leaving the stated Euclidean multiple.
Lemma. Congruently splitting every cell of \(\bar p_n\) into \(m\) equal subcells yields \(a_{nm}=ma_n\). Hence \(a_n=cn\) for a common \(c>0\).
Proof. The embedded tangent vector repeats each coordinate \(u_i/m\) exactly \(m\) times. Isometry gives \[ a_n\sum_i u_i^2 =a_{nm}\sum_{i=1}^n\sum_{j=1}^m(u_i/m)^2 =\frac{a_{nm}}m\sum_i u_i^2. \] Thus \(a_{nm}=ma_n\). Comparing two dimensions through a common multiple shows that \(a_n/n\) is constant.
For a rational point \(p_i=k_i/N\), split cell \(i\) into \(k_i\) equal subcells. The resulting distribution is uniform on \(N\) points. Pulling the uniform metric back gives \[ g_p(u,v)=c\sum_i\frac{u_iv_i}{p_i}. \] Continuity extends the formula from rational points to the whole interior. A fully formal proof also verifies compatibility among all simplex dimensions, but these lemmas contain the decisive mechanism.
What uniqueness does and does not assert
The theorem does not say that Fisher is the only metric one can put on one fixed parametric family. Uniqueness comes from requiring coherent metrics on all finite models and invariance under sufficient statistics. Infinite sample spaces need additional continuity assumptions. Quantum state spaces admit an entire Petz family of monotone metrics because noncommutativity changes the allowed morphisms.