Chernoff information links hypothesis testing error exponents with exponential geodesics between distributions. It is one of the clearest operational appearances of information geometry.
Chernoff coefficient
The Chernoff exponent is obtained by optimizing a log-normalizer along an exponential path.
Definition. For densities \(p,q\), \[ C(P,Q)=-\min_{0\le t\le1}\log\int p^{1-t}q^t\,\mathrm{d}\mu. \] The density proportional to \(p^{1-t}q^t\) lies on the exponential geodesic joining \(p\) and \(q\).
Remark. At \(t=1/2\), the same expression gives the Bhattacharyya coefficient. Chernoff information optimizes over the point on the exponential geodesic.
Testing exponent
Chernoff information is the best symmetric exponential error rate for testing two simple hypotheses.
Theorem. For testing \(P^n\) versus \(Q^n\) with equal priors, the optimal Bayes error probability satisfies \[ -\lim_{n\to\infty}\frac1n\log P_e^{(n)}=C(P,Q). \]
Proof. The likelihood-ratio test is optimal by Neyman-Pearson. Upper bounds follow from exponential Markov inequalities applied to likelihood ratios, producing \(\int p^{1-t}q^t\). Lower bounds use a change-of-measure argument at the tilted Chernoff distribution, showing that no test can beat the optimized exponent.
Chernoff point
The optimizing point balances the two KL divergences back to the endpoints.
Proposition. If \(t^*\in(0,1)\) is the optimizer and \(r_{t^*}\propto p^{1-t^*}q^{t^*}\), then \[ D(R_{t^*}\|P)=D(R_{t^*}\|Q) \] under differentiability and interior optimality.
Proof. Let \(\psi(t)=\log\int p^{1-t}q^t\,\mathrm{d}\mu\). The derivative is \(\psi'(t)=\mathbb E_{R_t}\log(q/p)\). Interior optimality gives \(\psi'(t^*)=0\). Expanding the two KL divergences from \(R_{t^*}\) to \(P\) and \(Q\) shows their difference is exactly \(\mathbb E_{R_{t^*}}\log(q/p)\).
Geometric interpretation
The Chernoff point is where the exponential geodesic meets a decision boundary of equal divergence.
Remark. This picture is especially clean in exponential families, where the Chernoff path is a straight line in natural parameters and the optimizing point balances normal directions to likelihood-ratio level sets.
Exponential interpolation between two laws
For mutually dominated \(P,Q\), define \[ Z(s)=\int p^{1-s}q^s\,\mathrm{d}\mu,\qquad \psi(s)=\log Z(s),\qquad 0\le s\le1, \] and the exponential arc \[ r_s(x)=\exp\{(1-s)\log p(x)+s\log q(x)-\psi(s)\}. \] Hölder's inequality gives \(Z(s)\le1\); \(\psi\) is convex.
Proposition. \[ \psi'(s)=E_{R_s}\log\frac qp,\qquad \psi''(s)=\operatorname{Var}_{R_s}\!\left(\log\frac qp\right)\ge0. \] If \(\log(q/p)\) is nonconstant, the minimizer \(s^*\) is unique whenever it is interior and satisfies \(E_{R_{s^*}}\log(q/p)=0\).
Proof. Differentiate the log normalizer under the integral. The first derivative is the tilted expectation; the second is its variance. Strict positivity of the variance gives strict convexity.
The Chernoff point is equidistant in KL
Theorem. If \(s^*\in(0,1)\) minimizes \(\psi\), then \[ C(P,Q)=-\psi(s^*) =D(R_{s^*}\|P)=D(R_{s^*}\|Q). \]
Proof. Since \(\log(r_s/p)=s\log(q/p)-\psi(s)\), \[ D(R_s\|P)=s\psi'(s)-\psi(s). \] Similarly, \[ D(R_s\|Q)=-(1-s)\psi'(s)-\psi(s). \] At the minimizer \(\psi'(s^*)=0\), both equal \(-\psi(s^*)\).
Operational testing exponent
Theorem (Chernoff). For testing \(P^n\) versus \(Q^n\) with fixed positive prior probabilities, the minimum Bayes error \(P_{e,n}^*\) satisfies \[ \lim_{n\to\infty}-\frac1n\log P_{e,n}^*=C(P,Q) =-\min_{0\le s\le1}\log\int p^{1-s}q^s\,\mathrm{d}\mu. \]
Achievability follows by exponential tilting and large deviations of the log-likelihood ratio; the matching converse changes measure to the Chernoff point, under which the decision boundary has zero mean. The symmetric choice \(s=1/2\) gives Bhattacharyya distance, an upper bound on Bayes error but not generally the optimal exponent.
Local Fisher approximation
For \(Q=P_{\theta+h}\) near \(P=P_\theta\), \[ C(P_\theta,P_{\theta+h}) =\frac18h^\top I_\theta h+o(\|h\|^2), \] with \(s^*=1/2+o(1)\). Thus the Chernoff exponent locally measures one eighth of squared Fisher length. Globally it follows the exponential arc and can differ substantially from Fisher--Rao distance.