TOPICS
Search

Amari-Kobayashi Conjecture

The Amari-Kobayashi conjecture is an extremal Fisher information problem for separately compressed correlated binary data. Let (X_i,Y_i), 1<=i<=n, be independent pairs of uniform signs with Pr(X_i=x,Y_i=y)=(1+rxy)/4 for x,y in {-1,1} and -1<r<1. Two encoders represented by Boolean functions produce f(X_1,...,X_n) and g(Y_1,...,Y_n), and the question is to maximize the Fisher information about r in their joint distribution (Amari 2011, 2023).

Chen et al. (2026) proved that, for r!=0, the optimum is

max_(1<=k<=n)(k^2r^(2k-2))/(1-r^(2k)).

It is attained by matching parity functions on the same k coordinates, namely the products of the selected X_i and of the corresponding Y_i. At r=0 the optimum is 1. The encoders are held fixed when differentiating with respect to r, although the optimizing choice can depend on its target value. Chen et al. (2026) report GPT-6 Astra assistance with an analytic proof replacing an earlier computer-certified scalar step.

See also

Fisher Information

Explore with Wolfram|Alpha

References

Amari, S.-i. "On Optimal Data Compression in Multiterminal Statistical Inference." IEEE Trans. Inform. Th. 57, 5577-5587, 2011. https://doi.org/10.1109/TIT.2011.2162270.Amari, S.-i. "Multiterminal Statistical Inference: An Unsolved Problem." In Mathematics Going Forward: Collected Mathematical Brushstrokes (Ed. J.-M. Morel and B. Teissier). Cham, Switzerland: Springer, pp. 361-366, 2023. https://doi.org/10.1007/978-3-031-12244-6_25.Chen, J.; Watanabe, S.; and Yu, L. "A Sharp Fisher-Information Inequality for Boolean Pairs: Proof of the Amari-Kobayashi Conjecture." 29 Oct 2026. https://arxiv.org/abs/2610.39355.

Cite this as:

Weisstein, Eric W. "Amari-Kobayashi Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Amari-KobayashiConjecture.html

Subject classifications