Jensen-$$\phi $$-entropy, g-mean mixture distributions and optimal information: application to binary models

Abstract This paper has two main objectives. First, we introduce a Jensen-type extension of the $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> -entropy, which naturally generalizes the classical Jensen–Shannon divergence and explores its connections with Bregman divergence. Several theoretical results are established that highlight the interplay between these entropy and divergence measures, particularly in the context of mixture distributions. Second, we propose the g-mean mixture distribution, a general class of statistical models that encompasses many classical mixtures, such as arithmetic, geometric, and $$(\alpha ,\lambda )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -mixtures, as special cases. We show that this class arises as the optimal solution to a family of optimization problems formulated in terms of Bregman divergence. Finally, we apply the Jensen- $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> -entropy to two binary models as an alternative to classical estimation methods, demonstrating promising empirical performance and potential for further theoretical investigation.