De Bruijn's identity is a celebrated result in information theory, linking the derivative of entropy with respect to the variance of Gaussian perturbation to the Fisher information. While it has been widely applied and rigorously derived under second moment or mild regularity assumptions, the necessary conditions ensuring the validity of the required exchanges of differentiation and integration have not been fully identified in the literature.
In this talk, we present necessary and sufficient conditions for de Bruijn's identity, as well as for continuity of entropy with respect to Gaussian perturbation. We explain their implications for characterizing the equality case in the Entropy Power Inequality (EPI) under minimal assumptions. We then turn to the question of stability in the EPI. After reviewing some existing negative and positive results, we conclude with a qualitative stability theorem in the weak convergence sense, valid under very mild assumptions.
The talk is based on joint work with Ioannis Kontoyiannis (Cambridge).