An Invariant Measure for Differential Entropy: From Kullback-Leibler Divergence to Scale-Invariant Information Theory

Shannon's differential entropy for continuous variables suffers from a fundamental limitation: it is not invariant under scale transformations. This makes entropy values dependent on the choice of measurement units rather than reflecting intrinsic properties of distributions. While Jaynes proposed the limiting density of discrete points (LDDP) as a theoretical solution, a concrete method for computing the required invariant measure has been lacking. This paper establishes a rigorous connection between Kullback-Leibler divergence and the invariant measure, providing theoretical proofs of invariance under affine transformations and a practical computational method. We prove that entropy normalized by the median of k-nearest neighbor distances is invariant under affine transformations (Theorems 1 and 2). The non-negativity of the resulting entropy has been validated empirically across all tested distribution families, though a complete theoretical proof remains an open question. This approach extends naturally to multivariate settings, enabling scale-invariant mutual information estimation. We provide open-source implementations in Julia (EntropyInvariant.jl) and Python (entropy_invariant) and demonstrate their advantages over traditional approaches, particularly for variables with disparate scales.