Distances Between Probability Distributions of Different Dimensions

Comparing probability distributions is an indispensable and ubiquitous task in machine learning statistics. The most common way to compare a pair of Borel measures compute metric between them, by far the widely used notions are Wasserstein total variation metric. next divergence this case almost every known divergences such as those Kullback–Leibler, Jensen–Shannon, Rényi, many more, special cases $f$ -divergence. Nevertheless these metrics may only be computed, fact, defined, when on spaces same dimension. How would one quantify, say, KL-divergence uniform distribution interval [−1, 1] Gaussian notation="LaTeX">$\mathbb {R}^{3}$ ? We show that give rise natural distances defined different dimensions, e.g., {R}^{m}$ another {R}^{n}$ where notation="LaTeX">$m, n$ distinct, so meaningful answer previous question.