Equivalence of distance-based and RKHS-based statistics in hypothesis testing

Source: Dev.to

Overview

The article explains that two popular tools for testing differences between groups—energy distance (a classic statistical measure) and Maximum Mean Discrepancy (MMD) (a kernel‑based method from machine learning)—are actually two representations of the same underlying concept when an appropriate kernel is chosen.

Key Insights

• Equivalence: A test formulated with energy distance can be expressed in the RKHS (Reproducing Kernel Hilbert Space) framework of MMD, and vice‑versa.
• Kernel Choice: While some kernels reproduce the traditional energy‑distance test, other kernels can increase statistical power, allowing detection of subtler differences.
• Consistency: The work clarifies conditions under which these tests are consistent (they will detect any true difference) and when they may fail.

Practical Implications

• Researchers can freely switch between the two formulations, leveraging tools and theory from both statistics and machine learning.
• By experimenting with different kernels, practitioners can tailor tests to specific data sets without learning an entirely new methodology.
• The connection provides simple, actionable tricks to improve test performance, leading to more robust conclusions.

Who Should Care

• Scientists conducting hypothesis tests on experimental data.
• Students learning about non‑parametric testing methods.
• Data enthusiasts interested in applying modern machine‑learning techniques to classical statistical problems.

Further Reading

Equivalence of distance-based and RKHS-based statistics in hypothesis testing