Maximum mean discrepancy with an RBF kernel

What it takes and gives

Runs with

Python 3.10 or later

Entry points

mmd_squared, permutation_test, median_bandwidth

Known answers

7 cases its tests reproduce

Limits and scope

Limits

Quadratic cost in the pooled sample size: a few thousand rows at most, and permutation tests multiply that by the number of permutations. One Gaussian kernel with one bandwidth; the median heuristic is a convention, not an optimal choice, and the test's power depends on it. Rejecting the null says the samples differ, not how or whether the difference matters.

Guarantees

The biased estimate is non-negative and 0 for identical samples; the unbiased estimate is reported as computed, including negative values. p = (1 + #{permuted statistic >= observed}) / (1 + B) is a valid p-value under exchangeability of the pooled sample. Permutations use a seeded Fisher-Yates shuffle driven by random() only.

Assumes

Rows within each sample are exchangeable; under the null both samples come from one distribution. The bandwidth is fixed before the permutations (the median heuristic uses the pooled sample, which permutations do not change).

Where it is shown

• Statistics for comparing resultsDeterministic skills

Related components

• Brier score decomposition and expected calibration error
• Split conformal prediction intervals
• Anytime-valid confidence sequence for a bounded mean
• Probabilistic and deflated Sharpe ratio
• Ledoit-Wolf covariance shrinkage
• Weighted Vendi score