The maximum mean discrepancy is the largest difference in expectation over the unit ball of a reproducing-kernel Hilbert space. Its biased empirical estimate keeps the diagonal terms,
Arguments
- x, y
The two samples, matrices with matching column counts. A vector is read as a single column.
- kernel
One of "rbf", "laplacian", "linear". A characteristic kernel (rbf, laplacian) makes MMD zero only when the distributions are equal; "linear" compares means alone and is blind to any difference that leaves the mean unchanged.
- B
Number of label permutations. Default 999.
- cdf
Optional function giving the null CDF of the statistic.
- gamma
Kernel bandwidth. Defaults to the median heuristic on the pooled sample, 1 / median squared pairwise distance. Ignored by the linear kernel.
- unbiased
Use equation (3) rather than the biased estimate.
Value
Named list with statistic, p_value,
kernel, gamma, B, m, n,
unbiased, null_statistics, method.
Details
$$MMD_b^2 = \frac{1}{m^2}\sum_{i,j} k(x_i,x_j) + \frac{1}{n^2}\sum_{i,j} k(y_i,y_j) - \frac{2}{mn}\sum_{i,j} k(x_i,y_j)$$
while the unbiased estimate of Gretton et al.'s equation (3) drops them, dividing the within-sample sums by \(m(m-1)\) and \(n(n-1)\) over \(i \ne j\).
The null has no simple closed form, so the p-value comes from permuting the group labels. Any bias is shared by the observed and permuted statistics and cancels in the ranking, which is why the biased form is a sound test statistic even though the unbiased one is the better estimate of \(MMD^2\).
Mirrors morie.fn.otmtest on the Python side.