This paper takes advantage of kernel method to test whether two r.v.s have the same distribution. The idea is very close to independence test based on kernel method. We know two r.v.s
\mathbb{E} f(x) = \mathbb{E} f(y) if their distribution is identical. Given a large enough function set, we may use\mathrm{MMD}(\mathcal{F}, x, y ) = \sup_{f \in \mathcal{F}} \mathbb{E} f(x) - \mathbb{E} f(y).Then we try some universal kernel and its corresponding RKHS. With some derivation, the empirical evalutation of MMD is based on\mathrm{MMD}^2(\mathcal{F}, x, y) = \mathbb{E} \langle \phi(x), \phi(x')\rangle + \mathbb{E} \langle \phi(y), \phi(y')\rangle - 2 \mathbb{E} \langle \phi(x), \phi(y)\rangle \approx \frac{1}{m^2} \sum_{i, j = 1}^m k(x_i, x_j) + \frac{1}{n^2} \sum_{i, j = 1}^n k(y_i, y_j) - 2 \frac{1}{mn} \sum_{i = 1}^m \sum_{j = 1}^n k(x_i, y_j).I am not sure whether a later work in NIPS 2008 is based on this, which will be scanned soon.
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