Pearson and Neyman χ²

The χ² divergences measure squared differences in probability, scaled by one of the two distributions. The Pearson χ² divergence divides by Q:

The Neyman χ² divergence divides by P instead:

Both are f-divergences: Pearson uses $f(t)=(t-1{)}^2$. They are asymmetric and unbounded — Pearson blows up where $Q(x)\to 0$ but $P(x)>0$. Pearson χ² is the statistic behind the classic chi-squared goodness-of-fit test, and it is the divergence minimised by Least-Squares GAN.