Hang, 2022). While this method provably converges on
the true density in the limit of inﬁnite data, ﬁnite sample
performance is inevitably rough and discontinuous. Smooth
results can be obtained by applying either kernel density
estimation (KDE) or maximum likelihood estimation (MLE)
within each leaf (Smyth et al., 1995; Gray and Moore, 2003;
Loh, 2009; Ram and Gray, 2011; Criminisi et al., 2012),
a version of which we develop further below. Existing
methods have mostly been limited to supervised trees rather
than unsupervised forests, and are often inefﬁcient in high
dimensions.
Another strategy, better suited to high-dimensional settings,
uses Chow-Liu trees (Chow and Liu, 1968) to learn a second-
order approximation to the underlying joint distribution
(Bach and Jordan, 2003; Liu et al., 2011; Rahman et al.,
2014). Whereas these methods estimate a series of bivariate
densities over the full support of the data, we attempt to
Adversarial Random Forests for Density Estimation and Generative Modeling