  Causal investigations in observational studies pose a great challenge in
research where randomized trials or intervention-based studies are not
feasible. We develop an information geometric causal discovery and inference
framework of "predictive asymmetry". For $(X, Y)$, predictive asymmetry enables
assessment of whether $X$ is more likely to cause $Y$ or vice-versa. The
asymmetry between cause and effect becomes particularly simple if $X$ and $Y$
are deterministically related. We propose a new metric called the Directed
Mutual Information ($DMI$) and establish its key statistical properties. $DMI$
is not only able to detect complex non-linear association patterns in bivariate
data, but also is able to detect and infer causal relations. Our proposed
methodology relies on scalable non-parametric density estimation using Fourier
transform. The resulting estimation method is manyfold faster than the
classical bandwidth-based density estimation. We investigate key asymptotic
properties of the $DMI$ methodology and a data-splitting technique is utilized
to facilitate causal inference using the $DMI$. Through simulation studies and
an application, we illustrate the performance of $DMI$.
