  Causal inference is perhaps one of the most fundamental concepts in science,
beginning originally from the works of some of the ancient philosophers,
through today, but also weaved strongly in current work from statisticians,
machine learning experts, and scientists from many other fields. This paper
takes the perspective of information flow, which includes the Nobel prize
winning work on Granger-causality, and the recently highly popular transfer
entropy, these being probabilistic in nature. Our main contribution will be to
develop analysis tools that will allow a geometric interpretation of
information flow as a causal inference indicated by positive transfer entropy.
We will describe the effective dimensionality of an underlying manifold as
projected into the outcome space that summarizes information flow. Therefore
contrasting the probabilistic and geometric perspectives, we will introduce a
new measure of causal inference based on the fractal correlation dimension
conditionally applied to competing explanations of future forecasts, which we
will write $GeoC_{y\rightarrow x}$. This avoids some of the boundedness issues
that we show exist for the transfer entropy, $T_{y\rightarrow x}$. We will
highlight our discussions with data developed from synthetic models of
successively more complex nature: then include the H\'{e}non map example, and
finally a real physiological example relating breathing and heart rate
function.
  Keywords: Causal Inference; Transfer Entropy; Differential Entropy;
Correlation Dimension; Pinsker's Inequality; Frobenius-Perron operator.
