The specific findings and empirical results derived in the study 'On Geometry of Information Flow for Causal Inference', as illustrated in the accompanying figure.

University of Michigan - Ann Arbor

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Slightly out of scope, but a very interesting read : 
[arxiv.org/abs/2002.02078](https://arxiv.org/abs/2002.02078) 

On Geometry of Information Flow for Causal Inference

If there exists a system X, we ask if the data contained in X is sufficient to predict future states of X, or if there is a system Y that can help improve our predictions of X. If there is, then we say X is not closed.

Granger Causality

Movement of information between systems. 

Information flow

Summary : Image

Given two dynamical systems x, y such that :
$$x' = f_1(x, y)$$
$$y' = f_2(x, y)$$
Where $a'$ denotes the next iteration/future version of $a$, transfer entropy is defined as : 
$$T_{y \rightarrow x} =  D_{KL}((p(x'|x))||p(x'|x,y) )$$


Transfer Entropy

Asymmetric Space Transfer Operator Theorem

Theorem 2

Conditional Correlation Dimensional Geometric Information flow

Correlation Dimensional Geometric information Flow

1. Identify causality through geometric methods. 2. Derivative of the function of the time series is related to transfer entropy. 3. Provide tool $$\textit{geoC}$$ to identify causality geometrically. 4. Correlation dimension can be used to determine causality.

Key Ideas of 'On Geometry of Information Flow for Causal Inference'

Main conclusions of the paper 'On Geometry of Information Flow for Causal Inference':

1. It develops a geometric interpretation of information flow as causal inference, measured by positive transfer entropy.
2. This geometric description allows for newer and more efficient computational methods for causal inference.
3. It also reveals the geometry of the underlying data.

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