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Where Det stands for Determinism, and Deg stands for Degeneracy. EI is the difference between the two. In the table, we also list the values of Det and Deg corresponding to the matrices.

Where Det stands for Determinism, and Deg stands for Degeneracy. EI is the difference between the two. In the table, we also list the values of Det and Deg corresponding to the matrices.

The first transition probability matrix is a [[Permutation]] matrix and is reversible; thus, it has the highest determinism, no degeneracy, and therefore the highest EI. The second matrix's first three states transition to each other with a 1/3 probability, resulting in the lowest determinism but also low degeneracy, yielding an EI of 0.81. The third matrix, despite having binary transitions, has high degeneracy because all three states transition to state 1, meaning we cannot infer their previous state. Thus, its EI equals that of the second matrix at 0.81.

The first transition probability matrix is a [[Permutation matrix]] and is reversible; thus, it has the highest determinism, no degeneracy, and therefore the highest EI. The second matrix's first three states transition to each other with a 1/3 probability, resulting in the lowest determinism but also low degeneracy, yielding an EI of 0.81. The third matrix, despite having binary transitions, has high degeneracy because all three states transition to state 1, meaning we cannot infer their previous state. Thus, its EI equals that of the second matrix at 0.81.

Although in the original literature <ref name=hoel_2013 />, EI was mostly applied to discrete-state Markov chains, [[Zhang Jiang]], [[Liu Kaiwei]], and [[Yang Mingzhe]] extended the definition to more general continuous-variable cases<ref name=zhang_nis /><ref name=yang_nis+ /><ref name=liu_exact />. This extension builds on EI's original definition by intervening on the cause variable X as a uniform distribution over a sufficiently large bounded interval, [math][-\frac{L}{2}, \frac{L}{2}]^n[/math]. The causal mechanism is assumed to be a conditional probability that follows a Gaussian distribution with a mean function [math]f(x)[/math] and covariance matrix [math]\Sigma[/math]. Based on this, the EI between the causal variables is then measured. The causal mechanism here is determined by the mapping [math]f(x)[/math] and the covariance matrix, which together define the conditional probability [math]Pr(y|x)[/math].

Although in the original literature <ref name=hoel_2013 />, EI was mostly applied to discrete-state Markov chains, [[Zhang Jiang]], [[Liu Kaiwei]], and [[Yang Mingzhe]] extended the definition to more general continuous-variable cases<ref name=zhang_nis /><ref name=yang_nis+ /><ref name=liu_exact />. This extension builds on EI's original definition by intervening on the cause variable X as a uniform distribution over a sufficiently large bounded interval, [math][-\frac{L}{2}, \frac{L}{2}]^n[/math]. The causal mechanism is assumed to be a conditional probability that follows a Gaussian distribution with a mean function [math]f(x)[/math] and covariance matrix [math]\Sigma[/math]. Based on this, the EI between the causal variables is then measured. The causal mechanism here is determined by the mapping [math]f(x)[/math] and the covariance matrix, which together define the conditional probability [math]Pr(y|x)[/math].