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. 2016 Nov 11;6:36831. doi: 10.1038/srep36831

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Copyright © 2016, The Author(s)

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Two graphs (a) and (b) are the Bayesian networks corresponding to the joint probabilities p(X_k = x_k, Y_k = y_k, Y_k+1 = y_k+1) = p(X_k = x_k)p(Y_k = y_k|X_k = x_k)p(Y_k+1 = y_k+1|X_k = x_k, Y_k = y_k) and p(X_m+1 = x_m+1, Y_m = y_m, Y_k+1 = y_k+1) = p(Y_m = y_m)p(Y_m+1 = y_m+1|Y_m = y_m)p(X_m+1 = x_m+1|Y_m+1 = y_m+1, Y_m = y_m), respectively (see also refs ¹⁵, ³⁷ and ⁶⁰). (a) Transfer entropy corresponds to the edge from X_k to Y_k+1 on the Bayesian network. If TE is zero, the edge from X_k to Y_k+1 vanishes, i.e., p(X_k = x_k, Y_k = y_k, Y_k+1 = y_k+1) = p(X_k = x_k)p(Y_k = y_k|X_k = x_k)p(Y_k+1 = y_k+1|Y_k = y_k). (b) Backward transfer entropy corresponds to the edge from Y_m to X_m+1 on the Bayesian network. If BTE is zero, the edge from Y_m to X_m+1 vanishes, i.e., p(X_m+1 = x_m+1, Y_m = y_m, Y_m+1 = y_m+1) = p(Y_m = y_m)p(Y_m+1 = y_m+1|Y_m = y_m)p(X_m+1 = x_m+1|Y_m+1 = y_m+1).

Inline graphic — Two graphs (a) and (b) are the Bayesian networks corresponding to the joint probabilities p(X_k = x_k, Y_k = y_k, Y_k+1 = y_k+1) = p(X_k = x_k)p(Y_k = y_k|X_k = x_k)p(Y_k+1 = y_k+1|X_k = x_k, Y_k = y_k) and p(X_m+1 = x_m+1, Y_m = y_m, Y_k+1 = y_k+1) = p(Y_m = y_m)p(Y_m+1 = y_m+1|Y_m = y_m)p(X_m+1 = x_m+1|Y_m+1 = y_m+1, Y_m = y_m), respectively (see also refs ¹⁵, ³⁷ and ⁶⁰). (a) Transfer entropy corresponds to the edge from X_k to Y_k+1 on the Bayesian network. If TE is zero, the edge from X_k to Y_k+1 vanishes, i.e., p(X_k = x_k, Y_k = y_k, Y_k+1 = y_k+1) = p(X_k = x_k)p(Y_k = y_k|X_k = x_k)p(Y_k+1 = y_k+1|Y_k = y_k). (b) Backward transfer entropy corresponds to the edge from Y_m to X_m+1 on the Bayesian network. If BTE is zero, the edge from Y_m to X_m+1 vanishes, i.e., p(X_m+1 = x_m+1, Y_m = y_m, Y_m+1 = y_m+1) = p(Y_m = y_m)p(Y_m+1 = y_m+1|Y_m = y_m)p(X_m+1 = x_m+1|Y_m+1 = y_m+1).