We have proved in Section 6.5 that if X1X2X3X4 forms a Markov chain, the I-Measure μ* always vanishes on the five atoms
$$\eqalign{ & {\widetilde X_1} \cap \widetilde X_2^c \cap {\widetilde X_3} \cap \widetilde X_4^c \cr & {\widetilde X_1} \cap \widetilde X_2^c \cap {\widetilde X_3} \cap \widetilde X_4^{} \cr & {\widetilde X_1} \cap \widetilde X_2^c \cap \widetilde X_3^c \cap {\widetilde X_4} \cr & {\widetilde X_1} \cap {\widetilde X_2} \cap \widetilde X_3^c \cap {\widetilde X_4} \cr & \widetilde X_1^c \cap {\widetilde X_2} \cap \widetilde X_3^c \cap {\widetilde X_4}. \cr} $$
Consequently, the I-Measure μ* is completely specified by the values of J.L* on the other ten nonempty atoms of F4, and the information diagram for four random variables forming a Markov chain can be displayed in two dimensions as in Figure 6.11.




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Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Raymond W. Yeung
    • 1
  1. 1.The Chinese University of Hong KongHong Kong

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