Directed information References Navigation menu"Causality, Feedback And Directed Information"cs/060807010.1109/TIT.2008.200984910.1109/TIT.2002.8061350912.487210.1109/TIT.2011.21362701109.129310.1109/TIT.2013.22481921302.397110.1109/TIT.2016.2604846

Information theory


information theoryinformationJames Masseyconditional mutual informationmutual informationcausalitycapacity of channelmemorylessgamblingcompressionreal-time control




Directed information, I(Xn→Yn)displaystyle I(X^nto Y^n), is a measure of information theory and it measures the amount of information that flows from the process Xndisplaystyle X^n to Yndisplaystyle Y^n, where Xndisplaystyle X^n denotes the vector X1,X2,...,Xndisplaystyle X_1,X_2,...,X_n and Yndisplaystyle Y^n denotes Y1,Y2,...,Yndisplaystyle Y_1,Y_2,...,Y_n. The term directed information was coined by James Massey and is defined as



I(Xn→Yn)=∑i=1nI(Xi;Yi|Yi−1)displaystyle I(X^nto Y^n)=sum _i=1^nI(X^i;Y_i,

where I(Xi;Yi|Yi−1)Y^i-1) is the conditional mutual information.


Note that if n=1displaystyle n=1, directed information becomes mutual information I(X;Y)displaystyle I(X;Y). Directed information has many applications in problems where causality plays an important role such as capacity of channel with feedback,[1][2] capacity of discrete memoryless networks with feedback,[3]gambling with causal side information,[4]compression with causal side information,[5]
and in real-time control communication settings.[6]





References




  1. ^ Massey, James (1990). "Causality, Feedback And Directed Information" (ISITA)..mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em


  2. ^ Permuter, Haim Henry; Weissman, Tsachy; Goldsmith, Andrea J. (February 2009). "Finite State Channels With Time-Invariant Deterministic Feedback". IEEE Transactions on Information Theory. 55 (2): 644–662. arXiv:cs/0608070. doi:10.1109/TIT.2008.2009849.


  3. ^ Kramer, G. (January 2003). "Capacity results for the discrete memoryless network". IEEE Transactions on Information Theory. 49 (1): 4–21. doi:10.1109/TIT.2002.806135.


  4. ^ Permuter, Haim H.; Kim, Young-Han; Weissman, Tsachy (June 2011). "Interpretations of Directed Information in Portfolio Theory, Data Compression, and Hypothesis Testing". IEEE Transactions on Information Theory. 57 (6): 3248–3259. arXiv:0912.4872. doi:10.1109/TIT.2011.2136270.


  5. ^ Simeone, Osvaldo; Permuter, Haim Henri (June 2013). "Source Coding When the Side Information May Be Delayed". IEEE Transactions on Information Theory. 59 (6): 3607–3618. arXiv:1109.1293. doi:10.1109/TIT.2013.2248192.


  6. ^ Charalambous, Charalambos D.; Stavrou, Photios A. (August 2016). "Directed Information on Abstract Spaces: Properties and Variational Equalities". IEEE Transactions on Information Theory. 62 (11): 6019–6052. arXiv:1302.3971. doi:10.1109/TIT.2016.2604846.








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