Entropy of intervening variable in Markov Chain
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Let's assume we are given discrete random variables $X$, $Z$, with some nonzero mutual information $I[X,Z] > 0$. I would like to understand the minimum entropy of variables $Y$ such that $X rightarrow Y rightarrow Z$ is a Markov chain. By the data processing inequality, $H[Y] geq I[X,Z]$. But can we also find some $Y$ for which this bound is close to tight, that is $H[Y]$ close to $I[X,Z]$?
More formally: Let $F$ be the set of variables $Y$ such that $X rightarrow Y rightarrow Z$ is a Markov chain. Can we upper-bound $inf_{Yin F} H[Y]$ in relation to $I[X,Z]$?
Or are there conditions under which such an upper bound holds?
I'd be interested in any kind of upper bound on $inf_{Yin F} H[Y]$, even if it is much larger than $I[X,Z]$, as long as it is better than $inf_{Yin F} H[Y] leq H[X], H[Z]$.
I have looked around the web but didn't find anything related. Any pointers (or an explanation why this isn't possible) would be greatly appreciated. Thanks!
markov-chains information-theory entropy
asked Nov 25 at 19:54
Michael Hahn
11
add a comment |
up vote
0
down vote
favorite
Let's assume we are given discrete random variables $X$, $Z$, with some nonzero mutual information $I[X,Z] > 0$. I would like to understand the minimum entropy of variables $Y$ such that $X rightarrow Y rightarrow Z$ is a Markov chain. By the data processing inequality, $H[Y] geq I[X,Z]$. But can we also find some $Y$ for which this bound is close to tight, that is $H[Y]$ close to $I[X,Z]$?
More formally: Let $F$ be the set of variables $Y$ such that $X rightarrow Y rightarrow Z$ is a Markov chain. Can we upper-bound $inf_{Yin F} H[Y]$ in relation to $I[X,Z]$?
Or are there conditions under which such an upper bound holds?
I'd be interested in any kind of upper bound on $inf_{Yin F} H[Y]$, even if it is much larger than $I[X,Z]$, as long as it is better than $inf_{Yin F} H[Y] leq H[X], H[Z]$.
I have looked around the web but didn't find anything related. Any pointers (or an explanation why this isn't possible) would be greatly appreciated. Thanks!
markov-chains information-theory entropy
asked Nov 25 at 19:54
Michael Hahn
11
I think you're looking at the Wyner common information. Check out ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1055346 , in particular the subsection marked 'second approach' on pg 164 (and, for that matter, the rest of the paper, it's quite nice). Wyner also mentions conditions for $H(Y) = I(X;Z)$ to hold, and there's another paper of Witsenhausen that may be interesting. I'm not sure if there are better bounds than the trivial ones above but given the time there must be progress on this. In any case the keyword should help you find an answer.
– stochasticboy321
Nov 27 at 2:57
Awesome thanks!
– Michael Hahn
Nov 27 at 19:54
You're welcome. I just remembered - a mid 2010s paper of Anantharam and Kamath did some fun stuff with hypercontractivity over the Gray-Wyner system. This might also be germane, esp. regarding better bounds.
– stochasticboy321
Nov 27 at 23:04
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let's assume we are given discrete random variables $X$, $Z$, with some nonzero mutual information $I[X,Z] > 0$. I would like to understand the minimum entropy of variables $Y$ such that $X rightarrow Y rightarrow Z$ is a Markov chain. By the data processing inequality, $H[Y] geq I[X,Z]$. But can we also find some $Y$ for which this bound is close to tight, that is $H[Y]$ close to $I[X,Z]$?
More formally: Let $F$ be the set of variables $Y$ such that $X rightarrow Y rightarrow Z$ is a Markov chain. Can we upper-bound $inf_{Yin F} H[Y]$ in relation to $I[X,Z]$?
Or are there conditions under which such an upper bound holds?
I'd be interested in any kind of upper bound on $inf_{Yin F} H[Y]$, even if it is much larger than $I[X,Z]$, as long as it is better than $inf_{Yin F} H[Y] leq H[X], H[Z]$.
I have looked around the web but didn't find anything related. Any pointers (or an explanation why this isn't possible) would be greatly appreciated. Thanks!
markov-chains information-theory entropy
asked Nov 25 at 19:54
Michael Hahn
11
Let's assume we are given discrete random variables $X$, $Z$, with some nonzero mutual information $I[X,Z] > 0$. I would like to understand the minimum entropy of variables $Y$ such that $X rightarrow Y rightarrow Z$ is a Markov chain. By the data processing inequality, $H[Y] geq I[X,Z]$. But can we also find some $Y$ for which this bound is close to tight, that is $H[Y]$ close to $I[X,Z]$?
More formally: Let $F$ be the set of variables $Y$ such that $X rightarrow Y rightarrow Z$ is a Markov chain. Can we upper-bound $inf_{Yin F} H[Y]$ in relation to $I[X,Z]$?
Or are there conditions under which such an upper bound holds?
I'd be interested in any kind of upper bound on $inf_{Yin F} H[Y]$, even if it is much larger than $I[X,Z]$, as long as it is better than $inf_{Yin F} H[Y] leq H[X], H[Z]$.
I have looked around the web but didn't find anything related. Any pointers (or an explanation why this isn't possible) would be greatly appreciated. Thanks!
markov-chains information-theory entropy
markov-chains information-theory entropy
asked Nov 25 at 19:54
Michael Hahn
11
asked Nov 25 at 19:54
Michael Hahn
11
edited Nov 25 at 20:05
asked Nov 25 at 19:54
Michael Hahn
11
asked Nov 25 at 19:54
Michael Hahn
11
asked Nov 25 at 19:54
Michael Hahn
11
11
I think you're looking at the Wyner common information. Check out ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1055346 , in particular the subsection marked 'second approach' on pg 164 (and, for that matter, the rest of the paper, it's quite nice). Wyner also mentions conditions for $H(Y) = I(X;Z)$ to hold, and there's another paper of Witsenhausen that may be interesting. I'm not sure if there are better bounds than the trivial ones above but given the time there must be progress on this. In any case the keyword should help you find an answer.
– stochasticboy321
Nov 27 at 2:57
Awesome thanks!
– Michael Hahn
Nov 27 at 19:54
You're welcome. I just remembered - a mid 2010s paper of Anantharam and Kamath did some fun stuff with hypercontractivity over the Gray-Wyner system. This might also be germane, esp. regarding better bounds.
– stochasticboy321
Nov 27 at 23:04
add a comment |
I think you're looking at the Wyner common information. Check out ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1055346 , in particular the subsection marked 'second approach' on pg 164 (and, for that matter, the rest of the paper, it's quite nice). Wyner also mentions conditions for $H(Y) = I(X;Z)$ to hold, and there's another paper of Witsenhausen that may be interesting. I'm not sure if there are better bounds than the trivial ones above but given the time there must be progress on this. In any case the keyword should help you find an answer.
– stochasticboy321
Nov 27 at 2:57
Awesome thanks!
– Michael Hahn
Nov 27 at 19:54
You're welcome. I just remembered - a mid 2010s paper of Anantharam and Kamath did some fun stuff with hypercontractivity over the Gray-Wyner system. This might also be germane, esp. regarding better bounds.
– stochasticboy321
Nov 27 at 23:04
I think you're looking at the Wyner common information. Check out ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1055346 , in particular the subsection marked 'second approach' on pg 164 (and, for that matter, the rest of the paper, it's quite nice). Wyner also mentions conditions for $H(Y) = I(X;Z)$ to hold, and there's another paper of Witsenhausen that may be interesting. I'm not sure if there are better bounds than the trivial ones above but given the time there must be progress on this. In any case the keyword should help you find an answer.
– stochasticboy321
Nov 27 at 2:57
I think you're looking at the Wyner common information. Check out ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1055346 , in particular the subsection marked 'second approach' on pg 164 (and, for that matter, the rest of the paper, it's quite nice). Wyner also mentions conditions for $H(Y) = I(X;Z)$ to hold, and there's another paper of Witsenhausen that may be interesting. I'm not sure if there are better bounds than the trivial ones above but given the time there must be progress on this. In any case the keyword should help you find an answer.
– stochasticboy321
Nov 27 at 2:57
Awesome thanks!
– Michael Hahn
Nov 27 at 19:54
Awesome thanks!
– Michael Hahn
Nov 27 at 19:54
You're welcome. I just remembered - a mid 2010s paper of Anantharam and Kamath did some fun stuff with hypercontractivity over the Gray-Wyner system. This might also be germane, esp. regarding better bounds.
– stochasticboy321
Nov 27 at 23:04
You're welcome. I just remembered - a mid 2010s paper of Anantharam and Kamath did some fun stuff with hypercontractivity over the Gray-Wyner system. This might also be germane, esp. regarding better bounds.
– stochasticboy321
Nov 27 at 23:04
add a comment |
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I think you're looking at the Wyner common information. Check out ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1055346 , in particular the subsection marked 'second approach' on pg 164 (and, for that matter, the rest of the paper, it's quite nice). Wyner also mentions conditions for $H(Y) = I(X;Z)$ to hold, and there's another paper of Witsenhausen that may be interesting. I'm not sure if there are better bounds than the trivial ones above but given the time there must be progress on this. In any case the keyword should help you find an answer.
– stochasticboy321
Nov 27 at 2:57
Awesome thanks!
– Michael Hahn
Nov 27 at 19:54
You're welcome. I just remembered - a mid 2010s paper of Anantharam and Kamath did some fun stuff with hypercontractivity over the Gray-Wyner system. This might also be germane, esp. regarding better bounds.
– stochasticboy321
Nov 27 at 23:04