Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Maximum Entropy in Gaussian Setting

  1. Jul 11, 2013 #1
    I have a doubt about the distribution of random variables that maximize the differential entropy in a set of inequalities. It is well known that the Normal distribution maximizes the differential entropy. I have the following set of inequalities:

    T1 < I(V;Y1|U)
    T2 < I(U;Y2)
    T3 < I(X1,X2;Y3|V)
    T4 < I(X1,X2;Y3)

    where, Y1=X1+N1, Y2=a*X1+N2, Y3=b*X1+X2+N3. N1,N2,N3 are Gaussian ~ N(0,1). The lower case a and b are positive real numbers a < b. U, V, X1 and X2 are random variables. I want to maximize that set of inequalities. I know the following:

    (i) From T4, h(Y3) maximum is when Y3 is Gaussian then X1 and X2 are Gaussian.

    (ii) From T2 we maximize it by having h(Y2) or h(a*X1+N2) maximum. From this by the Entropy Power Inequality (EPI) we bound -h(a*X1+N2|U) and have X1|U Gaussian.

    (iii) From T1 we maximize it by having h(Y1|U) or h(X1+N1|U) maximum which we can do as -h(a*X1+N2|U) in the part ii can be bounded having Y1 Gaussian (satisfying the maximum entropy theorem).

    The Question:

    From T3, can I assume that jointly Gaussian distribution will maximize h(Y3|V) or h(b*X1+X2+N3) having the assumptions i,ii,iii ?

    My aim is to show that jointly Gaussian distribution of U, V, X1 and X2 maximizes the set of inequalities. I hope anyone can help me out with this.
  2. jcsd
  3. Jul 11, 2013 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    I don't understand your notation. Is "T4" a number or is it only a designator for an expression? Are the vertical bars "|" to denote absolute values? - conditional probabilities?
  4. Jul 11, 2013 #3
    Thanks Stephen for your reply. Basically the set of inequalities is what is known in Information Theory as a rate region:
    T1 < I(V;Y1|U)
    T2 < I(U;Y2)
    T3 < I(X1,X2;Y3|V)
    T1+T2+T3 < I(X1,X2;Y3).
    T1, T2 adn T3 are the rates obtained when transmitting messages 1, 2 and 3. The I's are Mutual Informations and the vertical bars "|" indicate conditioning. For instance I(V;Y1|U) = h(Y1|U) - h(Y1|U,V) where h(x) is the differential entropy.
    My question is basically is after having assumed h(X1+N1|U) maximum implies (X1+N1|U) Gaussian in (iii), could I assume h(b*X1+X2+N3|V) maximum implies (b*X1+X2+N3|V) Gaussian???? I know if I hadn't assumed (i,ii,iii) this last question would be affirmative, but having (i,ii,iii) is it still true?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Maximum Entropy in Gaussian Setting
  1. Gaussian distribution! (Replies: 3)

  2. Gaussian function (Replies: 1)

  3. Gaussian function (Replies: 1)