Maximum Entropy Distribution Given Marginals

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Hi all,

I'm a pure mathematician (Graph Theory) who has to go through a Physics paper, and I am having trouble getting through a part of it. Maybe you guys can point me in the right direction:

Let P(x,y) be a joint distribution function.

Let H = - $\Sigma$_x,y P(x,y) log P(x,y), which is the Entropy.

We are given the marginal probability density functions q₁(x) and q₂(y).

To get the maximum entropy distribution consistent with the marginals, we solve this problem:

max H

subject to constraints,

$\Sigma$_y P(x,y) = q₁(x)
$\Sigma$_x P(x,y) = q₂(y)

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The Lagrangian is,

L = - $\Sigma$ _x,y (P(x,y) log P(x,y)) + $$\lambda$$₁ $\Sigma$_y (P(x,y) - q₁(x)) + $$\lambda$$₂ $\Sigma$_x (P(x,y) - q₂(y))

To find the stationary point of L, we differentiate it with respect to a particular P(x',y') while keeping the other variables constant.

I know how to do this for the first sum, which gives the solution -(1 + log P(x',y')).

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How do we differentiate the second and third sum with respect to P(x',y')?

The paper gave the solution to differentiating L with respect to P(x',y') as:

-(1 + log P(x',y')) + $$\lambda$$₁ + $$\lambda$$₂


Thanks!