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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 = - [itex]\Sigma[/itex]_{x,y}P(x,y) log P(x,y), which is the Entropy.

We are given the marginal probability density functions q_{1}(x) and q_{2}(y).

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

max H

subject to constraints,

[itex]\Sigma[/itex]_{y}P(x,y) = q_{1}(x)

[itex]\Sigma[/itex]_{x}P(x,y) = q_{2}(y)

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The Lagrangian is,

L = - [itex]\Sigma[/itex]_{x,y}(P(x,y) log P(x,y)) + [tex]\lambda[/tex]_{1}[itex]\Sigma[/itex]_{y}(P(x,y) - q_{1}(x)) + [tex]\lambda[/tex]_{2}[itex]\Sigma[/itex]_{x}(P(x,y) - q_{2}(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')) + [tex]\lambda[/tex]_{1}+ [tex]\lambda[/tex]_{2}

Thanks!

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# Maximum Entropy Distribution Given Marginals

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