- #1

Legendre

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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)

---------------------

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')).

---------------------

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!