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Lagrangian Multiplier with Matrices

  1. Sep 17, 2009 #1
    I'm trying to use calculus of variations to solve for the probability distribution with highest entropy for a given covariance matrix. I want to maximize this:

    [tex]H[p(\vec{x})] = -\int p(\vec{x})*ln(p(\vec{x}))d\vec{x}[/tex]

    with the following constraints:

    [tex]\int p(\vec{x}) = 1[/tex]
    [tex]\int \vec{x}p(\vec{x})d\vec{x} = \vec{u}[/tex]
    [tex]\int (\vec{x}-\vec{u})(\vec{x}-\vec{u})^{T}p(\vec{x})d\vec{x} = \Sigma[/tex]

    Using Lagrangian multipliers, the proposed maximization function is:

    [tex]F[p(\vec{x})] = -\int p(\vec{x})*ln(p(\vec{x}))d\vec{x} + \lambda_{1}(\int p(\vec{x})d\vec{x}-1) + \vec{m}^{T}(\int \vec{x}p(\vec{x})d\vec{x} - \vec{u}) + Tr\{L(\int (\vec{x}-\vec{u})(\vec{x}-\vec{u})^{T}p(\vec{x})d\vec{x} - \Sigma)\}[/tex]

    I understand that m is needed because there is D constraints imposed by the mean. And L is needed because there is DxD constraints imposed by the covariances. But what is the trace operator doing in there?

    Thanks for the help
  2. jcsd
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