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Fisher Information Matrix: Equivalent Expressions

  1. Jul 6, 2012 #1
    I don't understand the following step regarding the [itex](i,j)^{th}[/itex] element of the Fisher Information Matrix, [itex]\textbf{J}[/itex]:
    [tex]J_{ij}\triangleq\mathcal{E}\left\{ \frac{\partial}{\partial\theta_{i}}L_{\textbf{x}}(\textbf{θ})\frac{\partial}{\partial\theta_{j}}

    L_{\mathbf{x}}(\textbf{θ})\right\}
    \\
    =-\mathcal{E}\left\{ \frac{\partial^{2}}{\partial\theta_{i} \partial \theta_{j}}L_{\textbf{x}}(\textbf{θ})\right\}[/tex]

    which is given in (Eq. 8.26, on p. 926 of) "Optimum Array Processing" by Harry van Trees. I don't know if the details matter, but [itex]L_{\textbf{x}}[/itex] is the log-likelihood function and he is looking at the problem of estimating the non-random real vector, [itex]\textbf{θ}[/itex], from discrete observations of a complex Gaussian random vector, [itex]\textbf{x}[/itex].

    Am I missing something obvious? I'm not very sharp on partial derivatives.
     
    Last edited: Jul 6, 2012
  2. jcsd
  3. Jul 6, 2012 #2

    DrDu

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    L_X is minus the log of the pdf. Write down the explicit expression for the expectation values in terms of L_X and its derivatives and use partial integration.
     
  4. Jul 7, 2012 #3
    The expressions are rather intimidating functions of complex matrices. I don't think I want to try partial integration. I tried just evaluating the derivatives to see if they would come out the same, but I ran out of paper before I had even scratched the surface.

    If the result is specific to this problem, then I would be willing to take it on face value. It's just frustrating that I keep seeing the result stated without proof. It's as though it's too obvious to warrant a formal proof.
     
  5. Jul 9, 2012 #4

    DrDu

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  6. Jul 11, 2012 #5
  7. Jul 11, 2012 #6

    DrDu

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    6 month is completely inacceptable. As I said, this topic is contained in every book on introductory statistics.
    Elsewise it is a good idea to search for some lecture notes containing the problem: I used "fisher information lecture notes" and almost every lecture note contained a proof of the statement, e.g. the first one i got:
    http://ocw.mit.edu/courses/mathemat...ications-fall-2006/lecture-notes/lecture3.pdf
     
  8. Jul 14, 2012 #7
    This is obviously incorrect.

    This is no longer necessary. The topic is resolved.
     
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