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Math Help - MLE of amplitude rescaling

  1. #1
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    MLE of amplitude rescaling

    if y=ax+n where y is a d-dimensional vector, a is a scalar (amplitude rescaling factor), and n is a d-dimensional vector drawn from a zero mean gaussian, what is the MLE of a given x and y?

    the way i see it, this is equivalent to minimizing the sum of squared errors:
    \sum_{i=1}^d(y_i - ax_i)^2, taking the derivative with respect to a and setting it equal to zero i end up with a = \frac{x^Ty}{x^Tx}. for some reason, this seems too simple, and intuitively, doesn't make a whole lot of sense. does this look right?

    edit: n is drawn from a zero mean gaussian with vI covariance (v is scalar, I is the identity matrix).

    i think its the same, except you're minimizing \sum_{i=1}^dv(y_i - ax_i)^2, so a = v\frac{x^Ty}{x^Tx}

    edit 2: v cancels out, a = \frac{x^Ty}{x^Tx}
    Last edited by mr fantastic; May 31st 2009 at 05:10 AM. Reason: Added correct latex tags ($ don't work here)
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  2. #2
    Moo
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    Hello,

    I think you're correct. But I've never applied the MLE to a linear model...

    You may be interested by this part of a Wikipedia article : Linear model - Wikipedia, the free encyclopedia, which seems to confirm your result.

    The only difference is that you need the X in the article to be of full rank (for a nxm matrix, "full rank" means that its rank is min(n,m)). But here, I'm quite unsure what stands for X
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  3. #3
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    cool, thanks

    in my case x and y are just vectors so x^Ty and x^Tx are scalar values, no need to worry about rank. I got good results with this solution ( on a nearest neighbor classifier), so I'm assuming its correct

    thanks again.

    Quote Originally Posted by Moo View Post
    Hello,

    I think you're correct. But I've never applied the MLE to a linear model...

    You may be interested by this part of a Wikipedia article : Linear model - Wikipedia, the free encyclopedia, which seems to confirm your result.

    The only difference is that you need the X in the article to be of full rank (for a nxm matrix, "full rank" means that its rank is min(n,m)). But here, I'm quite unsure what stands for X
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