Results 1 to 2 of 2

Math Help - Data likelihood

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    2

    Data likelihood

    I'm trying to understand a paper which shows a linear model:

    \mathbf{x} = \mathbf{A}\mathbf{s} + \mathbf{\epsilon}\qquad\qquad(1)

    where \mathbf{x} is an L \times N vector, \mathbf{A} is an L \times M matrix, and \mathbf{s} is an M \times N vector. Additive Gaussian noise is represented by \mathbf{\epsilon} with variance \sigma^2.

    I'm trying to understand how it is that the data log likelihood has this form:

    \log P(\mathbf{x}|\mathbf{A},\mathbf{s}) \propto -\frac{1}{2 \sigma^2}(\mathbf{x}-\mathbf{A}\mathbf{s})^2\qquad\qquad(2)

    There is no indication of how the log likelihood in eqn (2) follows from (1), and I've seen it in a few papers already, so I'm assuming it's standard prerequisite knowledge of probability. I'm having trouble finding sources where I could look this kind of thing up.

    Another confusing aspect, is that (\mathbf{x}-\mathbf{A}\mathbf{s}) has dimensions L \times N, so how can you calculate its square, and how can the log likelihood be a scalar?
    Last edited by jmpeax; January 29th 2010 at 01:45 AM. Reason: Added variance term to epsilon.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Jan 2010
    Posts
    2
    For the benefit of others, I found a great source from Matrix normal distribution (Wikipedia) as the reference:

    Dawid, A.P. (1981). "Some matrix-variate distribution theory: Notational considerations and a Bayesian application". Biometrika 68 (1): 265274.

    which suggests that the squared notation in equation (2) isn't a matrix square operation, and one can see how (2) follows from (1) with something like

    \mathbf{x}|\mathbf{A},\mathbf{s} \sim \mathcal{N}(\mathbf{A}\mathbf{s},\sigma)

    since \mathbf{x} is a random variable of mean \mathbf{A}\mathbf{s} with variance \sigma^2 as depicted in (1). Seems like this is on the right track, but correct me if I am wrong.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: July 6th 2010, 06:33 PM
  2. find maximum likelihood estimator using given data
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: December 2nd 2009, 06:24 AM
  3. Likelihood and log-likelihood functions
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: April 29th 2009, 02:44 PM
  4. Likelihood Ratio Test & Likelihood Function
    Posted in the Advanced Statistics Forum
    Replies: 6
    Last Post: April 23rd 2009, 06:53 AM
  5. Replies: 0
    Last Post: March 30th 2008, 12:44 PM

Search Tags


/mathhelpforum @mathhelpforum