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Math Help - random vector expectations

  1. #1
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    Question random vector expectations

    I come across a question about calculating the expectation for random vectors
    and I have no idea how to do with it.
    Anyone could give me a hand? Thanks for your attention.

    The PDF of \mathbf{y}, conditioned on \mathbf{z}, is Gaussian with mean vector H\mathbf{z} and covariance matrix
    \sigma^2I-\alpha^2HH^T, i.e.,

    \mathrm{p}(\mathbf{y}|\mathbf{z}) =<br />
\mathcal{N}(\mathbf{y}|H\mathbf{z}, \sigma^2I-\alpha^2HH^T),

    and the PDF of \mathbf{z} and \mathbf{y}, given \mathbf{x}, are also both Gaussian:

    \mathrm{p}(\mathbf{z}|\mathbf{x}) =<br />
\mathcal{N}(\mathbf{z}|W\mathbf{x}, \alpha^2I).

    \mathrm{p}(\mathbf{y}|\mathbf{x}) =<br />
\mathcal{N}(\mathbf{y}|HW\mathbf{x}, \sigma^2I).

    We also have

    \mathrm{p}(\mathbf{y}|\mathbf{z}, \mathbf{x}) =<br />
\mathrm{p}(\mathbf{y}|\mathbf{z}).

    W, H, \alpha, \sigma are all known and I is an identity matrix.
    Now what is the conditional expectation of \mathbf{z}, given some
    observed \mathbf{y} and \mathbf{x}, i.e.,

    \mathrm{E}(\mathbf{z}|\mathbf{y}, \mathbf{x}) =<br />
\int_{\mathbb{R}^{n}}<br />
\mathbf{z}\cdot\mathrm{p}(\mathbf{z}|\mathbf{y},<br />
\mathbf{x})\mathrm{d}\mathbf{z}= \mathrm{?}

    The answer is something like

    \mathrm{E}(\mathbf{z}|\mathbf{y}, \mathbf{x}) = W\mathbf{x} +<br />
\frac{\alpha^2}{\sigma^2}H^T(\mathbf{y}-HW\mathbf{x}).

    But don't know how
    Last edited by tinwai; March 18th 2008 at 01:44 AM. Reason: add a condition which is missed earlier
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  2. #2
    Moo
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  3. #3
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    Quote Originally Posted by Moo View Post
    Hello,

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    Thank you for the information. I have re-written the question with math tags.
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