random vector expectations

**tinwai** · March 17th 2008, 05:45 AM

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.,

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

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

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

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

We also have

$\mathrm{p}(\mathbf{y}|\mathbf{z}, \mathbf{x}) = \mathrm{p}(\mathbf{y}|\mathbf{z})$ .

$W, H, \alpha, \sigma$ 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.,

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

The answer is something like

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

But don't know how

**Moo** · March 17th 2008, 10:01 AM

Hello,

Lift ?

There's no image

**tinwai** · March 18th 2008, 12:47 AM

Originally Posted by Moo

Hello,

Lift ?

There's no image

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

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