# random vector expectations

• Mar 17th 2008, 05:45 AM
tinwai
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 $\displaystyle \mathbf{y}$, conditioned on $\displaystyle \mathbf{z}$, is Gaussian with mean vector $\displaystyle H\mathbf{z}$ and covariance matrix
$\displaystyle \sigma^2I-\alpha^2HH^T$, i.e.,

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

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

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

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

We also have

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

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

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

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

But don't know how
• Mar 17th 2008, 10:01 AM
Moo
Hello,

Lift ? :p

There's no image
• Mar 18th 2008, 12:47 AM
tinwai
Quote:

Originally Posted by Moo
Hello,

Lift ? :p

There's no image

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