Math Help - random vector expectations

1. 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.,

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

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

2. Hello,

Lift ?

There's no image

3. Originally Posted by Moo
Hello,

Lift ?

There's no image
Thank you for the information. I have re-written the question with math tags.