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{?} $

The answer is something like

$\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