# Question about inverse-variance weighting

Suppose we want to make inference on an unobserved realization $x$ of a random variable $\tilde x$, which is normally distributed with mean $\mu_x$ and variance $\sigma^2_x$. Suppose there is another random variable $\tilde y$ (whose unobserved realization we’ll similarly call $y$) that is normally distributed with mean $\mu_y$ and variance $\sigma^2_y$. Let $\sigma_{xy}$ be the covariance of $\tilde x$ and $\tilde y$.

Now suppose we observe a signal on $x$,

where $\tilde u\sim\mathcal{N}(0,\phi_x^2)$, and a signal on $y$,

where $\tilde v\sim\mathcal{N}(0,\phi_y^2)$. Assume that $\tilde u$ and $\tilde v$ are independent.

What is the distribution of $x$ conditional on $a$ and $b$?

What I know so far:
Using inverse-variance weighting,

and

Since $x$ and $y$ are jointly drawn, $b$ should carry some information about $x$. Other than realizing this, I’m stuck. Any help is appreciated!

I’m not sure whether the inverse-variance weighting formulas apply here. However I think you might compute the conditional distribution of $x$ given $a$ and $b$ by assuming that $x$, $y$, $a$ and $b$ follow a joint multivariate normal distribution.
then, letting $a=x+u$ and $b=y+v$, you can find that
(Note that in the above it is implicitly assumed that $u$ and $v$ are independent between each other and also with $x$ and $y$.)
From this you could find the conditional distribution of $x$ given $a$ and $b$ using standard properties of the multivariate normal distribution (see here for example: http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Conditional_distributions).