Suppose we want to make inference on an unobserved realization x of a random variable ˜x, which is normally distributed with mean μx and variance σ2x. Suppose there is another random variable ˜y (whose unobserved realization we’ll similarly call y) that is normally distributed with mean μy and variance σ2y. Let σxy be the covariance of ˜x and ˜y.
Now suppose we observe a signal on x,
where ˜u∼N(0,ϕ2x), and a signal on y,
where ˜v∼N(0,ϕ2y). Assume that ˜u and ˜v are independent.
What is the distribution of x conditional on a and b?
What I know so far:
Using inverse-variance weighting,
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.
Specifically, if you assume (compatibly with what specified in the question) that
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).