# learning a gaussian distribution through dependent vairiable observations

Is it possible to infer the parameters of a gaussian random variable by sampling from a distribution that is linearly dependent on the variable of interest? For example:

y = Ax + n

With

x ~ N(u,S)

n ~ N(0,Q)

A : known constant matrix

Q : known constant matrix

Can we infer the mean (u) and covariance matrix (S) of x from observations (samples) from the y distribution?

• Are $A$ or $Q$ known? – Bridgeburners Jul 10 '17 at 14:47
• Yes. Sorry. I have edited the post to indicate that – ejlouw Jul 11 '17 at 6:21

Given the distributions of $x$ and $n$, we know that $y \sim N(Au, A'SA+Q)$. Given that $A$ and $Q$ are known, and assuming you know the sample mean and variance of $y$, then it's a system of 2 equations with 2 unknowns, which you can easily solve.