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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?

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  • $\begingroup$ Are $A$ or $Q$ known? $\endgroup$
    – Bridgeburners
    Jul 10, 2017 at 14:47
  • $\begingroup$ Yes. Sorry. I have edited the post to indicate that $\endgroup$
    – ejlouw
    Jul 11, 2017 at 6:21

1 Answer 1

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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.

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