Let $y$ denote a D-dimensional multivariate normal random variable with density $$y \sim N(\mu, \Sigma(\mu)) $$ such that the covariance $\Sigma$ is a deterministic non-linear function of $\mu$. Further assume a prior distribution for $\mu$ given by $$\mu \sim N(\rho, \Omega)$$.

I have two questions. (1) Is this prior and likelihood conjugate even with $\Sigma$ being a non-linear function of $\mu$? If the answer is "it depends" please give some insights as to what conditions on $\Sigma(\mu)$ would lead to conjugacy. (2) If the answer to any of the above is "Yes" please give the posterior $p(\mu|y)$ or $p(\mu, \Sigma|y)$. Actually, even if the answer is no, but there is an analytical closed form for the posterior I would appreciate knowing what it is.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.