Bayesian calculation of parameter multiplying normal

Question

I am interested in a model like:

$y_{i} = \sum_{k\in K}{\beta_{k} z_{k}}$, with $z_{k} \tilde{} N(\mu_{k}, \sigma_{k})$. where $\beta \equiv(\beta_{k})_{k\in K}$ is not known, but all else is.

I assume that there is a prior $\pi (\beta)$ of an arbitrary distribution and I want to calculate the posterior (of $\beta$), given an observation of $(y_{i})_{i\in N}$ (but not of $z_{k}$).

I wonder if there are conjugate prior distributions for this case (any other information might be useful as well). I understand that for unknown mean and variance normal occurrences you can use normal mean and inverse chi square variance priors, but here they must have (EDIT:) related distributions.

Although I'm interested in the general case, any information about the case $K = \{k\}$ would be helpful already.

Answer 1

The likelihood function is $p(y_i | \beta) = N(y_i; \sum_k \beta_k \mu_k, \sum_k \beta_k^2 \sigma_k)$, so a conjugate prior for $\beta$ would be proportional to this. Unfortunately, this is not a standard distribution unless $\mu=0$ and $K=\{k\}$, in which case it is a generalized Gamma distribution.