# Conjugate prior for parameter W when the likelihood is normal with mean and variance both functions of W

Suppose that $x$ is an observable scalar variable and $$x \thicksim N(W\mu_0,W^2\sigma_0^2)$$ Where $W$ is a parameter that must be estimated from data, and $\mu_0$ , $\sigma_0$ are known constants.

I want to solve the problem in Bayesian framework, so I consider $W$ as a random variable, and I have to choose a prior probability distribution for it.

What is a suitable conjugate prior for W in my problem? (I prefer the priors which belong to a known family of distributions).

• Could you give some information about what you are trying to model? – Pieter May 18 '16 at 19:34

The natural conjugate prior for this model is $$p(W) \propto W^{-a}\exp\left(-\frac{b}{W^2}+\frac{c}{W}\right)$$ and its posterior is $$p(W|x) \propto W^{-(a+1)}\exp\left(-\frac{(b+x^2/[2\sigma_0^2])}{W^2}+\frac{(c+x\mu_0/\sigma_0^2)}{W}\right).$$ It is not from a family of distributions that I know and I have no idea if, or for what parameter settings and data, this will be valid probability density.
This problem is related to the generalized inverse Gaussian distribution which is the conjugate prior for the model $$x|W \sim N(\alpha+\beta W,W)$$ the difference is that $W$ shows up in both the mean and the variance rather than $W^2$ in the variance.