# Joint prior distributions in WinBUGS

Suppose we have a hierarchical model summarised by the following:

$y_{i} \sim N(\mu_{i}, \sigma^{2})$, for $i = 1, \ldots, n$; (For these purposes, assume $\sigma^{2}$ is known)

where $\mu_{i} = \alpha + \beta x_{i}$ and each $x_{i}$ is a given predictor variable for $i = 1, \ldots, n$

Next,

$\alpha \sim N(\mu_{\alpha}, \sigma_{\alpha}^{2})$ and $\beta \sim N(\mu_{\beta}, \sigma_{\beta}^{2})$

Finally,

$p(\mu_{\alpha}, \mu_{\beta}, \sigma_{\alpha}^{2}, \sigma_{\beta}^{2}) \propto \frac{1}{\sigma^{2}_{\alpha}\sigma^{2}_{\beta}}$

How can this model be fitted in WinBUGS?

I know that the last two levels are given by:

for (i in 1:N){
y[i] ~ dnorm(mu[i], sigma.squared)
mu[i] <- alpha + beta*x[i]
}
alpha ~ dnorm(mu.alpha, sigma.squared.alpha)
beta ~ dnorm(mu.beta, sigma.squared.beta)


However, I'm not sure, how the joint prior for $\mu_{\alpha}, \mu_{\beta}, \sigma_{\alpha}^{2}, \sigma_{\beta}^{2}$ should be incorporated into the model.

In particular, how is the hyper-prior distribution represented in the WinBUGS code, how can I achieve values for $\mu_{\alpha}$ and $\mu_{\beta}$ from this distribution and how should $\sigma_{\alpha}^{2}$ and $\sigma_{\beta}^{2}$ be chosen?

In "vintage" WinBugs this may work:

mu.alpha ~ dnorm(0, 0.001)
mu.beta  ~ dnorm(0, 0.001)

sigma.squared.alpha ~ dgamma(0.001, 0.001)
sigma.squared.beta  ~ dgamma(0.001, 0.001)


The prior for the means shoulds be specifyed as

mu~dflat()


I haven't pratice bayesian for a long time, but I think a uniform prior on log(sigma) is equivalent to what you want. You could verify it with the following formula :

so for variances, this code should work :

log(sigma)~dflat()