# Creating a correlated prior

I would like to create a "weakly informative" prior distribution for a couple parameters. They both could theoretically take any value between 0 and 1, but I have reason to think that they should be negatively correlated. Is there any sort of standard joint distribution (a beta-analogue to the multivariate normal?) where I could easily specify such a distribution?

In looking around, I've read about the Dirichlet distribution, which seems to be almost perfect, but inappropriate because of it's requirement that the sum = 1.

• You could put $(\theta_1,\theta_2)=\left(\frac{\psi_1}{1+\psi_1},\frac{\psi_2}{1+\psi_2}\right)$ where $(\psi_1,\psi_2)$ is a bivariate Gaussian couple. You could also assign a prior for $\theta_1$ and for the conditional law $(\theta_2 \mid \theta_1)$. – Stéphane Laurent Sep 26 '12 at 3:02
• @StéphaneLaurent that seems like an answer to me :-). Maybe put some plots and you are done! – Néstor Sep 26 '12 at 5:45
• I don't think it needs plots. It already has the hallmark of a good answer: I feel a bit stupid for not thinking of it myself (the conditional bit, at least)! – Gregor - reinstate Monica Sep 26 '12 at 6:09

You could put for instance $(\theta_1,\theta_2)=\bigl(\frac{\psi_1}{1+\psi_1},\frac{\psi_2}{1+\psi_2}\bigr)$ where $(\psi_1,\psi_2)$ is a bivariate log-Gaussian couple. You could also assign a prior for $\theta_1$ and for the conditional law $(\theta_2 \mid \theta_1)$.
• I like the coniditional idea, but I must admit I'm confused by Gaussian couple answer. $\psi_1/(1+\psi_1)$ isn't restricted to [0,1], so I don't see how it applies. – Gregor - reinstate Monica Sep 27 '12 at 6:21