# What priors are typically used for the correlation parameter in a bivariate normal?

The Bivariate normal distribution contains a correlation parameter. I want to implement an MCMC sampler to sample from the posterior distribution of the parameters of the bivariate normal distribution. I have already selected the priors for the mean and variance parameters. However, I was not sure how to choose the prior for $$\rho\in(-1,1)$$, the correlation parameter.

Are there any standard choices such as the Jeffreys prior or other noninformative priors? If so, what are they?

• See: stats.stackexchange.com/questions/304684/… Obviously, this is a little simpler, when you just have a bivariate normal rather than more dimensions. Feb 3, 2022 at 14:38
• @Björn Very interesting. Too bad they do not give the prior on $\rho$ explicitly. Feb 3, 2022 at 14:39
• This master thesis considers a large number of priors for $\rho$. Feb 3, 2022 at 14:44
• @JarleTufto Perfect. I will accept it as an answer if you post it as such. Feb 3, 2022 at 14:47
• @Priorian "Link-only" answers are discouraged, see stats.meta.stackexchange.com/q/2487/77222, so the above comment will have to do for now. Feb 3, 2022 at 14:52

## 1 Answer

The Wishart distribution is the conjugate prior to the inverse of the covariance matrix of a multivariate normal distribution (it as a distribution over positive semi-definite matrices). So instead of using separate priors for the variance and correlation, you can use it as a prior for the entire (inverse of) covariance matrix.

• In the link you can find the marginal distributions of the matrix elements in the bivariate case - from that you can get an expression for $\rho$ as a (complicated) function of $\chi^2$ and normal random variables. I doubt however if this is useful in practice, and I'm not familiar with other useful priors for $\rho$. Feb 3, 2022 at 14:50