# Prior for covariance matrix?

Given a set of data $$\{(x_i\pm e_{x,i},\,y_i\pm e_{y,i})\}_i$$ (with uncorrelated uncertainties), I want to model it as a multivariate Gaussian function with an unknown mean $$\boldsymbol{\mu}$$ and a covariance matrix $$\boldsymbol{\Sigma}$$. What is the correct prior for them? What happens if I assume that mean of both variables are 0?

Context: I've tried MCMC without giving any prior by assuming the likelihood for each pair of points to be a Gaussian with covariance of $$\boldsymbol{\Sigma}_i = \boldsymbol{\Sigma} + \pmatrix{{e_{x,i}}^2& 0\\0 &{e_{y,i}}^2}$$ but it lead to the correlation coefficient of $$\pm$$1, so I thought it was the problem of incorrect prior.

• Can you expand on 'correct prior'? Usually the 'correct prior' is the prior distribution that honestly and accurately reflects your (prior) beliefs about the parameters of interest. Jun 17 at 8:18
• @jcken Sorry for unclear wording - I don't know statistics very well. What I want is the least informative prior...or at least something like that. I need to calculate the correlation for any arbitrary set of data in industrial scale, and in most cases I know nothing about them. The intuition I have is that the correlation, given the measurement uncertainties, cannot be exactly 1 or -1. And closer they are to the unity, more unlikely they become. Jun 17 at 8:43
• Here is a new paper on the topic Jun 17 at 19:25

Model the usual mean and variance for $$X$$ using normal and inverse gamma priors. Model the response $$Y$$ conditionally using a regression model, with normal priors for the intercept and slope, and the inverse gamma for the .
$$\beta = \text{Cov}(X,Y)/\text{Var}(X)$$