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, 2021 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, 2021 at 8:43
• Here is a new paper on the topic Jun 17, 2021 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 .
$$\beta = \text{Cov}(X,Y)/\text{Var}(X)$$