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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.

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  • $\begingroup$ 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. $\endgroup$
    – jcken
    Jun 17 at 8:18
  • $\begingroup$ @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. $\endgroup$
    – Hojin Cho
    Jun 17 at 8:43
  • $\begingroup$ Here is a new paper on the topic $\endgroup$ Jun 17 at 19:25
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The trick is that there's no need to parametrize the covariance matrix.

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 .

You can derive the covariance term using the following relation:

$$ \beta = \text{Cov}(X,Y)/\text{Var}(X)$$

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  • $\begingroup$ Could you provide a proof for the statements you made? I want to follow it so that I could check if this is strictly true in case of multivariate Gaussian fitting. $\endgroup$
    – Hojin Cho
    Jun 21 at 5:04

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