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I was wondering if there are any ways of modelling a regression with heteroscedastic normal errors in conjugate form using Bayesian Linear regression. I.e., is there a conjugate form for the model \begin{align} Y_{i} &= X_i\beta + \varepsilon_i \\ \varepsilon_i & \sim \mathcal{N}(0, \sigma_i), \end{align} where $\sigma_i$ need not be equal to $\sigma_j$? I know there are ways of doing this using MCMC or approximations, but I am specifically wondering if there is a closed form Bayesian solution to this. The error term also does not have to be normal - if there is a closed form assuming that $\varepsilon_i$ is $t$-distributed with mean $0$ and variance $\sigma_i$ (or any other spherical distribution), I would be more than happy, too.

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    $\begingroup$ Take a look at Equation (4) in this paper: $\endgroup$
    – huighlh
    Commented Jan 9, 2019 at 1:23
  • $\begingroup$ Are you assuming these $\sigma^2_i$s are known, or are you putting a prior on them? I'm assuming the second one, but just making sure. $\endgroup$
    – Taylor
    Commented Jan 9, 2019 at 2:37
  • $\begingroup$ Either would be interesting, but a prior formulation for them is ultimately what I would be looking for $\endgroup$
    – Jeremias K
    Commented Jan 9, 2019 at 13:00

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