# #StackBounty: #bayesian #mixed-model #prior Unbounded likelihoods for unpenalized mixed effects

### Bounty: 50

Most mixed effects models assume that the random effects \$gamma\$ follow a \$MVN(0, Sigma_{gamma})\$ distribution. In some cases, specific structure is put on \$Sigma_{gamma}\$. For now, let’s just assume it’s an unstructured covariance matrix.

Now, if we don’t apply any priors or penalties on \$Sigma_{gamma}\$, it seems to me that the likelihood is unbounded. In particular, if we set \$gamma = 0\$, then as \$det(Sigma_{gamma}) rightarrow 0\$, the contribution of the log-density of \$gamma\$ approaches infinity. As long as the other contributions to the likelihood are finite (which is typically the case), this implies the log-likelihood would be unbounded.

Clearly, for the MLE this creates an issue (although I know REML is a more popular alternative). For most MCMC algorithms, having unbounded log-densities can be problematic as well, even if the posterior is still proper.

How is this issue typically handled? Is there a canonical penalty/prior on \$Sigma_{gamma}\$?

Get this bounty!!!

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