why the log likelihood could be a positive value in mixed model?

[Le Wang]

When I try to fit a mixed model, stata outputs a positive logarithmic likelihood function value, which almost violates common sense. I wonder why this happens. Is there any reference to support this result? Looking forward to getting help.

[Maarten Buis]

A log-likelihood can be positive and that does not indicate a problem. I don't think there is any literature on that. It is one of those things that are surprising the first time you see it, but once you understand what is going on, it is pretty trivial, and it is hard to publish trivial stuff...

Often log-likelihoods are negative, which means that the likelihood is less than 1. If the log-likelihood is positive, then the likelihood is larger than 1. This can be surprising when you were introduced to the likelihood as "the probability of observing the data given the model and its parameters"; a probability cannot be larger than 1. However, when your dependent/explained/left-hand-side/y-variable is continuous, then the likelihood is no longer a probability but a probability density, and densities depend on the unit of the dependent variable. That way they can easily be larger than 1 and that would not indicate a problem.