Why are Bayesian mixed-effects models (e.g., brms) more able to estimate complex models than Frequentist mixed models (e.g., lme4)?

Question

It is commonly suggested that if you are having trouble getting your lme4, Frequentist mixed-effects model to converge, you can either (a) simplify and drop random effects in the model, or (b) pivot to Bayesian mixed-effects models using brms (https://m-clark.github.io/posts/2020-03-16-convergence/).

I often accept this as true that Bayesian mixed-effects models can estimate complex models with maximal random effects (Barr et al., 2013) that Frequentist models cannot. However, I am unclear on the reasons for why this is, specifically why Bayesian models can estimate more complex models than Frequentist? Is it primarily due to the prior regularizing the random effects and "biasing" them away from the boundaries, so that you don't get weird aberrancies like correlations of 1/-1 like you sometimes see in Frequentist lme4?

Answer 1

Random effects are used to capture correlations in the data, namely, within the same level of the corresponding grouping factors. The parameters that quantify the strength of these correlations are the variance components (i.e., the variances and covariances between the random effects). It is often the case in real data that these correlations are rather small in magnitude. Hence, some of these variance components are practically zero, which is on the boundary of their corresponding parameter space. This is one of the main reasons why you experience convergence problems under the frequentist approach. Under the Bayesian approach, you typically specify a prior for the variance-covariance matrix of the random effects that provides some "information" not present in the data.

Answer 2

When you “estimate” a Bayesian model most often what you do is you sample from the posterior distribution. Posterior is, by Bayes theorem, basically a product of the priors and the likelihood. If you have very little data, or it does not provide much valuable information, the posterior would be dominated by the priors. In extreme cases, you would be sampling just from the priors. If you are willing to accept your guesses about the parameters as “estimates”, then you can estimate the parameters of any model with no effort.

I am not saying that to ridicule the Bayesian approach, I'm a great fan of it. What I'm trying to say is that plugging in some model to the MCMC algorithm is the easiest part and your job is far from done at this stage. The least you need to do after it is to check if the results make sense, e.g. are not completely dominated by the priors (are completely “random”).