Plateaus can cause stability issues through floating point instabilities. For example a plateau that stretches out towards infinity will cause result in divergences around exp(700) when the range of double precision floating point ends. Plateaus that stretch towards a finite boundary can cause problems because of the implicit Jacobian – this happens for example with a beta model that concentrates too strongly at 0 or 1.
That said I’m not sure why there would be a plateau in the posterior density function here provided that there are decent priors on the hierarchical population location and scale. There are decent priors, right…?
My guess is that this actually is a curvature problem due to the varying informativeness of the likelihood functions. In particular the hierarchies for the parameters with peaked likelihood functions probably need to be centered while the hierarchical for the parameters with non-peaked likelihood functions probably need to be non-centered.
This isn’t an issue of sampling efficiency. You’d see the same issues with Monte Carlo estimators that use exact samples – the problem is that the Monte Carlo and Markov chain Monte Carlo estimator variance scales with the true variance of the expectand (function whose expectation value is being taken). Heavier tails imply larger variances and noisier estimators.