Non-centered parameterization #

Syntax #

parameterization info: non-centered

Usage #

When you see this sort of funnel-like shape in your funnel plots,

In the lower panel of the funnel traces, you can see that \(\sigma_{\beta_0}\) drifts into smaller values and gets a bit “stuck”. Basically, the sampler can’t efficiently explore that parameter space. While stuck, the slopes \(\beta_j\) compress / squish together, thereby creating the problematic funnel we saw in the first figure.

It’s possible to escape this funnel with a small reparameterization, called the “non-centered” parameterization. We won’t go into detail here, but the basic problem, and solution, is described by others. The outputs used to find evidence of a funnel (and hence a clue that the non-centered parameterization may be needed), are found here.

Resources #

Approachable, less-technical material #

• Statistical rethinking ( Citation: McElreath, 2020 McElreath, R. (2020). Statistical rethinking: A bayesian course with examples in r and stan. Chapman; Hall/CRC. )
• “Why hierarchical models are awesome, tricky, and Bayesian”
• This parameterization also appears in discussions by the Stan and greta communities: here and here.

Technical / mathy material #

• A General Framework for the Parametrization of Hierarchical Models by Papaspiliopoulos et al. ( Citation: 2007 Papaspiliopoulos, O., Roberts, G. & Sköld, M. (2007). A general framework for the parametrization of hierarchical models. Statistical Science. 59–73. )
• Hamiltonian Monte Carlo for Hierarchical Models by Betancourt and Girolami ( Citation: 2015 Betancourt, M. & Girolami, M. (2015). Hamiltonian monte carlo for hierarchical models. Current trends in Bayesian methodology with applications, 79(30). 2–4. )
• Appendix D in Monnahan et al. ( Citation: 2017 Monnahan, C., Thorson, J. & Branch, T. (2017). Faster estimation of bayesian models in ecology using hamiltonian monte carlo. Methods in Ecology and Evolution, 8(3). 339–348. )

References #