convergence in stan

[Linas Mockus]

Hi,

I have a question about convergence in stan. Rhat of a fit is 1 and looks like mixing is good (warmup=20000, n.draws=10000), but there are regions of almost straight lines between the nicely mixing regions (see the attached, page 1).  Those regions with nearly straight lines create humps in the posterior (page 3).  The predictive posterior (green lines .25/.75 quantiles and median blue line on page 4 looks reasonable.  Red line is MAP).  In order to find the MAP Differential Evolution optimization procedure was needed - 2 compartmental PK model is nonlinear and had multiple modes.  In this model I have used multiplicative error (the stan code attached).  For complete picture r code is attached as well.

Can anybody explain/suggest what is going on?

Thanks a lot,
Linas

[Michael Betancourt]

You’ll want to focus your attention on the sigma_theta trace — everything stalls when

sigma_theta hits zero.  What’s happening here is that HMC (and just about everything)

is not geometrically ergodic for your model, and getting stuck like this is a manifestation 

of the lack of geometric ergodicity.  You’ll also note that n_divergent becomes non-zero

in this region as well, which is an HMC specific manifestation of the problem.

This is particularly important as geometric ergodicity is what guarantees that Monte Carlo 

estimates converge to the true expectations (via the Monte Carlo Central Limit Theorem), 

and in a case like this you have to be very dubious of the estimates.  Note that algorithms

like Random Walk Metropolis and the Gibbs sampler suffer from the same problem, only

they’re usually not fast enough to be run for enough iterations to see the pathology manifest.

For more on the problem and possible solutions see http://arxiv.org/abs/1312.0906.

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<population.pdf><2comp_multerr_h.stan><2comp_multerr_h.r>

[Linas Mockus]

Michael, thanks for the insight.  I have used theta=theta_sigma*theta_var+theta_mu and now the model runs very smoothly.

The model is a hierarchical 2 compartmental PK model and data includes blood plasma measurements over some period of time for many individuals.  I wonder if I were to apply non-hierarchical version of the same model to single individual (it has only 8 data points) how many parameters the model can handle?  The 2 compartmental model has 6 parameters including error term.  Can I use the same trick which introduces one more parameter (note that I have only 8 data points) and still get convergence?

Thank you,
Linas

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[Michael Betancourt]

>
> The model is a hierarchical 2 compartmental PK model and data includes blood plasma measurements over some period of time for many individuals. I wonder if I were to apply non-hierarchical version of the same model to single individual (it has only 8 data points) how many parameters the model can handle? The 2 compartmental model has 6 parameters including error term. Can I use the same trick which introduces one more parameter (note that I have only 8 data points) and still get convergence?

What do you mean by non-hierarchical? Are you taking the hyper parameter posterior from the
first fit and using that as a prior for the new individual? So long as your prior is well-behaved you
can handle as little data as possible. Also, in terms of parameterizations the non-centered
parameterization (which sped up your model) works best with sparse data so would continue
to be the best choice for a single individual.