Persistent divergent transitions in simple model

Modeling
1

(This refers to the same model mentioned in “Error codes from rstan::optimzing”.) I have a fairly simple model that has been giving me a lot of divergent transitions, and I can’t seem to make them go away, despite

  • cranking up adapt_delta to 0.9999,
  • doing a lot of reparameterization,
  • modifying priors in various ways.

The model is basically this:

\mu_t = a + b \cdot t
a_t \sim \mbox{AR(1) process}
y_t = \mu_t + a_t \cdot \mu_t ^ x

All of x, a, b, and the AR(1) parameters are being estimated, with x restricted to the interval (0,1) and the autoregressive coefficient \phi restricted to (0,1). The parameterization used is a, b, \mathrm{logit}(\phi), \mathrm{logit}(x), and \log(\sigma_{\epsilon} \cdot \mu_u^x), where u is the midpoint of the time series.

Besides my original data set, I’ve been testing it on synthetic data generated from the intended model.

Here is a pairs plot from a run that used iter=4000, adapt_delta=0.9999, and max_treedepth=15. I’ve managed to reduce posterior correlations to moderate levels. There is a bit of funnel behavior in the interaction between b and logit_phi, but it’s not all that strong, and there are plenty of divergent transitions out in the broad part. Any suggestions?

pairsplot
pairsplot927×803 197 KB

2

Dear Kevin,

could you post the Stan model, data and calling instructions? That would be so helpful in analyzing the problem.

3

Here they are.
linear-ar1.stan is the stan model file.
example-data.csv is the data.
example.R is a (simplified) script to run the model.
linear-ar1.stan (2.3 KB)
example.R (1.1 KB)
example-data.csv (4.9 KB)

4

Your model is quite unusual. You use Jacobian adjustments, but changing of your model slightly
wouldn’t need that. I think it’s the preferred way, see AR(1) in Stan’s Usermanual for details.

One statement that really needs improvement is

  real a;
  real b;

following:

real log_mu_mid = log(a + tmid * b);

You have to specify:

real<lower=0> a;
real<lower=0> b;

Due to time restrictions, I’m unable to fully check your Jacobian Adjustments. I believe that the model needs to be slightly modified in terms of the Stan Usermanual.

5

Thanks for catching the missing restrictions on a and b. Once I put those in, the divergent transitions went away.

BTW, I think what you’re seeing as unusual in the model is what I had to do to allow for missing data. The zero-centered AR(1) equation to use when there are no data points between t and t+n is

a_{t+n} \sim \mathrm{normal}(\phi^n a_t, \sigma_n),
\sigma_n = \sigma_{\epsilon} \cdot \left(\sum_{i=0}^{n-1} \phi^{2i}\right)^{1/2}.

6

Are you trying to perfom a stochastic trend ar(1) process?

Cause the x parameter might affect your model fit.