Low effective sample size but good R-hat is this a problem?

Question

I am using Stan (Hamiltonian Monte-Carlo) to run a highly paramaterized model. One of the parameters in particular has a very low effective sample size (n_eff < .10*number of retained draws), but has an acceptable R-hat value 1.0006.

In my mind this means the chains are mixing well, but the retained samples are still highly correlated from their respective previous draws.

I have been conducting simulation work, the obtained posterior mean for the parameter is accurate, although the distribution is wider than the other parameters.

Should I be concerned about the very low effective sample size?

Other than reparamaterizing my model, are there other suggestions for resolving the low effective sample size for this parameter?

Things I have tried so far:

1. Longer burn-in
2. Increasing adapt_delta
3. Thinning (although it is my understanding thinning is not useful in HMC)
4. Using conjugate prior distributions

Answer 1

In BDA3 on page 285, they mention that $\hat{R}$ is an estimate of "the factor by which the scale of the current distribution for [a particular univariate parameter] might be reduced if the simulations were continued in the limit $n \to \infty$." Because your $\hat{R}$ is so close to $1$, this means that it's unlikely that running the simulation any longer will be of much use.

You mention that, for a particular parameter, it appears the "distribution is wider than the other parameters." This might be a property of the marginal posterior of that parameter, which is fine. A few of your suggestions might actually be invalidating some of the theory you're using to justify your estimates. For example, I wouldn't fiddle with the burn-ins too much, and changing the priors after you don't like simulation results is generally a no-no.

Edit: Regarding your request for more information about effective sample size, close after the portion I cited above in the same book, they discuss $\hat{R}$'s connection with effective sample size $n_{\text{eff}}$ as well as recommended settings for both. In particular, they mention that it might suffice to have as few as $10$ or $100$ as an effective sample size, and only aiming for more if an increased amount of precision is desired. Here's a quote:

As a default rule, we suggest running the simulation until $\hat{n}_{\text{eff}}$ is at least $5m$... Having an effective sample size of $10$ per sequence should typically correspond to stability of all the simulated sequences.

Here, $m$ is the "number of chains (after splitting)" BDA3 pg. 284 you're simulating . Notice that in particular that the desired number they suggest does not depend on the number of draws, retained or otherwise. The reason why these two metrics are "typically" related is because their formulas make use of the same quantities.

However, they are getting at the dispersion of two different quantities. $\hat{R}$ measures the potential reduction in scale for the posterior distribution, while effective sample size gives you the reduction in the variance of the estimate for an expected value of this distribution. In the ideal case, we wouldn't have to resort to sampling, and we could just derive, say, the posterior expected value; here, we would have zero error on the latter, but a possibly very large number for the former. For more information take a look at page 286 and 287.

Answer 2

An old thread, but still likely relevant to many users. Instead of increasing adapt_delta, you can try decreasing it to, say, 0.8 or a value that still does not lead to divergent iterations. My understanding is this increases the mixing and, thus, may increase the ESS.