log posterior effective sample size question

[Andria Dawson]

Hi all,

I know it is standard to think about parameter effective sample size (ESS), and sometimes even run chains until you get a certain number of ESSs for parameters. Stan computes the ESS for the log posterior (LP), and I was wondering if there is there any literature that talks about being able to use this LP ESS as some sort of measure of overall sample quality as we do for parameter ESS. 

For example, sometimes I get high parameter ESS, but low LP ESS, and I'm trying to figure out exactly what I learn from this (related to model and/or sample quality).

Cheers, and thanks for your help,

Andria 

[Andrew Gelman]

We typically think of lp__ as just one more quantity of interest.  Poor convergence on an quantity of interest represents poor convergence in the distribution.  (If any man anywhere is a slave, none of us is free.)   So I think you have poor mixing here.

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[Michael Betancourt]

This is a really tricky issue and it requires being somewhat pedantic about MCMC estimators.

The first thing to keep in mind is that autocorrelation and effective sample size are all dependent

on the choice of a function.  Yes the entire distribution of the Markov chain is converging to the

target posterior,

T^{n} delta_{0} -> pi

but we have no way to measuring that.  So instead we monitor how expectations are converging,

1/N sum_{n = 1}^{N} f(x_{n}) -> E_{pi} [ f ],

with the f-dependent autocorrelation and effective sample size.  One consequence of this is that

expectations of some functions converge faster than others — for example the expectation of a

constant converges immediately!  Intuitively, the more a function varies the more sensitive it will

be to fine detailed structure in the posterior and hence slower it will be to converge.  So seeing

variation in the effective sample sizes (including the log posterior density which we can consider

as just another function) is not technically pathological.

But then comes the second consideration — we don’t know the autocorrelation and effective

sample size exactly and we have to estimate them.  Estimating autocorrelation is really, really

hard, especially when the autocorrelations are large.  One consequence of this is that when

expectations are converging really slowly the autocorrelation and effective sample size estimators

are typically huge overestimates.  Consequently, seeing variation in the effective sample sizes

can be due to poor estimators.

So which is it?  Well in practice we’ve seen the latter more than the former which, combined with

our perpetual yet friendly skepticism of MCMC, makes us take low log posterior density ESS

seriously.  But like so many MCMC diagnostics it’s not always a pathology so we’d require more

details to say more.

Sorry for the ambiguity, doing MCMC well is hard!