Single ESS value as min(ess_bulk, ess_tail)

[paul-buerkner]

Currently, we have one rhat value as default (worst case of rhat_bulk and rhat_tail) but always two ESS values, ess_bulk and ess_tail, without a single ess generic. Could it make sense to add such an ess generic and make the default min(ess_bulk, ess_tail)?

Specifically, @avehtari what would you think about this?

[jgabry]

I like this idea! Curious what @avehtari thinks.

[avehtari]

A more natural single ESS would be the minimum of all possible quantile ESSs, but that would increase the computation cost. If we anyway compute ess_bulk and ess_tail, minimum of those would be cheap, but the explanation of the value is longer. Rhat is inherently more abstract and thus I think someone not understanding details of the underlying min(rhat_bulk, rhat_tail) is not a big issue, but I'm worried that a single ESS might be interpreted as ESS for any expectation (or quantile) (and maybe even interpreted as the number of effective draws). ESS is somewhere between Rhat and MCSE, and it's clear we wouldn't report just one MCSE. So I'm worried, how a single ESS would be interpreted and it would require care how it's displayed and documented.

[paul-buerkner]

I understand. That makes sense to me. Perhaps, we still want to define an ess generic with default, even if we do not display it by default. For how many and which quantiles shall we try to compute the ESS before taking the minimum?

[avehtari]

As $summary() is often by default computed for every variable, I'm worried about the increased computational cost of computing minimum ESS over many quantiles in such default operation. Minimum of Bulk-ESS and Tail-ESS could be justified as an approximation of ESS over many quantiles. We could have also a function that would compute minimum over e.g. 5, 7 or 9 quantiles that users could call explicitly.