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I was going through the Stan documentation which can be downloaded from here. I was particularly interested in their implementation of the Gelman-Rubin diagnostic. The original paper Gelman & Rubin (1992) define the the potential scale reduction factor (PSRF) as follows:

Let $X_{i,1}, \dots , X_{i,N}$ be the $i$th Markov chain sampled, and let there be overall $M$ independent chains sampled. Let $\bar{X}_{i\cdot}$ be the mean from the $i$th chain, and $\bar{X}_{\cdot \cdot}$ be the overall mean. Define, $$W = \dfrac{1}{M} \sum_{m=1}^{M} {s^2_m}, $$ where $$s^2_m = \dfrac{1}{N-1} \sum_{t=1}^{N} (\bar{X}_{m t} - \bar{X}_{m \cdot})^2\,. $$ And define $B$ $$B = \dfrac{N}{M-1} \sum_{m=1}^{M} (\bar{X}_{m \cdot} - \bar{X}_{\cdot \cdot})^2 \,.$$

Define $$\hat{V} = \left(\dfrac{N-1}{N} \right)W + \left( \dfrac{M+1}{MN} \right)B\,.$$ The PSRF is estimate with $\sqrt{\hat{R}}$ where $$ \hat{R} = \dfrac{\hat{V}}{W} \cdot \dfrac{df+3}{df+1}\,,$$ where $df = 2\hat{V}/Var(\hat{V})$.

The Stan documentation on page 349 ignores the term with $df$ and also removes the $(M+1)/M$ multiplicative term. This is their formula,

The variance estimator is $$\widehat{\text{var}}^{+}(\theta \, | \, y) = \frac{N-1}{N} W + \frac{1}{N} B\,.$$ Finally, the potential scale reduction statistic is defined by $$ \hat{R} = \sqrt{\frac{\widehat{\text{var}}^{+}(\theta \, | \, y) }{W}}\,. $$

From what I could see, they don't provide a reference for this change of formula, and neither do they discuss it. Usually $M$ is not too large, and can often be as low so as $2$, so $(M+1)/M$ should not be ignored, even if the $df$ term can be approximated with 1.

So where does this formula come from?

EDIT: I have found a partial answer to the question "where does this formula come from?", in that the Bayesian Data Analysis book by Gelman, Carlin, Stern, and Rubin (Second edition) has exactly the same formula. However, the book does not explain how/why it is justifiable to ignore those terms?

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  • $\begingroup$ There is no published paper on it yet, and the formula will probably change in the next few months anyway. $\endgroup$ – Ben Goodrich May 31 '18 at 18:28
  • $\begingroup$ @BenGoodrich Thanks for the comment. Can you say anything more on the motivation of using this formula? And why exactly will the formula change? $\endgroup$ – Greenparker May 31 '18 at 18:55
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    $\begingroup$ The current split R-hat formula is the way it is mostly to make it apply to the case where there there is only one chain. The coming changes are mostly to deal with the fact that the underlying marginal posterior distribution may not be normal or have a mean and / or variance. $\endgroup$ – Ben Goodrich Jun 1 '18 at 14:31
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    $\begingroup$ @BenGoodrich Yes, I get why STAN does split Rhat. But even in that case $M = 2$, and so the constant $(M+1)/M = 3/2$ which is not ignorable. $\endgroup$ – Greenparker Jun 2 '18 at 13:28
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I followed the specific link given for Gelman & Rubin (1992) and it has $$ \hat{\sigma} = \frac{n-1}{n}W+ \frac{1}{n}B $$ as in the later versions, although $\hat{\sigma}$ replaced with $\hat{\sigma}_+$ in Brooks & Gelman (1998) and with $\widehat{\rm var}^+$ in BDA2 (Gelman et al, 2003) and BDA3 (Gelman et al, 2013).

BDA2 and BDA3 (couldn't check now BDA1) have an exercise with hints to show that $\widehat{\rm var}^+$ is unbiased estimate of the desired quantity.

Gelman & Brooks (1998) has equation 1.1 $$ \hat{R} = \frac{m+1}{m}\frac{\hat{\sigma}_+}{W} - \frac{n-1}{mn}, $$ which can be rearranged as $$ \hat{R} = \frac{\hat{\sigma}_+}{W} + \frac{\hat{\sigma}_+}{Wm}- \frac{n-1}{mn}. $$ We can see that the effect of second and third term are negligible for decision making when $n$ is large. See also the discussion in the paragraph before Section 3.1 in Brooks & Gelman (1998).

Gelman & Rubin (1992) also had the term with df as df/(df-2). Brooks & Gelman (1998) have a section describing why this df corretion is incorrect and define (df+3)/(df+1). The paragraph before Section 3.1 in Brooks & Gelman (1998) explains why (d+3)/(d+1) can be dropped.

It seems your source for the equations was something post Brooks & Gelman (1998) as you had (d+3)/(d+1) there and Gelman & Rubin (1992) had df/df(-2). Otherwise Gelman & Rubin (1992) and Brooks & Gelman (1998) have equivalent equations (with slightly different notations and some terms are arranged differently). BDA2 (Gelman, et al., 2003) doesn't have anymore terms $\frac{\hat{\sigma}_+}{Wm}- \frac{n-1}{mn}$. BDA3 (Gelman et al., 2003) and Stan introduced split chains version.

My interpretation of the papers and experiences using different versions of $\hat{R}$ is that the terms which have been eventually dropped can be ignored when $n$ is large, even when $m$ is not. I also vaguely remember discussing this with Andrew Gelman years ago, but if you want to be certain of the history, you should ask him.

Usually M is not too large, and can often be as low so as 2

I really do hope that this is not often the case. In cases where you want to use split-$\hat{R}$ convergence diagnostic, you should use at least 4 chains split and thus have M=8. You may use less chains, if you already know that in your specific cases the convergence and mixing is fast.

Additional reference:

  • Brooks and Gelman (1998). Journal of Computational and Graphical Statistics, 7(4)434-455.
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  • $\begingroup$ Yes it has the same $\hat{\sigma}^2$ as you mention, but their $\hat{R}$ statistic is $(\hat{\sigma}^2 + B/mn)/W * df_{term}$ (look at the equation on top of page 495 in the Stat Science official version), which introduces the $(m+1)/m$ term I was talking about. In addition, look at the code and description in the R package coda, which has had the GR diagnostic since 1999. $\endgroup$ – Greenparker Jun 10 '18 at 21:20
  • $\begingroup$ I'm confused. The article via the link you provided and the article from Stat Science web pages has only pages 457-472.I didn't check now, but years ago and last year when I checked coda, it didn't have the current recommended version. $\endgroup$ – Aki Vehtari Jun 11 '18 at 7:00
  • $\begingroup$ Note that I edited my answer. Gelman & Brooks (1998) has that (m+1)/m term more clearly, and it seems you missed the last term which mostly cancels the effect of (m+1)/m term for decision making. See that paragraph before section 3.1. $\endgroup$ – Aki Vehtari Jun 11 '18 at 7:48
  • $\begingroup$ Sorry about that, that was a typo. It's page 465, and Gelman and Rubin have the same exact definition as Brooks and Gelman (which you state above). Equation 1.1 in Brooks and Gelman is exactly what I wrote down as well (when you rearrange some terms). $\endgroup$ – Greenparker Jun 11 '18 at 8:10
  • $\begingroup$ "We can see that the effect of second and third term are negligible for decision making when n is large", so what you are saying is that the expression in BDA and hence STAN comes from essentially ignoring these terms for large n? $\endgroup$ – Greenparker Jun 11 '18 at 8:13

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