Explain why R-hat and ESS do not apply to the sampler diagnostics

src/reference-manual/analysis.qmd

@@ -327,6 +327,32 @@ making the sample of interest more Gaussian and hence the $\hat{R}$
 statistic more accurate.
 
 
+#### Sampler diagnostics {-}
+
+The convergence statements above are about functions of the
+parameters.  The sampler diagnostics `accept_stat__`, `treedepth__`,
+`n_leapfrog__`, and `divergent__` also depend on the step size and the
+metric.  Each chain adapts its own step size and metric during warmup
+and keeps them fixed during sampling, so their values can differ
+between chains.  For example, a smaller step size usually gives a
+higher acceptance statistic, a larger tree depth, and more leapfrog
+steps.  The stationary distribution of these sampler diagnostics can
+therefore differ between chains, whereas $\hat{R}$ and the multi-chain
+effective sample size assume the same stationary distribution in all
+chains.  The column `stepsize__` gives the step size of each
+iteration, which is constant in each chain unless step size jitter is
+on.
+
+The step size and the metric change the proposals of the sampler but
+not the target distribution.  The parameters and `lp__`, a function of
+the parameters, have the same stationary distribution in all chains.
+The column `energy__` is the value of the Hamiltonian at the draw: the
+negative of `lp__` plus the kinetic energy.  Because Stan draws a new
+momentum in each iteration, the kinetic energy has the same
+distribution for any metric.  The statistic $\hat{R}$ and the effective
+sample size thus apply to the parameters, `lp__`, and `energy__`.
+
+
 #### Final words on convergence monitoring {-}
 
 "Convergence" is a global property and holds for all integrable