Please keep to third person and remove the "our".

Which is it, mean or median? I've never quite understood this concept myself, so I'm not sure. But with many distributions medians and means can be pretty far apart with enough skew.

It would be better if these were links.

I think this should say that it's a Markov chain, but then that's not true during warmup.

I would say that the divergence is referring to not preserving the Hamiltonian in the Hamiltonian simulation, as I'm not sure people understand what is supposed to be diverging here.

I'd lead with the "For a given number" sentence as that is the real explanation.

The warmup iterations can be saved, but they shouldn't be used for inference (i.e., calculating means or quantiles).

I'd link this to say that the maximum number of leapfrog steps is 2^(max treedepth), and each leapfrog step requires a log density and gradient evaluation.

usually if you hit the limit, the problem is that you need to parameterize. Very rarely is increasing the max depth to 11 or 12 or 13 the right answer.

I wouldn't say there's uncertainty in the samples. The uncertainty we care about is in the estimate and what would happen under new samples. As such, MCSE is an estimate of the standard deviation of the posterior mean estimator under replicated data.

The 0.01 value only holds if the estimate is 1. So this needs to be removed. MCSE is relative the estimate. For example, 0.0012 is reported to two digits of accuracy and requires MCSE be below 0.0001. But this isn't how people report. People regularly write things like 0.32 +/- 0.05 where 0.05 is the standard error.

diagnostic -> statistic of multiple chains

It doesn't evaluate, it tests. And it's not a foolproof test in that there are false positives and false negatives.

The two types of variation do not become equal. Rhat is essential sqrt(1 + between/within). The idea is that the between variation goes to 0 and the within stays constant at its nominal value based on the model, so that Rhat converges to 1.

Charles Margossian's nested R-hat paper is super useful for understanding what R-hat is doing.

What's happening is that the within chain variation scales the between chain variation to make R-hat unit free.

They're not mathematically equivalent in that they define two different densities (otherwise there wouldn't be much point). The idea is that the posteriors have enough information to derive each others random variables of interest.

Stan can always represent a model in infinitely many different ways.

Centering a predictor does not leave the model the same---it shifts things from coefficients into the intercept in a regression, for example. The point is that the centered posterior draws can be transformed to non-centered posterior draws.

Is this really the 5% and 95% quantiles? I don't know.

You're describing what people call "burn in" despite Gelman's objections that it's a bad analogy. In Stan, warmup critically does adaptation. As such, it doesn't form a Markov chain. We just assume that the adaptation will also take care of burn in.

There are three warmup phases. Phase I attempts to burn in with a unit mass matrix. Phase II attempts to estimate the mass matrix. Phase III does a final tuning of step size.