Quick Google search reveals ... I'm not sure what. From this post on the Stan Google group, the following quotes are pertinent in answer to a question about (I thought) using HMC together with another type of sampler for one of the parameters. It seems the questioner has either a hierarchical or discrete model (both bad for Stan) and that in the later quotes it appears I could alternate Stan with RWMetropolis. Read on...
But then the questioner asks something else:
"It is fairly common that a published algorithm uses HMC as a kernel in combination with standard Metropolis-Hastings or Gibbs kernels... it would be surprising if this were _never_ a good idea..."
To which an answer goes:
"HMC is making a joint update of all the model parameters which is what is important. This leaves two issues: hierarchical modes are hard and discrete models are harder.... You can certainly use Stan as a sampling API and stick it into a bigger sampler, but without addressing the fundamental problems with discrete variables and conditional updates you're only going to be able to go so far."
Then Bob Carpenter also says:
"The technical problem right now in plugging samples back into Stan is that there's not an easy way to pull the adapted mass matrix out and then plug it back in. We'll be making that easier going forward, probably as part of Stan 3, which is not on the near term horizon yet, but we're working on design."
That's what he means when he says "there is not a mechanism by which you can pass "data" that is really an unknown that gets updated some other way". More from Bob Carpenter later:
"Yes --- you can interleave steps from Markov chains with the right stationarity properties (i.e., the stationary distribution of the chain is the posterior p(theta)). So you can alternate differential evolution and RWM or HMC or NUTS and the combination still has the right stationarity properties...The theory is that you need to ensure that the stationary distribution of the Markov chain is the posterior p(theta). If that holds, as it does for differential evolution and NUTS or HMC (or RWM), then you can."