Background
I’m currently working on a project which will utilize methods found in the appendix of this paper. The paper outlines how Gibbs sampling can be used to estimate the prevalence (\pi) of a disease, the sensitivity (S) , and the specificity (C) from the results of a diagnostic test for which there is no gold standard to compare against. Details can be found in the “One Diagnostic Test” section of the paper.
The method proceeds as follows:

Choose priors of the prevalence, sensitivity, and specificity. This paper uses beta priors for all three. Sample an initial point from each prior.

Using the samples from step 1, draw “pseudodata” from the conditional distribution.
Let a and b be the observed number of positive and negative test results. Let Y1 be the number of true positives and Y2 be the number of false negatives. If we knew the true prevalence, sensitivity, and specificity, then (from the appendix)…
Y1 \vert a , \pi, S, C \sim \operatorname{Binom} \left( a, \dfrac{\pi S}{\pi S + (1\pi)(1C)} \right)
Y2 \vert a , \pi, S, C \sim \operatorname{Binom} \left( b, \dfrac{\pi (1S)}{\pi (1S) + (1\pi)C} \right)
We don’t have the true prevalence, sensitivity, and specificicty, so we use the samples drawn from the prior in step 1 to obtain inital estimates of Y1 and Y2.
If we knew the true Y1 and Y2 then, then the conditional distribution of the prevalence, sensitivity, and specificity would be
\pi \vert a,b,Y1,Y2, \alpha_{\pi} \beta_{\pi} \sim \operatorname{Beta}(Y1+Y2 + \alpha_{\pi}, a+bY1Y2+\beta_{\pi})
S \vert Y1,Y2, \alpha_{S} \beta_{S} \sim \operatorname{Beta}(Y1+S, Y2+\beta_S)
C \vert a,b,Y1,Y2, \alpha_{C} \beta_{C} \sim \operatorname{Beta}(bY2+ \alpha_C, a  Y1 + \beta_C)
Here, \alpha_x, \beta_x are the parameters for the prior for metric x (i.e. prevalence, sensitivity, specificity). We don’t know the true Y1 and Y2, so we use the samples drawn from above.
 Repeat step 2, drawing from each conditional distribution and then using those draws to generate further daws.
It is quite easy to code this up in python or R, but I would like to use Stan so that I can utilize some of the diagnostics and parallelization in the rstan
package.
Question
This problem doesn’t require the use of HMC. Can I still use Stan for this problem? My inital thoughts are “No”, due to this thread, but I thought I would double check. I know I can use the algorithm = 'fixed param'
argument in rstan::stan
to draw from a generated quantities block. However, I’m not sure how to make stan perform the sort of sampling I’ve described.
Your time and insight is appreciated.