**A**n interesting question (with no clear motivation) on X validated wondering why a Gibbs sampler produces NAs… Interesting because multi-layered:

- The attached R code indeed produces NAs because it calls the Negative Binomial Neg(x¹,p) random generator with a zero success parameter, x¹=0, which automatically returns NAs. This can be escaped by returning a one (1) instead.
- The Gibbs sampler is based on a Bin(x²,p) conditional for X¹ and a Neg(x¹,p) conditional for X². When using the most standard version of the Negative Binomial random variate as the number of failures, hence supported on 0,1,2…. these two conditionals are incompatible, i.e., there cannot be a joint distribution behind that returns these as conditionals, which makes the limiting behaviour of the Markov chain harder to study. It however seems to converge to a distribution close to zero, which is not contradictory with the incompatibility property: the stationary joint distribution simply does not enjoy the conditionals used by the Gibbs sampler as its conditionals.
- When using the less standard version of the Negative Binomial random variate understood as a number of attempts for the conditional on X², the two conditionals are compatible and correspond to a joint measure proportional to , however this pmf does not sum up to a finite quantity (as in the original Gibbs for Kids example!), hence the resulting Markov chain is at best null recurrent, which seems to be the case for p different from ½. This is unclear to me for p=½.

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