## null recurrent = zero utility?

Posted in Books, R, Statistics with tags , , , , , , , on April 28, 2022 by xi'an

The stability result that the ratio

$\dfrac{\sum^T_{t=1} f(\theta^{(t)})}{\sum^T_{t=1} g(\theta^{(t)})}\qquad(1)$

converges holds for a Harris π-null-recurrent Markov chain for all functions f,g in L¹(π) [Meyn & Tweedie, 1993, Theorem 17.3.2] is rather fascinating. However, it is unclear it can be useful in simulation environments, as for the integral priors we have been studying over the years with Juan Antonio Cano and Diego Salmeron Martinez. Above, the result of an experiment where I simulated a Markov chain as a Normal random walk in dimension one, hence a Harris π-null-recurrent Markov chain for the Lebesgue measure λ, and monitored the stabilisation of the ratio (1) when using two densities for f and g,  to its expected value (1, shown by a red horizontal line). There is quite a variability in the outcome (repeated 100 times),  but the most intriguing is the quick stabilisation of most cumulated averages to values different from 1. Even longer runs display this feature

which I would blame on the excursions of the random walk far away from the central regions for both f and g, that is on long sequences where zeroes keep being added to numerator and denominators in (1). As far as integral approximation is concerned, this is not very helpful!

## Gibbs sampling with incompatible conditionals

Posted in Books, Kids, R, Statistics with tags , , , , , , on July 23, 2019 by xi'an

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

1. 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.
2. 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.
3. 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 $x_1^{-1} {x_1 \choose x_2} p^{x_2} (1-p)^{x_1-x_2}$, 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.