This 2023 paper of Perepolkin, Goodrich, and Sahlin vaguely relates to our insufficient Gibbs work in that a Bayesian analysis is conducted based solely on quantile summaries. Except that here the input is the entire cdf, or the—inverse cdf—quantile function, or—its derivative—the quantile density function, instead of the probability density function—used as the likelihood in the posterior. Which is a non-problem from a mathematical perspective since all these functions describe the same probability distribution. Which makes the following quote rather puzzling (in its obviousness).
“We aim to show that the quantile-based Bayesian inference using the intermediate depths leads to the same posterior beliefs as the conventional density-based inference.”
The authors still make a big case of the difference, obviously, to the point of proposing a different notation for Y~F. But using the same symbol f for different densities. The formal expression of the posterior based on the quantile function actually requires the cdf function and the density or the quantile density to be available (at least in a numerical sense), witness eqn (11).
The paper still could hold some interest in its computational component. Relating to ABC, obviously, since distributions defined by quantiles and cdfs often come as benchmarks for ABC, when the associated pdf/likelihood is unavailable. Witness g-and-k distributions (with the caveat MCMC can be implemented in this case). Unfortunately, the paper entirely relies on numerical inversion (with a puzzling comment that MCMC rejection gets higher with numerical inversion, p6). And only mentions ABC in the conclusion, possibly to pacify a referee’s comment. And actually consider that “quantile parameterized quantile distributions don’t lend themselves easily as sampling distributions due to the special nature of their parameterization” (p4). Hence making me wonder at the overall relevance of the entire endeavour….
