The tenets of quantile-based inference in Bayesian models

Bayesian inference can be extended to probability distributions defined in terms of their inverse distribution function, i.e. their quantile function. This applies to both prior and likelihood. *Quantile-based likelihood* is useful in models with sampling distributions which lack an explicit probability density function. *Quantile-based prior* allows for flexible distributions to express expert knowledge. The principle of *quantile-based* Bayesian inference is demonstrated in the univariate setting with a Govindarajulu likelihood, as well as in a *parametric quantile regression*, where the error term is described by a quantile function of a Flattened Skew-Logistic distribution.