The tenets of quantile-based inference in Bayesian models

Abstract

This paper extends the application of Bayesian inference to probability distributions defined in terms of its quantile function. We describe the method of *indirect likelihood* to be used in the Bayesian models with sampling distributions which lack an explicit cumulative distribution function. We provide examples and demonstrate the equivalence of the "quantile-based" (indirect) likelihood to the conventional "density-defined" (direct) likelihood. We consider practical aspects of the numerical inversion of quantile function by root-finding required by the indirect likelihood method. In particular, we consider a problem of ensuring the validity of an arbitrary quantile function with the help of Chebyshev polynomials and provide useful tips and implementation of these algorithms in Stan and R. We also extend the same method to propose the definition of an *indirect prior* and discuss the situations where it can be useful

Quantile-parameterized distributions for expert knowledge elicitation

“…Several distributions, such as the Generalized Lambda Distribution (GLD) (Freimer et al, 1988;Ramberg and Schmeiser, 1974), the g-and-k distribution (Haynes et al, 1997;Haynes and Mengersen, 2005;Jacob, 2017;Prangle, 2017), the g-and-h distribution (Field and Genton, 2006;Mac Gillivray, 1992;Rayner and MacGillivray, 2002), and the Wakeby distribution (Jeong-Soo, 2005;Rahman et al, 2015;Tarsitano, 2005b), have been extensively studied and documented in the literature. These distributions are defined by noninvertible quantile functions (Perepolkin et al, 2021b). However, the research on quantile-parameterized distributions remains relatively unexplored.…”

“…is referred to as the density quantile function (Parzen, 1979) or p-pdf (Gilchrist, 2000). The relationships between these functions are concisely illustrated in the probability function Möbius loop (Figure 1), as described in Perepolkin et al (2021b).…”

“…The root value p ∈ (p a , p b ) of the function g(p) can be found using any of the bracketing root-finding algorithms (Perepolkin et al, 2021b). It can then be substituted into the following expressions to find the lower a and upper b limit parameters of the triangular distribution: Kotz and Van Dorp (2004) provide an algorithm for fitting a four-parameter generalization of the triangular distribution called the Two-Sided Power Distribution (TSP), using three quantile-probability pairs and a mode value.…”

“…For invertible distributions, the inverse quantile function is the cumulative distribution function (CDF) Q −1 (y i |θ) = F (y i |θ), otherwise, the inverse can be computed numerically as F (y i |θ) = Q −1 (y i |θ) (Perepolkin et al, 2021b). Drovandi and Pettitt (2011) show that the joint density of a single (multivariate) observation (y i , .…”

“…When a quantile-based prior specification is used, only the multivariate normal log-density needs to be added because the Jacobian for the marginal QF transformation is reciprocal to the DQF of the prior (Perepolkin et al, 2021b).…”