Using fractional derivatives to derive marginal densities

TL;DR

This paper introduces a new analytical method for deriving marginal densities using fractional derivatives of moment-generating functions, requiring modest assumptions about the likelihood and prior functions.

Contribution

The paper proposes a novel fractional derivative approach for marginal density derivation, expanding analytical tools in probabilistic modeling.

Findings

Method requires prior moment-generating functions to be finite, continuous, and differentiable.

Provides probabilistic and statistical insights into the fractional derivative approach.

Applicable under specific likelihood function forms, with modest assumptions.

Abstract

This paper presents a novel method for analytical derivations of marginal densities using the fractional derivatives of moment-generating functions. Although the method requires likelihood functions to take specific forms, its assumptions are otherwise modest. It only requires that the prior moment-generating functions exist, are finite, and are continuous and differentiable at certain points. We also present the probabilistic and statistical insights behind this method.