Exact posterior distribution for nonconjugate Pareto models

Abstract

In Bayesian analysis, the so-called conjugate models allow obtaining the posterior distribution in exact form, in the sense that the posterior quantities can explicitly be written in a computable form. However, this class of models only involves a few structures, with some specific prior distribution for every data distribution. Although approximate methods such as numerical integration and MCMC are very efficient in Bayesian inference, little attention has been devoted to alternative views. In particular, the well-known conjugate Pareto-Gamma model is very restrictive, since the prior information can be expressed only through a Gamma distribution. In this work, we use special functions to extend the class of possible prior distributions that allow obtaining the posterior distribution in exact form. We give results that allow the use of any prior distribution within the broad class of H-functions. An example is provided to illustrate the theory.

Appendix: Proofs

Proof

The likelihood of a Pareto distribution is given by

$$\begin{aligned} {L}_{x}\left( \theta \right) ={\theta }^{k}{x}_{m}^{k\theta }\prod _{i=1}^{k}\frac{1}{{x}_{i}^{\theta +1}}\times I\left( t\ge {x}_{m}\right) ={k}_{1}{\theta }^{k}{e}^{-\theta {\sum }_{i=1}^{k}\textrm{log}\left( \frac{{x}_{i}}{{x}_{m}}\right) }\times I\left( t\ge {x}_{m}\right) , \end{aligned}$$

(1)

where $t=\min (x_i)$ and $k_1=1/\prod _{i=1}^k x_i$. We write the unnormalized posterior moments in terms of H-functions,

$$\begin{aligned} I\left( r\right)= & {} {\int }_{0}^{\infty }{\theta }^{r}{L}_{x}\left( \theta \right) p\left( \theta \right) d\theta ={k}_{1}{\int }_{0}^{\infty }{\theta }^{r+{k}}{e}^{-\theta {\sum }_{i=1}^{k}\textrm{log}\left( \frac{{x}_{i}}{{x}_{m}}\right) p\left( \theta \right) d\theta ,}\nonumber \\= & {} {k}_{1}{k}_{2}{\int }_{0}^{\infty }{\theta }^{r+k+{\alpha }_{2}}{H}_{0~1}^{1~0}\left[ \theta \sum _{i=1}^{k}\textrm{log}\left( \frac{{x}_{i}}{{x}_{m}}\right) \Bigg |\begin{array}{c}-\\ \left( \textrm{0,1}\right) \end{array}\right] {H}_{M~N}^{P~Q}\left[ {c\theta }^{d}\Bigg |\begin{array}{c}{}_{1}{\left( {c}_{j},{C}_{j}\right) }_{P}\\ {}_{1}{\left( {d}_{j},{D}_{j}\right) }_{Q}\end{array}\right] d\theta . \end{aligned}$$

The result follows by Theorem 3, by letting $\alpha _{1}=k+1$, $m=q=1$, $n=p=0$, $a=\sum _{i=1}^k\log \left( \frac{x_i}{x_m}\right)$, $b=1$ and $(b_{1},B_{1})=(0,1)$. $\square$

Proof of Theorem 4

Considering the posterior distribution (11),

$$\begin{aligned} P\left( \theta \le \vartheta |x\right)= & {} \frac{{k}_{2}}{I\left( 0\right) {\prod }_{i=1}^{k}{x}_{i}}{\int }_{0}^{\vartheta }{\theta }^{k+{\alpha }_{2}}{e}^{-\theta {\sum }_{i=1}^{k}\textrm{log}\left( \frac{{x}_{i}}{{x}_{m}}\right) }{H}_{M~N}^{P~Q}\left[ {c\theta }^{d}\Bigg |\begin{array}{c}{}_{1}{\left( {c}_{j},{C}_{j}\right) }_{P}\\ {}_{1}{\left( {d}_{j},{D}_{j}\right) }_{Q}\end{array}\right] \nonumber \\= & {} \frac{{k}_{2}}{I\left( 0\right) {\prod }_{i=1}^{k}{x}_{i}}\!\sum _{j=0}^{\infty }\frac{{\left( \!-{\sum }_{i=1}^{k}\textrm{log}\!\left( \frac{{x}_{i}}{{x}_{m}}\right) \right) }^{j}}{j!}\!{\int }_{0}^{\vartheta }{\theta }^{k+{\alpha }_{2}+j}{H}_{M~N}^{P~Q}\!\left[ {c\theta }^{d}\Bigg |\begin{array}{c}{}_{1}{\left( {c}_{j},{C}_{j}\right) }_{P}\\ {}_{1}{\left( {d}_{j},{D}_{j}\right) }_{Q}\end{array}\!\right] d\theta . \end{aligned}$$

From Definition 1, letting U be the integral above, we have

$$\begin{aligned} U= & {} \frac{1}{2\pi i}{\int }_{L}\phi \left( s\right) {c}^{s}{\int }_{0}^{\vartheta }{\theta }^{k+{\alpha }_{2}+j-ds}d\theta ds,\nonumber \\= & {} \frac{1}{2\pi i}{\int }_{L}\phi \left( s\right) {c}^{s}\frac{{\vartheta }^{k+{\alpha }_{2}+j-ds+1}}{k+{\alpha }_{2}+j-ds+1}d\theta ds,\\= & {} {\vartheta }^{k+{\alpha }_{2}+j+ds+1}\frac{1}{2\pi i}{\int }_{L}\phi \left( s\right) \frac{\Gamma \left( k+{\alpha }_{2}+j-ds+1\right) }{\Gamma \left( k+{\alpha }_{2}+j-ds+2\right) }{\left( {c\vartheta }^{d}\right) }^{-s}ds. \end{aligned}$$

Hence the result follows from Definition 1. $\square$

Proof of Theorem 5

The posterior predictive distribution is

$$\begin{aligned} p\left( \widetilde{x}|x\right) ={\int }_{0}^{\infty }{L}_{\widetilde{x}}\left( \theta \right) p\left( \theta |x\right) d\theta =\frac{1}{I\left( 0\right) }{\int }_{0}^{\infty }{L}_{\widetilde{x}}\left( \theta \right) {L}_{x}\left( \theta \right) p\left( \theta \right) d\theta \end{aligned}$$

If we combine $L_{\widetilde{x}}(\theta )$ and $L_{\textit{x}}(\theta )$, we have

$$\begin{aligned} {L}_{\widetilde{x},x}^{^{\prime }}\left( \theta \right) ={L}_{\widetilde{x}}\left( \theta \right) {L}_{x}\left( \theta \right) ={k}_{1}^{^{\prime }}{\theta }^{{\theta }_{1}^{^{\prime }}-1}{H}_{0~1}^{1~0}\left[ a^{\prime }\theta \Bigg |\begin{array}{c}-\\ \left( \textrm{0,1}\right) \end{array}\right] , \end{aligned}$$

where $k_1'=(\widetilde{x}\prod _{=1}^k x_i)^{-1},\,\alpha _{1}'=k+2$, $a'=\log \left( \widetilde{x}/x_{m}\right) +\sum _{i=1}^k\log \left( x_{i}/x_{m}\right)$ and $b'=1$.

It follows that we have the same situation as in Theorem 4, that is a likelihood function $L_{\widetilde{x},\textit{x}}'(\theta )$ and the prior distribution $p(\theta )$, hence we can apply Theorem 3 with $r=0$ and the result follows. $\square$