Exponential and bayesian conjugate families: Review and extensions

Abstract

Bayesian Inference, Conjugate Parameterisation, Enriched Prior, Extended Conjugate Family, Posterior Linearity, Quadratic Variance Function, Reducibility, Skewness Vector,

On conjugate families and Jeffreys priors for von Mises–Fisher distributions

“…observations from the vMF family $\mathrm{scriptF}$, the likelihood function for $\lambda$ is given by $Lfalse(\lambda |s,nfalse)=e^{s\prime \theta false(\lambda false)-\mathrm{italicnM}false(\theta false(\lambda false)false)}=C_d{false(false\parallel \theta false(\lambda false)false\parallel false)}^n{e}^{s\prime \theta false(\lambda false)},$where $s=x_1+\cdots +x_n$ is the resultant of the sample. Following Gutiérrez-Peña and Smith (1997, Definition 3.1), the standard conjugate family for $\mathrm{scriptF}$ relative to $\lambda$, denoted by ${\mathrm{scriptC}}_{\lambda }false(\mathrm{scriptF}false)$, has densities $\pi false(\lambda |s,\nu false)\propto Lfalse(\lambda |s,\nu false).$Using such a prior with parameters s 0 and ${\nu }_0$ will result in a posterior with parameters $s_n=s_0+x_1+\cdots +x_n$ and ${\nu }_n={\nu }_0+n$. As clearly $\frac{s_n}{{\nu }_n}=\frac{{\nu }_0}{{\nu }_0+n}\frac{s_0}{{\nu }_0}+\frac{n}{{\nu }_0+n}\overline{x},$${\nu }_0$ can be interpreted as the prior sample size, and …”

“…As for $\theta =\kappa \mu$ we have $d\theta =a_d{\kappa }^{d-1}d\kappa \mathrm{italicdU}false(\mu false)$ (where a d is the area of the unit hypersphere). The latter may be more natural as reference measure, turning the standard conjugate family relative to the polar coordinates ($\mu$, $\kappa$) parameterization into the (obvious generalization) of what Gutiérrez-Peña and Smith (1997) call the DY-conjugate family for $\mathrm{scriptF}$ relative to the parameterization.…”

“…This is not a coincidence: in fact, one can alternatively establish Theorem 1 (and equivalently, Theorem 2 for $\alpha =d-1$) without explicit convergence computations using the general results of Diaconis and Ylvisaker (1979), see also Gutiérrez-Peña and Smith (1997, Theorem 3.1). Let $\mu$ be a probability measure on (the Borel sets of) ${\mathrm{double-struckR}}^d$ with bounded support $\mathrm{scriptS}$ and consider the natural exponential family through $\mu$ with density $ffalse(x|\theta false)=e^{\theta \prime x-Mfalse(\theta false)}$, where $e^{Mfalse(\theta false)}=\int e^{\theta \prime x}d\mu false(xfalse)$, and the standard conjugate family with densities $\pi false(\theta |s,\nu false)=e^{s\prime \theta -\nu Mfalse(\theta false)}$ (with respect to the Lebesgue measure).…”

“…Guttorp and Lockhart (1988) introduce a Bayesian approach for finding the direction of a signal based on developing standard (e.g., Gutiérrez-Peña and Smith, 1997, Definition 3.1) conjugate priors for the von Mises (vM) distribution (i.e., for d =2) using the canonical ($\theta$) parameterization. Damien and Walker (1999) present a full Bayesian analysis of circular data using the vM distribution by employing standard conjugate priors for the polar coordinates ($\mu$, $\kappa$) parameterization, and developing a Gibbs sampler for this family of distributions.…”

Duality Induced from Conjugacy in the Curved Exponential Family

“…It has attracted the interest of many researchers including Morris (1983), Veronese (1992, 2001), Gutiérrez-Peña (1997), Gutiérrez-Peña and Smith (1997), Ibrahim and Chen (2000), Yanagimoto and Ohnishi (2005a, b) and Ohnishi and Yanagimoto (2007). An advantage of the conjugate analysis is that its simple inferential scheme visualizes an essence of Bayesian inference.…”