Rebayes: an R package for empirical bayes mixture methods

Abstract

Models of unobserved heterogeneity, or frailty as it is commonly known in survival analysis, can often be formulated as semiparametric mixture models and estimated by maximum likelihood as proposed by Robbins (1950) and elaborated by Kiefer and Wolfowitz (1956). Recent developments in convex optimization, as noted by Koenker and Mizera (2014b), have led to dramatic improvements in computational methods for such models. In this vignette we describe an implementation contained in the R package REBayes with applications to a wide variety of mixture settings: Gaussian location and scale, Poisson and binomial mixtures for discrete data, Weibull and Gompertz models for survival data, and several Gaussian models intended for longitudinal data. While the dimension of the nonparametric heterogeneity of these models is inherently limited by our present gridding strategy, we describe how additional fixed parameters can be relatively easily accommodated via profile likelihood. We also describe some nonparametric maximum likelihood methods for shape and norm constrained density estimation that employ related computational methods.

Poisson mean vector estimation with nonparametric maximum likelihood estimation and application to protein domain data

“…for given grid points b j , 1 There are many approaches to solve the optimization in (3.15), for example, the EM algorithm, the convex optimization in [18] etc. In our numerical studies, we use the Pmix function in the R packages "REBayes" [11] to obtain G n .…”

Nonparametric mean and variance adaptive classification rule for high‐dimensional data with heteroscedastic variances

“…represents the density function of V 𝑗 given 𝜎 2 . For 𝑗 = 1, … , p, we can characterize these two Bayes rules from model ( 7) ∼ (10).…”

“… $$$$ {\hat{G}}_0=\underset{G\in \mathcal{G}}{\mathrm{argmax}}\frac{1}{p}\sum \limits_{j=1}^p\left[\log \left\{\int {f}_{{\hat{F}}_0}\left({X}_j,{V}_j\mid \mu \right) dG\left(\mu \right)\right\}\right], $$$$ where $$$ {f}_{F_0}\left({X}_j,{V}_j\mid \mu \right)=\int f\left({X}_j,{V}_j\mid \mu, {\sigma}^2\right){dF}_0\left({\sigma}^2\right) $$$ and $$$ f\left({X}_j,{V}_j\mid \mu, {\sigma}^2\right) $$$ is expressed as the product of the conditional density function of $$$ {X}_j $$$ given $$$ \mu $$$ and $$$ {\sigma}^2 $$$ and the conditional density function of $$$ {V}_j $$$ given $$$ {\sigma}^2 $$$. Although, the convex optimization problem (14) is infinite‐dimensional, various computationally efficient algorithms have been developed over the years [10, 12, 16]. Following Lindsay [20]; Koenker and Mizera...…”

“…is expressed as the product of the conditional density function of X 𝑗 given 𝜇 and 𝜎 2 and the conditional density function of V 𝑗 given 𝜎 2 . Although, the convex optimization problem ( 14) is infinite-dimensional, various computationally efficient algorithms have been developed over the years [10,12,16]. Following Lindsay [20]; Koenker and Mizera [16]; Dicker and Zhao [6]; Feng and Dicker [8]; Park et al [21], we approximate F0 and Ĝ0 with fine grid points support set.…”