gdpDC: Generalized double pareto shrinkage prior with unpenalized...

In abnormally-distributed/Bayezilla: Bayesian Models With JAGS and Several Utilities for Data Exploration and Model Evaluation.

Description

The generalized double pareto shrinkage prior of Armagan, Dunson, & Lee (2013). For the binomial and poisson likelihoods the same pseudovariance functions used in all other models in this package are used. This variant grants the allowance for a set of covariates that are not penalized. For example, you may wish to include variables such as age and gender in all models so that the coefficients for the other variables are penalized while controlling for these. This is a common need in research.


This model is parameterized similarly to the Bayesian LASSO of Park & Casella (2008), Normal-Exponential-Gamma prior of Griffin, J. E. and Brown, P. J. (2011), and adaptive Bayesian LASSO of Leng, Tran and David Nott (2018). The key feature is that this model explicitly utilizes generalized double pareto priors through a scale mixture of normals, while the Bayesian LASSO utilizes double exponential priors through a scale mixture of normals. The Bayesian adaptive LASSO also utilizes double exponential just as the BLASSO, but has coefficient specific shrinkage parameters. The NEG-BLASSO utilizes (as the name suggests) normal-exponential-gamma priors, which behave very similarly to the GDP. Both the NEG and GDP distributions have a peak at zero, just as the double exponential distribution, but have very long, student-t-like tails.

In this model, the coefficient specific shrinkage parameters are given gamma distributions that with shape and rate parameters (alpha and zeta, respectively) each with independent gamma(4, 8) hyperpriors, which are very concentrated near 1.

To quote directly from Armagan, Dunson, & Lee (2013):


As α grows, the density becomes lighter tailed, more peaked and the variance becomes smaller, while as ζ grows, the density becomes flatter and the variance increases. Hence if we increase α, we may cause unwanted bias for large signals, though causing stronger shrinkage for noise-like signals; if we increase ζ we may lose the ability to shrink noise-like signals, as the density is not as pronounced around zero; and finally, if we increase α and η at the same rate, the variance remains constant but the tails become lighter, converging to a Laplace density in the limit. This leads to over-shrinking of coefficients that are away from zero. As a typical default specification for the hyperparameters, one can take α = ζ = 1. This choice leads to Cauchy-like tail behavior, which is well-known to have desirable Bayesian robustness properties.


The reason for not just fixing the values at 1 is that I have observed that this does not always result in much, if any, shrinkage, and that using hyperpriors results in much better sampling (better chain mixing and less autocorrelation). Furthermore, hyperpriors allow the data to speak as to which values are best.

Model Specification:



The marginal probability density function for the coefficients is of the form

Which makes the implied prior on the coefficients

Usage

gdpDC(formula, design.formula, data, family = "gaussian",
  log_lik = FALSE, iter = 10000, warmup = 1000, adapt = 2000,
  chains = 4, thin = 1, method = "parallel", cl = makeCluster(2),
  ...)

Arguments

formula	the model formula
design.formula	formula for the design covariates.
data	a data frame.
family	one of "gaussian" (the default), "binomial", or "poisson"
log_lik	Should the log likelihood be monitored? The default is FALSE.
iter	How many post-warmup samples? Defaults to 10000.
warmup	How many warmup samples? Defaults to 2500.
adapt	How many adaptation steps? Defaults to 5000.
chains	How many chains? Defaults to 4.
thin	Thinning interval. Defaults to 1.
method	Defaults to "parallel". For an alternative parallel option, choose "rjparallel" or. Otherwise, "rjags" (single core run).
cl	Use parallel::makeCluster(# clusters) to specify clusters for the parallel methods. Defaults to two cores.
...	Other arguments to run.jags.

Value

a runjags object

References

Armagan, A., Dunson, D. B., & Lee, J. (2013). Generalized Double Pareto Shrinkage. Statistica Sinica, 23(1), 119–143.

Examples

gdpDC()

abnormally-distributed/Bayezilla documentation built on Oct. 31, 2019, 1:57 a.m.