solve_for_hiershrink_scale: Numerical-based solution to the scale parameter

In umich-biostatistics/AdaptiveBayesianUpdates: Adaptive Bayesian Updates

Numerical-based solution to the scale parameter

Description

This function calculates a numerical-based solution to the quantity c / sigma in the equation at the end of first paragraph on page e50. It is intended to be provided as the value for beta_orig_scale and beta_aug_scale in the functions glm_sab, glm_nab, and glm_standard

Usage

solve_for_hiershrink_scale(
  target_mean,
  npar,
  local_dof = 1,
  global_dof = 1,
  slab_dof = Inf,
  slab_scale = 15,
  n,
  tol = .Machine$double.eps^0.5,
  max_iter = 100,
  n_sim = 2e+05,
  slab_precision = NULL
)

Arguments

target_mean	(pos. reals): the desired prior number of effective parameters (tilde xi_eff in Boonstra and Barbaro). An error will be thrown if target_mean > npar
npar	(pos. integers): the number of covariates.
local_dof	(pos. integer) number indicating the degrees of freedom for lambda_j. Boonstra and Barbaro always used local_dof = 1. Choose a negative value to tell the function that there are no local hyperparameters.
global_dof	(pos. integer) number indicating the degrees of freedom for tau. Boonstra and Barbaro always used global_dof = 1. Choose a negative value to tell the function that there is no global hyperparameter.
slab_dof	see slab_scale
slab_scale	(pos. real) these control the slab-part of the regularized horseshoe. Specifically, in the notation of Boonstra and Barbaro, d^2~InverseGamma(slab_dof/2, slab_scale^2*slab_dof/2). In Boonstra and Barbaro, d was fixed at 15, and you can achieve this by leaving these at their default values of slab_dof = Inf and slab_scale = 15.
n	(pos. integer) sample size of the study
tol	(pos. real) numerical tolerance for convergence of solution
max_iter	(pos. integer) maximum number of iterations to run without convergence before giving up
n_sim	(pos. integer) number of simulated draws from the underlying student-t hyperpriors to calculate the Monte Carlo-based approximation of the expectation.
slab_precision	(pos. real) the slab-part of the regularized horseshoe, this is equivalent to (1/d)^2 in the notation of Boonstra and Barbaro. If specified, it is assumed that you want a fixed slab component and will take precedence over any provided values of slab_dof and slab_scale; slab_precision is provided for backwards compatibility but will be going away in a future release, and the proper way to specify a fixed slab component with with precision 1/d^2 for some number d is through slab_dof = Inf and slab_scale = d.

Details

If the outcome of interest is binary, then sigma doesn't actually exist as a real parameter, and it will be set equal to 2 inside glm_sab, glm_nab, or glm_standard. If the outcome of interest is continuous, then sigma is equipped with its own weak prior. In either case, it is not intended that the user scale by sigma "manually".

Value

A list containing the following named elements:

• scalethe solution, interpreted as c / sigma

• diff_from_targetthe difference between the numerical value and the target, should be close to zero

• iternumber of iterations performed

• prior_numthe resulting value of m_eff corresponding to this solution

umich-biostatistics/AdaptiveBayesianUpdates documentation built on April 26, 2024, 2:11 a.m.