get_theta_aunif {sdPrior}

Find Scale Parameter for Hyperprior for Variances Where the Standard Deviations have an Approximated (Differentiably) Uniform Distribution.

Description

This function implements a optimisation routine that computes the scale parameter \theta of the prior \tau^2 (corresponding to a differentiably approximated version of the uniform prior for \tau) for a given design matrix and prior precision matrix such that approximately P(|f(x_{k}|\le c,k=1,\ldots,p)\ge 1-\alpha

Usage

get_theta_aunif(alpha = 0.01, method = "integrate", Z, c = 3,
  eps = .Machine$double.eps, Kinv)

Arguments

alpha

denotes the 1-\alpha level.

method

with integrate as default. Currently no further method implemented.

Z

the design matrix.

c

denotes the expected range of the function.

eps

denotes the error tolerance of the result, default is .Machine$double.eps.

Kinv

the generalised inverse of K.

Value

an object of class list with values from uniroot.

Author(s)

Nadja Klein

References

Nadja Klein and Thomas Kneib (2015). Scale-Dependent Priors for Variance Parameters in Structured Additive Distributional Regression. Working Paper.

Andrew Gelman (2006). Prior Distributions for Variance Parameters in Hierarchical Models. Bayesian Analysis, 1(3), 515–533.

Examples

set.seed(123)
library(MASS)
# prior precision matrix (second order differences) 
# of a spline of degree l=3 and with m=20 inner knots
# yielding dim(K)=m+l-1=22
K <- t(diff(diag(22), differences=2))%*%diff(diag(22), differences=2)
# generalised inverse of K
Kinv <- ginv(K)
# covariate x
x <- runif(1)
Z <- matrix(DesignM(x)$Z_B,nrow=1)
theta <- get_theta_aunif(alpha = 0.01, method = "integrate", Z = Z, 
                            c = 3, eps = .Machine$double.eps, Kinv = Kinv)$root