get_theta_aunif: 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.

the design matrix.

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.

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

Run this code

# NOT RUN {
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

# }

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