probit_horseshoe_regression: Gibbs Sampler for Probit Regression with Horseshoe Prior

In dcbdan/s525: Implementation of common Bayesian models

Description

The probit regression model with horseshoe prior is given by

y_i|π_i \sim Bernoulli(π_i),

π_i = Φ(x_i^tβ),

[β_j|λ_j] \sim N(0, λ_j^2), j=2,...,p,

p(β_1) \propto 1,

[λ_j|A] \sim C^{+}(0, A), j=2,...,p,

A \sim Uniform(0, 10).

where Φ is given by the gaussian CDF.

The implemented parameter-expanded model is given by

y_i^* = x_i^tβ + ε_i,

ε_i \sim N(0, 1),

y_i = I(y_i^* > 0).

The half-Cauchy parameter expansion is also used; given by

[η_j|γ_j] \sim Gamma(\frac{1}{2}, γ_j),

[γ_j] \sim Gamma(\frac{1}{2}, \frac{1}{A^2})

and η_j = λ_j^{-2} , τ_A = A^{-2} , The full conditionals are given by:

[y_i^*|y_i, β, X] \sim sgn(y_i, y_i^*)N(x_i^tβ, 1)

where sqn is 1 if both arguments are of the same sign and zero otherwise,

[β|Y^*, X, η] \sim \mathcal{N}(Q^{-1}l, Q^{-1})

where Q = X'X+diag(0, 1/η_2, ..., 1/η_p) and l = X'Y^*,

[η_j|β_j, γ_j] \sim \mathrm{exp}(\frac{β_j^2}{2} + γ_j),

[γ_j|η_j, τ_A] \sim \mathrm{exp}(η_j + τ_A),

[τ_A|γ] \sim \mathrm{Gamma}( \frac{p - 2}{2}, ∑ γ_i)\mathrm{I}_{(\frac{1}{100}, ∞)}.

Usage

probit_horseshoe_regression <- function(
  Y,
  X,
  niter,
  init = NULL)

Arguments

Y	n by 1 vector of ones and zeros
X	n by p predictor matrix, where p > 1 and the first column of X is all 1.
niter	number of gibbs sampling iterations
init	Initial starting values for beta. If NULL, beta is set to zero.

Details

This function returns a niter x p matrix of values where p is the second dimension of the predictor matrix X. The returned matrix contains all generated values of the gibbs sampling markov chain.

Examples

 print("TODO")

dcbdan/s525 documentation built on May 19, 2019, 10:48 p.m.