compute_GVA: Compute the Full-Covariance Gaussian VB Empirical Likelihood...

In VBel: Variational Bayes for Fast and Accurate Empirical Likelihood Inference

Compute the Full-Covariance Gaussian VB Empirical Likelihood Posterior

Description

Requires a given data set, moment conditions and parameter values and returns a list of the final mean and variance-covariance along with an array of the in-between calculations at each iteration for analysis of convergence

Usage

compute_GVA(
  mu0,
  C0,
  h,
  delthh,
  delth_logpi,
  z,
  lam0,
  rho,
  epsil,
  a,
  SDG_iters = 10000,
  AEL_iters = 500,
  verbosity = 500
)

Arguments

mu0	p x 1 initial vector of Gaussian VB mean
C0	p x p initial lower triangular matrix of Gaussian VB Cholesky
h	User-defined moment-condition function. Note that output should be an (n-1) x K matrix where K is necessarily \geq p
delthh	User-defined first-order derivative of moment-condition function. Note that output should be a K x p matrix of h(zi,th) with respect to theta
delth_logpi	User-defined first-order derivative of log-prior function. Note that output should be a p x 1 vector
z	Data matrix, n-1 x d matrix
lam0	Initial vector for Lagrange multiplier lambda
rho	Scalar numeric beteen 0 to 1. ADADELTA accumulation constant
epsil	Positive numeric scalar stability constant. Should be specified with a small value
a	Positive scalar adjustment constant. For more accurate calculations, small values are recommended
SDG_iters	Number of Stochastic Gradient-Descent iterations for optimising mu and C. Default: 10,000
AEL_iters	Number of iterations using Newton-Raphson for optimising AEL lambda. Default: 500
verbosity	Integer for how often to print updates on current iteration number. Default:500

Value

A list containing:

1. mu_FC: VB Posterior Mean at final iteration. A vector of size p x 1

2. C_FC: VB Posterior Variance-Covariance (Cholesky) at final iteration. A lower-triangular matrix of size p x p

3. mu_FC_arr: VB Posterior Mean for each iteration. A matrix of size p x (SDG_iters + 1)

4. C_FC_arr: VB Posterior Variance-Covariance (Cholesky) for each iteration. An array of size p x p x (SDG_iters + 1)

Author(s)

Weichang Yu, Jeremy Lim

References

Yu, W., & Bondell, H. D. (2023). Variational Bayes for Fast and Accurate Empirical Likelihood Inference. Journal of the American Statistical Association, 1–13. \Sexpr[results=rd]{tools:::Rd_expr_doi("doi:10.1080/01621459.2023.2169701")}

Examples

# -----------------------------
# Initialise Inputs
# -----------------------------
# Generating 30 data points from a simple linear-regression model
set.seed(1)
x    <- runif(30, min = -5, max = 5)
vari <- rnorm(30, mean = 0, sd = 1)
y    <- 0.75 - x + vari
lam0 <- matrix(c(0,0), nrow = 2)
z    <- cbind(x, y)

# Specify moment condition functions for linear regression and its corresponding derivative
h    <- function(z, th) {
    xi     <- z[1]
    yi     <- z[2]
    h_zith <- c(yi - th[1] - th[2] * xi, xi*(yi - th[1] - th[2] * xi))
    matrix(h_zith, nrow = 2)
}

delthh <- function(z, th) {
    xi <- z[1]
    matrix(c(-1, -xi, -xi, -xi^2), 2, 2)
}

# Specify derivative of log prior
delth_logpi <- function(theta) { -0.0001 * mu0 }

# Specify AEL constant and Newton-Rhapson iteration
a         <- 0.00001
AEL_iters <- 10

# Specify initial values for GVA mean vector and Cholesky
reslm <- lm(y ~ x)
mu0   <- matrix(unname(reslm$coefficients),2,1)
C0    <- unname(t(chol(vcov(reslm))))

# Specify details for ADADELTA (Stochastic Gradient-Descent)
SDG_iters <- 50
epsil     <- 10^-5
rho       <- 0.9

# -----------------------------
# Main
# -----------------------------
result <-compute_GVA(mu0, C0, h, delthh, delth_logpi, z, lam0, 
rho, epsil, a, SDG_iters, AEL_iters)

VBel documentation built on April 4, 2025, 2:24 a.m.