VBel (version 1.1.6)

compute_GVA: 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,
  SGD_iters = 10000,
  AEL_iters = 500,
  verbosity = 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 (SGD_iters + 1)

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

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 x K matrix where K is necessarily \(\geq\) p. Input format for h should be (zi, th) were zi corresponds to the ith observation's data and th is the parameter vector

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. Input format for h should be (zi, th) were zi corresponds to the ith observation's data and th is the parameter vector

delth_logpi

User-defined first-order derivative of log-prior function. Note that output should be a p x 1 vector. Input format for h should be (th) the parameter vector

z

Data matrix, n 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

SGD_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. To turn off, set to 0. Default:500

Author

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. 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(zi, th) {
    xi     <- zi[1]
    yi     <- zi[2]
    h_zith <- c(yi - th[1] - th[2] * xi, xi*(yi - th[1] - th[2] * xi))
    matrix(h_zith, nrow = 2)
}

delthh <- function(zi, th) {
    xi <- zi[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)
SGD_iters <- 50
epsil     <- 10^-5
rho       <- 0.9

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