compute_GVA {VBel}

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(
  mu,
  C,
  h,
  delthh,
  delth_logpi,
  z,
  lam0,
  rho,
  elip,
  a,
  iters,
  iters2,
  fullCpp,
  verbosity
)

Arguments

mu

Column vector, initial value of Gaussian VB mean

C

Lower Triangular Matrix, initial value of Gaussian VB Cholesky

h

User-defined moment-condition function, outputs a k x 1 matrix containing the kth row of h. Function must take two arguments: zi and theta for h(zi,th)

delthh

User defined function, outputs k x p Jacobian matrix of h(zi,th) with respect to theta

delth_logpi

User-defined function, outputs p x 1 matrix, derivative of log prior function

z

Data matrix, n-1 x d matrix

lam0

Initial guess for lambda, k x 1 matrix

rho

Scalar (between 0 to 1, reccomended to be close to 1) ADADELTA accumulation constant

elip

Scalar numeric stability constant. Should be specified with a small value

a

Scalar AEL constant, must be >0, small values recommended

iters

Number of iterations for GVA (default:10,000)

iters2

Number of iterations for Log AEL (default:500)

fullCpp

Bool whether to calculate the main section in cpp (TRUE) or only partially (FALSE, doing all the AEL calculations in R before handing values to cpp) (default: TRUE)

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 (iters + 1)

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

Author(s)

Wei Chang 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. doi:10.1080/01621459.2023.2169701

Examples

set.seed(1)
x    <- runif(30, min = -5, max = 5)
elip <- rnorm(30, mean = 0, sd = 1)
y    <- 0.75 - x + elip
lam0 <- matrix(c(0,0), nrow = 2)
th   <- matrix(c(0.8277, -1.0050), nrow = 2)
a    <- 0.00001
z    <- cbind(x, y)
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)
}

n     <- 31
reslm <- lm(y ~ x)
mu    <- matrix(unname(reslm$coefficients),2,1)
C_0   <- unname(t(chol(vcov(reslm))))

delth_logpi <- function(theta) { -0.0001 * mu }
elip        <- 10^-5
iters       <- 10
iters2      <- 50
rho         <- 0.9

# -----------------------------
# Main
# -----------------------------
ansGVARcppHalf <-compute_GVA(mu, C_0, h, delthh, delth_logpi, z, lam0, 
rho, elip, a, iters, iters2, fullCpp = FALSE)
ansGVARcppPure <-compute_GVA(mu, C_0, h, delthh, delth_logpi, z, lam0, 
rho, elip, a, iters, iters2, fullCpp = TRUE)