Package 'VBel'

Title: Variational Bayes for Fast and Accurate Empirical Likelihood Inference
Description: Computes the Gaussian variational approximation of the Bayesian empirical likelihood posterior. This is an implementation of the function found in Yu, W., & Bondell, H. D. (2023) <doi:10.1080/01621459.2023.2169701>.
Authors: Wei-Chang Yu [aut] , Jeremy Lim [cre, aut]
Maintainer: Jeremy Lim <[email protected]>
License: GPL (>= 3)
Version: 1.0.1
Built: 2025-02-25 02:59:39 UTC
Source: https://github.com/jlimrasc/vbel

Help Index

Variational Bayes for Fast and Accurate Empirical Likelihood Inference

Description

Computes the Gaussian variational approximation of the Bayesian empirical likelihood posterior. This is an implementation of the function found in Yu, W., & Bondell, H. D. (2023) <doi:10.1080/01621459.2023.2169701>.

Details

The DESCRIPTION file:

Package: VBel
Type: Package
Title: Variational Bayes for Fast and Accurate Empirical Likelihood Inference
Version: 1.0.1
Date: 2024-05-28
Authors@R: c( person("Wei-Chang", "Yu", , "[email protected]", role = c("aut"), comment = c(ORCID = "0000-0002-0399-3779")), person("Jeremy", "Lim", , "[email protected]", role = c("cre", "aut")) )
Description: Computes the Gaussian variational approximation of the Bayesian empirical likelihood posterior. This is an implementation of the function found in Yu, W., & Bondell, H. D. (2023) <doi:10.1080/01621459.2023.2169701>.
License: GPL (>= 3)
Imports: Rcpp (>= 1.0.12), stats
LinkingTo: Rcpp, RcppEigen
Encoding: UTF-8
Roxygen: list(markdown = TRUE)
RoxygenNote: 7.3.1
URL: https://github.com/jlimrasc/VBel
BugReports: https://github.com/jlimrasc/VBel/issues
Suggests: mvtnorm, testthat (>= 3.0.0)
Config/testthat/edition: 3
Repository: https://jlimrasc.r-universe.dev
RemoteUrl: https://github.com/jlimrasc/vbel
RemoteRef: HEAD
RemoteSha: bf6f67f2952e5e5ce3f58ad4f5f17d3ab5cfc217
Author: Wei-Chang Yu [aut] (<https://orcid.org/0000-0002-0399-3779>), Jeremy Lim [cre, aut]
Maintainer: Jeremy Lim <[email protected]>

Index of help topics: 

VBel-package            Variational Bayes for Fast and Accurate
                        Empirical Likelihood Inference
compute_AEL             Compute the Adjusted Empirical Likelihood
compute_GVA             Compute the Full-Covariance Gaussian VB
                        Empirical Likelihood Posterior
diagnostic_plot         Check the convergence of a data set computed by
                        'compute_GVA'

Author(s)

Wei-Chang Yu [aut] (<https://orcid.org/0000-0002-0399-3779>), Jeremy Lim [cre, aut]

Maintainer: Jeremy Lim <[email protected]>

References

https://www.tandfonline.com/doi/abs/10.1080/01621459.2023.2169701

See Also

compute_AEL() for choice of R and/or C++ computation of AEL

compute_GVA() for choice of R and/or C++ computation of GVA

diagnostic_plot() for verifying convergence of computed GVA data

Examples

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

Compute the Adjusted Empirical Likelihood

Description

Evaluates the AEL for a given data set, moment conditions and parameter values

Usage

compute_AEL(th, h, lam0, a, z, iters, useR_forz, returnH)

Arguments

th

Vector or scalar theta
h

User-defined function, outputs array
lam0

Initial vector for lambda
a

Scalar constant
z

n-1 by d matrix
iters

Number of iterations using Newton-Raphson for estimation of lambda (default: 500)
useR_forz

Bool whether to calculate the function first in R (True) or call the function in C (False) (default: True)
returnH

Whether to return calculated values of h, H matrix and lambda

Value

A numeric value for the Adjusted Empirical Likelihood function computed evaluated at a given theta value

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

# Generate toy variables
set.seed(1)
x     <- runif(30, min = -5, max = 5)
elip  <- rnorm(30, mean = 0, sd = 1)
y     <- 0.75 - x + elip

# Set initial values for AEL computation
lam0  <- matrix(c(0,0), nrow = 2)
th    <- matrix(c(0.8277, -1.0050), nrow = 2)
a     <- 0.00001
iters <- 10

# Define Dataset and h-function
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)
}
ansAELRcpp <- compute_AEL(th, h, lam0, a, z, iters, useR_forz = TRUE)

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)

Check the convergence of a data set computed by compute_GVA

Description

Plots mu and variance in a time series plot to check for convergence of the computed data (i.e. Full-Covariance Gaussian VB Empirical Likelihood Posterior)

Usage

diagnostic_plot(dataList, muList, cList)

Arguments

dataList

Named list of data generated from compute_GVA
muList

Array of indices of mu_arr to plot. (default:all)
cList

Matrix of indices of variance to plot, 2xn matrix, each row is (col,row) of variance matrix

Value

Matrix of variance of C_FC

Examples

# Generate toy variables
seedNum <- 100
set.seed(seedNum)
n       <- 100
p       <- 10
lam0    <- matrix(0, nrow = p)

# Calculate z
mean    <- rep(1, p)
sigStar <- matrix(0.4, p, p) + diag(0.6, p)
z       <- mvtnorm::rmvnorm(n = n-1, mean = mean, sigma = sigStar)

# Calculate intermediate variables
zbar    <- 1/(n-1) * matrix(colSums(z), nrow = p)
sumVal  <- matrix(0, nrow = p, ncol = p)
for (i in 1:p) {
zi      <- matrix(z[i,], nrow = p)
sumVal  <- sumVal + (zi - zbar) %*% matrix(zi - zbar, ncol = p)
}
sigHat  <- 1/(n-2) * sumVal

# Initial values for GVA
mu_0    <- matrix(zbar, p, 1)
C_0     <- 1/sqrt(n) * t(chol(sigHat))

# Define h-function
h       <- function(zi, th) { matrix(zi - th, nrow = 10) }

# Define h-gradient function
delthh  <- function(z, th) { -diag(p) }

# Set other initial values
delth_logpi <- function(theta) {-0.0001 * theta}
elip    <- 10^-5
T       <- 5 # Number of iterations for GVA
T2      <- 5 # Number of iterations for AEL
rho     <- 0.9
a       <- 0.00001

ansGVA <-compute_GVA(mu_0, C_0, h, delthh, delth_logpi, z, lam0, rho, elip, 
a, T, T2, fullCpp = TRUE)

diagnostic_plot(ansGVA)
diagnostic_plot(ansGVA, muList = c(1,4))
diagnostic_plot(ansGVA, cList = matrix(c(1,1, 5,6, 3,3), ncol = 2))