Variational Bayes for fast and accurate empirical likelihood inference

This package allows you to run GVA on a data set in R and C++ for faster computation (for 10,000 iterations of GVA: 40.23s for partial R and cpp and 28.5s for purely cpp computation).

This package also allows you to run AEL on a data set in R and C++ for faster computation (for 500 iterations of AEL: 0.2s for purely cpp and 0.1s for R and cpp with pre-z calculation).

To install this package in Mac requires a Fortran compiler (through
its RcppEigen dependency). Chances are, your current Fortran compiler is
not up-to-date. To update your Fortran compiler, simply follow the steps
here:

- In your Mac App Store, search “Xcode” and install.
- Open Terminal application. Type in

`xcode-select --install`

and follow the instructions.

3. Click on the link here.
Download the gfortan dmg file according to your MacOS version.

4. Open the dmg file, run the gfortran installer, follow all the
instructions.

An alternative recommended method is to use the packet manager Homebrew:

1. Check if you have homebrew with

`$ brew doctor`

If you don’t have it installed, use the following code from the Homebrew webiste. Check the website that it hasn’t changed since. It will ask for your user password (you won’t see characters as you type). Follow the instructions.

`/bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/HEAD/install.sh)"`

2. Install GFortran using gcc (contains GFortran).

`brew install gcc`

```
# Install the development version from GitHub:
install.packages("devtools")
devtools::install_github("jlimrasc/VBel")
```

```
library(VBel)
# 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
# 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)
}
# Define h-gradient function
delthh <- function(z, th) {
xi <- z[1]
matrix(c(-1, -xi, -xi, -xi^2), 2, 2)
}
# Set initial values for GVA computation
n <- 31 # Number of rows in z
reslm <- lm(y ~ x)
mu <- matrix(unname(reslm$coefficients),2,1)
C_0 <- unname(t(chol(vcov(reslm))))
rho <- 0.9
# Set other variables for GVA
delth_logpi <- function(theta) {-0.0001 * mu}
elip <- 10^-5
T <- 10 # Number of iterations for GVA
T2 <- 500 # Number of iterations for AEL
# Excecute functions
ansAELRcpp <- compute_AEL(th, h, lam0, a, z, T2)
ansGVARcppPure <-compute_GVA(mu, C_0, h, delthh, delth_logpi, z, lam0, rho, elip, a, T, T2)
diagnostic_plot(ansGVARcppPure) # Plot the results to check for convergence
```