Empirical Likelihood-based Approximate Bayesian Computation

Description

Perform empirical likelihood-based posterior approximation for ABC problems.

Usage

abcel(
  data.obs,
  theta0,
  rep,
  m,
  n,
  burn_in,
  data.func,
  summ.func,
  prior.func,
  proposal.func = NULL,
  transform.func = NULL,
  fixed.summ.num = TRUE,
  print_interval = 1000,
  plot_interval = 0,
  which_plot = NULL,
  k = NULL,
  ...
)

Arguments

Details

By default, the sampler performs an adaptive random walk on the parameters with N(0,initial.cov) as its proposal density for the first initial_interval steps. Subsequently, the variance of the proposal Normal density will be updated based on the previously sampled parameters.

When defining a new proposal or a new transformation, be sure to insert the ellipsis argument.

Value

A list with the following components:

size: Size of the data.

logpost: The logarithm of the posterior numerator.

theta: A matrix of sampled theta values.

differential_entropy: A vector of computed differential entropies.

acceptance_rate: Stepwise acceptance rates.

References

Chaudhuri, S., Ghosh, S., & Pham, K. C. (2020). On an Empirical Likelihood-Based Solution to the Approximate Bayesian Computation Problem. Statistical Analysis and Data Mining, Vol 17(5):e11711.

Examples

## Not run: 
 data.gk1<-function(theta,n=1000){
   p<-runif(n,min=0,max=1)
   h<-(1-exp(-theta[3]*qnorm(p)))/(1+exp(-theta[3]*qnorm(p)))
   x<-theta[1]+theta[2]*(1+0.8*h)*(1+qnorm(p)^2)^theta[4]*qnorm(p)
   return(x)
 }
 summary.gk1<-function(obs){
   summary=c(mean(obs),quantile(obs,p=c(.25,.5,.75)))
   return(summary)
 }
 prior.gk1<-function(theta){
   pr=c(dunif(theta[1],min=0,max=10,log=TRUE),
        dunif(theta[2],min=0,max=10,log=TRUE),
        dunif(theta[3],min=0,max=10,log=TRUE),
        dunif(theta[4],min=0,max=10,log=TRUE)
   )
   return(pr)
 }
 # run the algorithm
 # parameters
 rep<-100
 m<-40
 n<-1000
 ## target theta and the observed data
 theta.t<-c(3,1,2,.5)
 pr.t <- prior.gk1(theta.t)
 d=length(theta.t)
 data.obs<-data.gk1(theta.t,n=n)
 initial.cov=diag(d)
 diag(initial.cov)=c(1,1,1,1)*10^(-7)
 # initialize mean and variance for the initial theta
 marginal.mean.A<-3.003707
 marginal.mean.B<-1.012046
 marginal.mean.g<-2.017939
 marginal.mean.k<-0.4894453
 mean0<-matrix(0,4,1)
 mean0[1,]<-marginal.mean.A
 mean0[2,]<-marginal.mean.B
 mean0[3,]<-marginal.mean.g
 mean0[4,]<-marginal.mean.k
 marginal.var.A<-0.0002019947
 marginal.var.B<-0.0009072782
 marginal.var.g<-0.004954367
 marginal.var.k<-0.001020683
 initial.cov<-matrix(0,4,4)
 initial.cov[1,1]<-marginal.var.A
 initial.cov[2,2]<-marginal.var.B
 initial.cov[3,3]<-marginal.var.g
 initial.cov[4,4]<-marginal.var.k
 theta0<-MASS::mvrnorm(1,mean0,initial.cov/100)
 abcel <- function(data.obs=data.obs,
                   theta0=theta0,
                   rep=rep,
                   m=m,
                   n=n,
                   burn_in=burn_in,
                   data.func=data.gk1,
                   summ.func=summary.gk1,
                   prior.func=prior.gk1,
                   print_interval=1000,
                   plot_interval=0,
                   which_plot=NULL)

## End(Not run)

New Entropy Estimator

Description

Entropy estimator based on k-nearest neighbors.

Usage

entEst(summaries, k = NULL, terms = NULL, wts = NULL)

Arguments

Value

Estimated differential entropy.

Examples

x=matrix(rnorm(2500),nrow=100)
entropy.true=.5*25*log(2*pi*exp(1))
entropy=entEst(t(x))