abc (version 2.2.2)

expected.deviance: Expected deviance

Description

Model selection criterion based on posterior predictive distributions and approximations of the expected deviance.

Usage

expected.deviance(target, postsumstat, kernel = "gaussian", subset=NULL,
print=TRUE)

Value

A list with the following components:

expected.deviance

The approximate expected deviance.

dist

The Euclidean distances for summary statistics simulated a posteriori.

Arguments

target

a vector of the observed summary statistics.

postsumstat

a vector, matrix or data frame of summary statistics simulated a posteriori.

kernel

a character string specifying the kernel to be used when. Defaults to "gaussian". See density for details.

subset

a logical expression indicating elements or rows to keep. Missing values in postsumstat are taken as FALSE.

print

prints out what percent of the distances have been zero.

Details

This function implements an approximation for the expected deviance based on simulation performed a posteriori. Thus, after the posterior distribution of parameters or the posterior model probabilities have been determined, users need to re-simulate data using the posterior. The Monte-Carlo estimate of the expected deviance is computed from the simulated data as follows: \(D=-\frac{2}{n}\sum_{j=1}^{n}\log(K_\epsilon(\parallel s^j-s_0\parallel))\), where n is number of simulations, \(K\) is the statistical kernel, \(\epsilon\) is the error, i.e. difference between the observed and simulated summary statistics below which simualtions were accepted in the original call to postpr, the \(s^j\)'s are the summary statistics obtained from the posterior predictive simualtions, and \(s_0\) are the observed values of the summary statistics. The expected devaince averaged over the posterior distribution to compute a deviance information criterion (DIC).

References

Francois O, Laval G (2011) Deviance information criteria for model selection in approximate Bayesian computation arXiv:0240377.

Examples

## Function definitions
skewness <- function(x) {
sk <- mean((x-mean(x))^3)/(sd(x)^3)
return(sk)
}
kurtosis <- function(x) {  
k <- mean((x-mean(x))^4)/(sd(x)^4) - 3  
return(k)
}

## Observed summary statistics
obs.sumstat <- c(2.004821, 3.110915, -0.7831861, 0.1440266) 

## Model 1 (Gaussian)
## ##################
## Simulate data
theta <- rnorm(10000, 2, 10)
zeta <- 1/rexp(10000, 1)
param <- cbind(theta, zeta)
y <- matrix(rnorm(200000, rep(theta, each = 20), sd = rep(sqrt(zeta),
each = 20)), nrow = 20, ncol = 10000)

## Calculate summary statistics
s <- cbind(apply(y, 2, mean), apply(y, 2, sd), apply(y, 2, skewness),
apply(y, 2, kurtosis))

## ABC inference
gaus <- abc(target=obs.sumstat, param = param, sumstat=s, tol=.1, hcorr =
FALSE, method = "loclinear")
param.post <- gaus$adj.values

## Posterior predictive simulations
postpred.gaus <- matrix(rnorm(20000, rep(param.post[,1], each = 20), sd
= rep(sqrt(param.post[,2]), each = 20)), nrow = 20, ncol = 1000)
statpost.gaus <- cbind(apply(postpred.gaus, 2,
mean),apply(postpred.gaus, 2, sd),apply(postpred.gaus,
2,skewness),apply(postpred.gaus, 2,kurtosis))

# Computation of the expected deviance
expected.deviance(obs.sumstat, statpost.gaus)$expected.deviance
expected.deviance(obs.sumstat, statpost.gaus, kernel =
"epanechnikov")$expected.deviance

## Modele 2 (Laplace)
## ##################
## Simulate data
zeta <- rexp(10000)
param <- cbind(theta, zeta)
y <- matrix(theta + sample(c(-1,1),200000, replace = TRUE)*rexp(200000,
rep(zeta, each = 20)), nrow = 20, ncol = 10000)

## Calculate summary statistics
s <- cbind( apply(y, 2, mean), apply(y, 2, sd), apply(y, 2, skewness),
apply(y, 2, kurtosis))

## ABC inference
lapl <- abc(target=obs.sumstat, param = param, sumstat=s, tol=.1, hcorr =
FALSE, method = "loclinear")
param.post <- lapl$adj.values

## Posterior predictive simulations
postpred.lapl <- matrix(param.post[,1] + sample(c(-1,1),20000, replace =
TRUE)*rexp(20000, rep(param.post[,2], each = 20)), nrow = 20, ncol =
1000)
statpost.lapl <- cbind(apply(postpred.lapl, 2,
mean),apply(postpred.lapl, 2, sd),apply(postpred.lapl,
2,skewness),apply(postpred.lapl, 2,kurtosis))

## Computation of the expected deviance
expected.deviance(obs.sumstat, statpost.lapl)$expected.deviance
expected.deviance(obs.sumstat, statpost.lapl, kernel =
"epanechnikov")$expected.deviance