expected.deviance {abc} | R Documentation |
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)
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 |
subset |
a logical expression indicating elements or rows to
keep. Missing values in |
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).
Value
A list with the following components:
expected.deviance |
The approximate expected deviance. |
dist |
The Euclidean distances for summary statistics simulated a posteriori. |
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