ritorno da Roma

morning run in Roma

ABC for copula estimation

Clara Grazian and Brunero Liseo (di Roma) have just arXived a note on a method merging copulas, ABC, and empirical likelihood. The approach is rather hybrid and thus not completely Bayesian, but this must be seen as a consequence of an ill-posed problem. Indeed, as in many econometric models, the model there is not fully defined: the marginals of iid observations are represented as being from well-known parametric families (and are thus well-estimated by Bayesian tools), while the joint distribution remains uncertain and hence so does the associated copula. The approach in the paper is to proceed stepwise, i.e., to estimate correctly each marginal, well correctly enough to transform the data by an estimated cdf, and then only to estimate the copula or some aspect of it based on this transformed data. Like Spearman’s ρ. For which an empirical likelihood is computed and aggregated to a prior to make a BCel weight. (If this sounds unclear, each BEel evaluation is based on a random draw from the posterior samples, which transfers some uncertainty in the parameter evaluation into the copula domain. Thanks to Brunero and Clara for clarifying this point for me!)

At this stage of the note, there are two illustrations revolving around Spearman’s ρ. One on simulated data, with better performances than a nonparametric frequentist solution. And another one on a Garch (1,1) model for two financial time-series.

I am quite glad to see an application of our BCel approach in another domain although I feel a tiny bit uncertain about the degree of arbitrariness in the approach, from the estimated cdf transforms of the marginals to the choice of the moment equations identifying the parameter of interest like Spearman’s ρ. Especially if one uses a parametric copula which moments are equally well-known. While I see the practical gain in analysing each component separately, the object created by the estimated cdf transforms may have a very different correlation structure from the true cdf transforms. Maybe there exist consistency conditions on the estimated cdfs… Maybe other notions of orthogonality or independence could be brought into the picture to validate further the two-step solution…

ABC in Roma

Back in Roma after my ABC week last year, for the ABC in Rome workshop! The attendance is quite in par with the sizes of the previous audiences and the program is close to my own interests—unsurprisingly since I took part to the scientific committee! Hence talks on papers that have already been discussed on the ‘Og for most of them:

• Dennis Prangle on semi-automatic ABC model choice
• Oliver Ratman on acccurate ABC
• Judith Rousseau on model choice consistency
• Richard Everitt on latent MRFs
• myself (in replacement of Kerrie Mengersen) on (A)BC empirical likelihood
• Gael Martin on unscented Kalman filters for noisy diffusions (on which we worked last summer at Monash)
• Gérard Biau on ABC as knn
• Sarah Filippi on sequential ABC
• Nicolas Chopin on EP-ABC
• Daniel Wegman on speeding up ABC
• Anthony Lee on geometrically ergodic ABC
• Darren Wilkinson on intractable Markov process

(It almost sounds as if I had written the program by myself, but this is not the case, promised!) So from my own personal and egoistic perspective, the poster session was more surprising, with 18 posters ranging from theoretical extensions to applications. I actually wished it had lasted a wee bit longer as I did not have time to listen to all presenters before they vanished to the dinning room upstairs, but I appreciated very much the few exchanges I had. A fully enjoyable meeting then!!! I am definitely looking forward the next edition of ABC in [pick your capital], ABC in Sydney (2014) and ABC in Helsinki (2015) being already in the planning…

Here are my slides, just slightly updated from the previous version:

ABC in Roma [R lab #2]

Here are the R codes of the second R lab organised by Serena Arima in supplement of my lectures (now completed!). This morning I covered ABC model choice and the following example is the benchmark used in the course (and in the paper) about the impact of summary statistics. (Warning! It takes a while to run…)

n.iter=10000
n=c(10,100,1000)
n.sims=100
prob.m1=matrix(0,nrow=n.sims,ncol=length(n))
prob.m2=prob.m1

### Simulation from Normal model
for(sims in 1:n.sims){

## True data generation from the Normal distribution and summary statistics
for(i in 1:length(n)){
 y.true=rnorm(n[i],0,1)
stat.true=c(mean(y.true),median(y.true),var(y.true))

## ABC algorithm
# Sample from the prior
mu=rnorm(n.iter,0,2)
dist.m1=rep(NA,n.iter)
dist.m2=rep(NA,n.iter)

for(j in 1:n.iter){
# Data generation under both models
# Normal model
y.m1=rnorm(n[i],mu[j])
stat.m1=c(mean(y.m1),median(y.m1),var(y.m1))
# computing the distance
dist.m1[j]=sum((stat.m1-stat.true)^2)
# Laplace model
y.m2=mu[j]+sample(c(-1,1),n[i],rep=TRUE)*rexp(n[i],rate=sqrt(2))
stat.m2=c(mean(y.m2),median(y.m2),var(y.m2))
# computing the distance
dist.m2[j]=sum((stat.m2-stat.true)^2)
}

# select epsilon as 1% distance quantile
epsilon=quantile(c(dist.m1,dist.m2),prob=0.01)

# compute the posterior probability that data come from
# the model
prob.m1[sims,i]=sum(dist.m1<epsilon)/(2*n.iter*0.01)
}}

### Simulation from the Laplace model
for(sims in 1:n.sims){

## True data generation from the Laplace distribution and summary statistics
 for(i in 1:length(n)){
y.true=sample(c(-1,1),n[i],rep=TRUE)*rexp(n[i],rate=sqrt(2))
stat.true=c(mean(y.true),median(y.true),var(y.true))

## ABC algorithm
# Sample from the prior
mu=rnorm(n.iter,0,2)
dist.m1=rep(NA,n.iter)
dist.m2=rep(NA,n.iter)

for(j in 1:n.iter){
# Data generation under both models
# Normal model
y.m1=rnorm(n[i],mu[j])
stat.m1=c(mean(y.m1),median(y.m1),var(y.m1))
# computing the distance
dist.m1[j]=sum((stat.m1-stat.true)^2)
# Laplace model
y.m2=mu[j]+sample(c(-1,1),n[i],rep=TRUE)*rexp(n[i],rate=sqrt(2))
stat.m2=c(mean(y.m2),median(y.m2),var(y.m2))
# computing the distance
dist.m2[j]=sum((stat.m2-stat.true)^2)
}

# select epsilon as 1% distance quantile
epsilon=quantile(c(dist.m1,dist.m2),prob=0.01)

# compute the posterior probability that data come from
# the model
prob.m2[sims,i]=sum(dist.m2<epsilon)/(2*n.iter*0.01)
}
}

# Visualize the results
y.true=sample(c(-1,1),n[i],rep=TRUE)*rexp(n[i],rate=sqrt(2))
stat.true=c(mean(y.true),median(y.true),var(y.true))

## ABC algorithm
# Sample from the prior
mu=rnorm(n.iter,0,2)
dist.m1=rep(NA,n.iter)
dist.m2=rep(NA,n.iter)

for(j in 1:n.iter){
# Data generation under both models
# Normal model
y.m1=rnorm(n[i],mu[j])
stat.m1=c(mean(y.m1),median(y.m1),var(y.m1))
# computing the distance
dist.m1[j]=sum((stat.m1-stat.true)^2)
# Laplace model
y.m2=mu[j]+sample(c(-1,1),n[i],rep=TRUE)*rexp(n[i],rate=sqrt(2))
stat.m2=c(mean(y.m2),median(y.m2),var(y.m2))
# computing the distance
dist.m2[j]=sum((stat.m2-stat.true)^2)
}

# select epsilon as 1% distance quantile
epsilon=quantile(c(dist.m1,dist.m2),prob=0.01)

# compute the posterior probability that data come from
# the model
prob.m2[sims,i]=sum(dist.m2<epsilon)/(2*n.iter*0.01)
}
}

# Visualize the results
y.true=sample(c(-1,1),n[i],rep=TRUE)*rexp(n[i],rate=sqrt(2))
stat.true=c(mean(y.true),median(y.true),var(y.true))

## ABC algorithm
# Sample from the prior
mu=rnorm(n.iter,0,2)
dist.m1=rep(NA,n.iter)
dist.m2=rep(NA,n.iter)

for(j in 1:n.iter){
# Data generation under both models
# Normal model
y.m1=rnorm(n[i],mu[j])
stat.m1=c(mean(y.m1),median(y.m1),var(y.m1))
# computing the distance
dist.m1[j]=sum((stat.m1-stat.true)^2)
# Laplace model
y.m2=mu[j]+sample(c(-1,1),n[i],rep=TRUE)*rexp(n[i],rate=sqrt(2))
stat.m2=c(mean(y.m2),median(y.m2),var(y.m2))
# computing the distance
dist.m2[j]=sum((stat.m2-stat.true)^2)
}

# select epsilon as 1% distance quantile
epsilon=quantile(c(dist.m1,dist.m2),prob=0.01)

# compute the posterior probability that data come from
# the model
prob.m2[sims,i]=sum(dist.m2<epsilon)/(2*n.iter*0.01)
}}

# Visualize the results
boxplot(data.frame(prob.m1[,1],prob.m2[,1]),
names=c("M1","M2"),main="N=10")
boxplot(data.frame(prob.m1[,2],prob.m2[,2]),
names=c("M1","M2"),main="N=100")
boxplot(data.frame(prob.m1[,3],prob.m2[,3]),
names=c("M1","M2"),main="N=1000")

Once again, I had a terrific time teaching this “ABC in Roma” course, and not only for the immense side benefit of enjoy Roma in a particularly pleasant weather (for late February).