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After reading the arXiv paper by Korattikara, Chen and Welling, I wondered about the expression of the acceptance step of the Metropolis-Hastings algorithm as a mean of log-likelihoods over the sample. More specifically the long sleepless nights at the hospital led me to ponder the rather silly question of the impact of replacing mean by median. I thus tried running a Metropolis-Hastings algorithm with the substitute and it (of course!) let to a nonsensical answer, as shown by the above graph. The true posterior is the one for a normal model and the histogram indicates a lack of convergence of the Markov chain to this posterior even though it does converge to some posterior. Here is the R code for this tiny experiment:

#data generation
N=100
x=rnorm(N)
#HM steps
T=10^5
theta=rep(0,T)
curlike=dnorm(x,log=TRUE)
for (t in 2:T){
prop=theta[t-1]+.1*rnorm(1)
proplike=dnorm(x,mean=prop,log=TRUE)
u=runif(1)
bound=log(u)-dnorm(prop,sd=10,log=TRUE)+
dnorm(theta[t-1],sd=10,log=TRUE)
if (median(proplike)-median(curlike)>bound/N){
theta[t]=prop;curlike=proplike
} else { theta[t]=theta[t-1]}
}