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Second day at the Indo-French Centre for Applied Mathematics and the workshop. Maybe not the most exciting day in terms of talks (as I missed the first two plenary sessions by (a) oversleeping and (b) running across the campus!). However I had a neat talk with another conference participant that led to [what I think are] interesting questions… (And a very good meal in a local restaurant as the guest house had not booked me for dinner!)

To wit: given a target like

the simulation of λ can be demarginalised into the simulation of

where z is a latent (and artificial) variable. This means a Gibbs sampler simulating λ given z and z given λ can produce an outcome from the target (*). Interestingly, another completion is to consider that the z_{i}‘s are U(0,y_{i}) and to see the quantity

as an unbiased estimator of the target. What’s quite intriguing is that the quantity remains the same but with different motivations: (a) demarginalisation versus unbiasedness and (b) z_{i} Exp(λ) versus z_{i} U(0,y_{i}). The stationary is the same, as shown by the graph below, the core distributions are [formally] the same, … but the reasoning deeply differs.

Obviously, since unbiased estimators of the likelihood can be justified by auxiliary variable arguments, this is not in fine a big surprise. Still, I had not though of the analogy between demarginalisation and unbiased likelihood estimation previously.Here are the R procedures if you are interested:

n=29
y=rexp(n)
T=10^5
#MCMC.1
lam=rep(1,T)
z=runif(n)*y
for (t in 1:T){
lam[t]=rgamma(1,shap=2,rate=1+sum(z))
z=-log(1-runif(n)*(1-exp(-lam[t]*y)))/lam[t]
}
#MCMC.2
fam=rep(1,T)
z=runif(n)*y
for (t in 1:T){
fam[t]=rgamma(1,shap=2,rate=1+sum(z))
z=runif(n)*y
}