the Flatland paradox [#2]
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Another trip in the métro today (to work with Pierre Jacob and Lawrence Murray in a Paris Anticafé!, as the University was closed) led me to infer—warning!, this is not the exact distribution!—the distribution of x, namely
since a path x of length l(x) will corresponds to N draws if N-l(x) is an even integer 2p and p undistinguishable annihilations in 4 possible directions have to be distributed over l(x)+1 possible locations, with Feller’s number of distinguishable distributions as a result. With a prior π(N)=1/N on N, hence on p, the posterior on p is given by
Now, given N and x, the probability of no annihilation on the last round is 1 when l(x)=N and in general
which can be integrated against the posterior. The numerical expectation is represented for a range of values of l(x) in the above graph. Interestingly, the posterior probability is constant for l(x) large and equal to 0.8125 under a flat prior over N.
Getting back to Pierre Druilhet’s approach, he sets a flat prior on the length of the path θ and from there derives that the probability of annihilation is about 3/4. However, “the uniform prior on the paths of lengths lower or equal to M” used for this derivation which gives a probability of length l proportional to 3l is quite different from the distribution of l(θ) given a number of draws N. Which as shown above looks much more like a Binomial B(N,1/2).
However, being not quite certain about the reasoning involving Fieller’s trick, I ran an ABC experiment under a flat prior restricted to (l(x),4l(x)) and got the above, where the histogram is for a posterior sample associated with l(x)=195 and the gold curve is the potential posterior. Since ABC is exact in this case (i.e., I only picked N’s for which l(x)=195), ABC is not to blame for the discrepancy! Here is the R code that goes with the ABC implementation:
#observation: elo=195 #ABC version T=1e6 el=rep(NA,T) N=sample(elo:(4*elo),T,rep=TRUE) for (t in 1:T){ #generate a path paz=sample(c(-(1:2),1:2),N[t],rep=TRUE) #eliminate U-turns uturn=paz[-N[t]]==-paz[-1] while (sum(uturn>0)){ uturn[-1]=uturn[-1]*(1- uturn[-(length(paz)-1)]) uturn=c((1:(length(paz)-1))[uturn==1], (2:length(paz))[uturn==1]) paz=paz[-uturn] uturn=paz[-length(paz)]==-paz[-1] } el[t]=length(paz)} #subsample to get exact posterior poster=N[abs(el-elo)==0]
Filed under: Books, Kids, R, Statistics, University life Tagged: ABC, combinatorics, exact ABC, Flatland, improper priors, Larry Wasserman, marginalisation paradoxes, paradox, Pierre Druilhet, random walk, subjective versus objective Bayes, William Feller
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