Archive for non-central chi-square distribution

optimal Gaussian zorbing

Posted in Books, Kids, R, Statistics with tags , , , , , , on August 30, 2022 by xi'an

A zorbing puzzle from the Riddler: cover the plane with four non-intersecting disks of radius one towards getting the highest probability (under the standard bivariate Normal distribution).

As I could not see a simple connection between the disks and the standard Normal, beyond the probability of a disk being given by a non-central chi-square cdf (with two degrees of freedom), I (once again) tried a random search by simulated annealing, which ended up with a configuration like the above, never above 0.777 using a pedestrian R code like

for(t in 1:1e6){# move the disk centres
 Ap=A+vemp*rnorm(2)
 Bp=B+vemp*rnorm(2)
 while(dist(rbind(Ap,Bp))<2)Bp=B+vemp*rnorm(2)
 Cp=C+vemp*rnorm(2)
 while(min(dist(rbind(Ap,Bp,Cp)))<2)Cp=C+vemp*rnorm(2)
 Dp=D+vemp*rnorm(2)
 while(min(dist(rbind(Ap,Bp,Cp,Dp)))<2)Dp=D+vemp*rnorm(2)
 #coverage probability
 pp=pchisq(1,df=2,ncp=Ap%*%Ap)+pchisq(1,df=2,ncp=Bp%*%Bp)+
    pchisq(1,df=2,ncp=Cp%*%Cp)+pchisq(1,df=2,ncp=Dp%*%Dp)
 #simulated annealing step
 if(log(runif(1))<(pp-p)/sqrt(temp)){
   A=Bp;B=Cp;C=Dp;D=Ap;p=pp
   if (sol$val<p) sol=list(val=pp,pos=rbind(A,B,C,D))}
 temp=temp*.9999}

I also tried a simpler configuration where all disk centres were equidistant from a reference centre, but this led to a lower “optimal” probability. I was looking forward the discussion of the puzzle, to discover if anything less brute-force was possible! But there was no deeper argument there beyond the elimination of other “natural” configurations (and missing the non-central χ² connection!). Among these options, having two disks tangent at (0,0) were optimal. But the illustration was much nicer:

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arXiv recent postings

Posted in Statistics, University life with tags , , , , , , , on June 14, 2013 by xi'an

As I glanced thru the recent arXiv postings in statistics, I found an overload of potential goodies, many more than I could handle in a reasonable time (unless I give up the NYT altogether!)…

For instance, Paulo Marques—a regular ‘Og commenter—wrote about an extension of Chib’s formula when the posterior is approximated in a non-parametric manner. (This was an idea I had toyed with at a time without pursuing it anywhere…)   I wonder at the stability of the approximation for two reasons: (i) Chib’s or Bayes’ formula does not require an expectation as the ratio is constant in θ; averaging over realisations of θ could have negative effects on the stability of the approximation (or not); (ii) using a non-parametric estimate will see the performances of Chib’s approximation plummet as the dimension of θ increases, most likely.

Another post is by Nogales, Pérez, and Monfort and deals with a Monte Carlo formula for conditional expectations that leaves me quite puzzled… More than excited about a potential application in ABC. Indeed, the idea is to replace the conditioning on the (hard) equality with a soft ball around the equality with a tolerance level ε. Since the authors do not seem particularly interested in ABC, I do no see the point in this technique. The first example they present does not account for the increasing computational effort as ε decreases. The second part of the paper deals with the best invariant estimator of the position parameter of a general half-normal distribution, using two conditional expectations in Proposition 2. I wonder how the method compares with bridge sampling and MCMC methods. As the paper seems to have gone beyond the Monte Carlo issue at this stage. And focus on this best invariant estimation…

Then Feroz, Hobson,  Cameron and Pettitt posted a work on importance nested sampling, on which I need to spend more time given the connection to our nested sampling paper with Nicolas. The paper builds on the Multinest software developped by Feroz, Hobson and co-authors. (Whose above picture is borrowed from.)

A side post on the moments of the partial non-central chi-square distribution function by Gill, Segura and Temme, since I always liked the specifics of this distribution… No comments so far!

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