Archive for simulation

random wake

Posted in Books, Kids, R, Statistics with tags , , , , , , on December 27, 2017 by xi'an

Just too often on X validated, one sees questions displaying a complete ignorance of the basics that makes one purposelessly wonder what is the point of trying to implement advanced methods when missing the necessary background. And just as often, I reacted to the question by wondering out loud about this… In the current case, the question was about debugging an R code for a mixture of two exponential distributions and the random walk Metropolis algorithm that comes with it. Except that the Normal noise was replaced with a Uniform U(0,1) noise, leading to a most obviously transient Markov chain.I eventually corrected the R code, which returned a rather nicely label-switching output. And which did not necessarily help with the comprehension of the fundamentals.

A of A

Posted in Books, Kids, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on November 30, 2017 by xi'an

Next June, at the same time as the ISBA meeting in Edinburgh, which is slowly taking shape, there will be an Analysis of Algorithms (AofA) meeting in Uppsala (Sweden) with Luc Devroye as the plenary Flajolet Award speaker. The full name of the conference is the 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms. While it is unfortunate the two conferences take place at the same time (and not in the same location), this also provides a continuity of conferences with the following week MCqMC in Rennes and the subsequent week summer school in simulation in Warwick (with Art Owen as the LMS Lecturer).

About our summer school, I want to point out that, thanks to several sponsors, we will be able to provide a consequent number of bursaries for junior researchers. This should be an additional incentive for attendees of the previous week Young Bayesian meeting (BAYSM) to remain the extra days nearby Warwick and attend this fantastic opportunity. Other instructors are Nicolas Chopin, Mark Huber and Jeff Rosenthal!

golden Bayesian!

Posted in Statistics with tags , , , , , , , , , on November 11, 2017 by xi'an

Why is it necessary to sample from the posterior distribution if we already KNOW the posterior distribution?

Posted in Statistics with tags , , , , , , , , on October 27, 2017 by xi'an

I found this question on X validated somewhat hilarious, the more because of the shouted KNOW! And the confused impression that because one can write down π(θ|x) up to a constant, one KNOWS this distribution… It is actually one of the paradoxes of simulation that, from a mathematical perspective, once π(θ|x) is available as a function of (θ,x), all other quantities related with this distribution are mathematically perfectly and uniquely defined. From a numerical perspective, this does not help. Actually, when starting my MCMC course at ENSAE a few days later, I had the same question from a student who thought facing a density function like

f(x) ∞ exp{-||x||²-||x||⁴-||x||⁶}

was enough to immediately produce simulations from this distribution. (I also used this example to show the degeneracy of accept-reject as the dimension d of x increases, using for instance a Gamma proposal on y=||x||. The acceptance probability plunges to zero with d, with 9 acceptances out of 10⁷ for d=20.)

easy riddle

Posted in Books, Kids, R with tags , , , , , on July 12, 2017 by xi'an

From the current Riddler, a problem that only requires a few lines of code and a few seconds of reasoning. Or not.

N households each stole the earnings from one of the (N-1) other households, one at a time. What is the probability that a given household is not burglarised? And what are the expected final earnings of each household in the list, assuming they all start with $1?

The first question is close to Feller’s enveloppe problem in that

\left(1-\frac{1}{N-1}\right)^{N-1}

is close to exp(-1) for N large. The second question can easily be solved by an R code like

N=1e3;M=1e6
fina=rep(1,N)
for (v in 1:M){
 ordre=sample(1:N)
 vole=sample(1:N,N,rep=TRUE)
 while (min(abs(vole-(1:N)))==0)
  vole[abs(vole-(1:N))==0]=sample(1:N,
     sum(vole-(1:N)==0))
 cash=rep(1,N)
 for (t in 1:N){
  cash[ordre[t]]=cash[ordre[t]]+cash[vole[t]];cash[vole[t]]=0}
 fina=fina+cash[ordre]}

which returns a pretty regular exponential-like curve, although I cannot figure the exact curve beyond the third burglary. The published solution gives the curve

{\frac{N-2}{N-1}}^{999}\times 2+{\frac{1}{N-1}}^{t-1}\times{\frac{N-1}{N}}^{N-t}\times\frac{N}{N-1}

corresponding to the probability of never being robbed (and getting on average an extra unit from the robbery) and of being robbed only before robbing someone else (with average wealth N/(N-1)).

Le Monde puzzle [#1013]

Posted in Books, Kids with tags , , , , , on June 23, 2017 by xi'an

A purely arithmetic Le Monde mathematical puzzle:

An operation þ applies to all pairs of natural integers with the properties

0 þ (a+1) = (0 þ a)+1, (a+1) þ (b+1)=(a þ b)+1, 271 þ 287 = 77777, 2018 þ 39 = 2018×39

Find the smallest integer d>287 such that there exists c<d leading to c þ d = c x d, the smallest integer f>2017 such that 2017 þ f = 2017×40. Is there any know integer f such that f þ 2017 = 40×2017?

The major appeal in this puzzle (where no R programming seems to help!) is that the “data” does not completely defines the operation  þ ! Indeed, when a<b, it is straightforward to deduce that a þ b = (0 þ 0)+b, hence solving the first two questions by deriving (0 þ 0)=270×287 [with d=315 and f=2017×40-270×287], but the opposed quantity b þ a is not defined, apart from (2018-39) þ 0. This however brings a resolution since

(2018-39) þ 0 = 2017×39 and (2018-39+2017) þ 2017 = 2017×39+2017 = 2017×40

leading to f=2018-39+2017=3996.

convergence of MCMC

Posted in Statistics with tags , , , , , , , , , on June 16, 2017 by xi'an

Michael Betancourt just posted on arXiv an historical  review piece on the convergence of MCMC, with a physical perspective.

“The success of these of Markov chain Monte Carlo, however, contributed to its own demise.”

The discourse proceeds through augmented [reality!] versions of MCMC algorithms taking advantage of the shape and nature of the target distribution, like Langevin diffusions [which cannot be simulated directly and exactly at the same time] in statistics and molecular dynamics in physics. (Which reminded me of the two parallel threads at the ICMS workshop we had a few years ago.) Merging into hybrid Monte Carlo, morphing into Hamiltonian Monte Carlo under the quills of Radford Neal and David MacKay in the 1990’s. It is a short entry (and so is this post), with some background already well-known to the community, but it nonetheless provides a perspective and references rarely mentioned in statistics.