Archive for the Statistics Category

the unbounded likelihood problem

Posted in Books, Statistics, Travel, University life on April 2, 2015 by xi'an

Following my maths of the Lindley-Jeffreys paradox post, Javier (from Warwick) pointed out a recent American Statistician paper by Liu, Wu and Meeker about Understanding and addressing the unbounded likelihood problem. (I remember meeting some of the authors when visiting Ames three years ago.) As often when reading articles in The American Statistician, I easily find reasons do disagree with the authors. Here are some.

“Fisher (1912) suggest that a likelihood defined by a product of densities should be proportional to the probability of the data.”

First, I fail to understand why an unbounded likelihood is an issue. (I also fail to understand the above quote: in a continuous setting, there is no such thing as the probability of the data. Only its density.) Especially when avoiding maximum likelihood estimation. The paper is quite vague as to why this is a statistical problem. They take as one category discrete mixture models. While the likelihood explodes around each observation (in the mean direction) this does not prevent the existence of convergent solutions to the likelihood equations. Or of Bayes estimators. Nested sampling itself manages this difficulty.

Second, I deeply dislike the baseline that everything is discrete or even finite, including measurement and hence continuous densities should be replaced with probabilities, called correct likelihood in the paper. Of course, using probabilities removes any danger of hitting an infinite likelihood. But it also introduces many layers of arbitrary calibration, incl. the scale of the discretisation. Like, I do not think there is any stability of the solution when the discretisation range Δ goes to zero, if the limiting theorem of the authors holds. But they do not seem to see this as an issue. I think it would make more sense to treat Δ as another parameter.

As an aside, I also find surprising the classification of the unbounded likelihood models in three categories, one being those “with three or four parameters, including a threshold parameter”. Why on Earth 3 or 4?! As if it was not possible to find infinite likelihoods with more than four parameters…

Le Monde puzzle [#905]

Posted in Books, Kids, R, Statistics, University life with tags , , , on April 1, 2015 by xi'an

A recursive programming  Le Monde mathematical puzzle:

Given n tokens with 10≤n≤25, Alice and Bob play the following game: the first player draws an integer1≤m≤6 at random. This player can then take 1≤r≤min(2m,n) tokens. The next player is then free to take 1≤s≤min(2r,n-r) tokens. The player taking the last tokens is the winner. There is a winning strategy for Alice if she starts with m=3 and if Bob starts with m=2. Deduce the value of n.

Although I first wrote a brute force version of the following code, a moderate amount of thinking leads to conclude that the person given n remaining token and an adversary choice of m tokens such that 2m≥n always win by taking the n remaining tokens:

optim=function(n,m){

 outcome=(n<2*m+1)
 if (n>2*m){
   for (i in 1:(2*m))
     outcome=max(outcome,1-optim(n-i,i))
   }
 return(outcome)
}

eliminating solutions which dividers are not solutions themselves:

sol=lowa=plura[plura<100]
for (i in 3:6){
 sli=plura[(plura>10^(i-1))&(plura<10^i)]
 ace=sli-10^(i-1)*(sli%/%10^(i-1))
 lowa=sli[apply(outer(ace,lowa,FUN="=="),
                1,max)==1]
 lowa=sort(unique(lowa))
 sol=c(sol,lowa)}

which leads to the output

> subs=rep(0,16)
> for (n in 10:25) subs[n-9]=optim(n,3)
> for (n in 10:25) if (subs[n-9]==1) subs[n-9]=1-optim(n,2)
> subs
 [1] 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
> (10:25)[subs==1]
[1] 18

Ergo, the number of tokens is 18!

MCMskv, Lenzerheide, Jan. 5-7, 2016

Posted in Mountains, Kids, Statistics, University life, Travel, pictures, R with tags , , , , , , , , , , , , , , , , on March 31, 2015 by xi'an

moonriseFollowing the highly successful [authorised opinion!, from objective sources] MCMski IV, in Chamonix last year, the BayesComp section of ISBA has decided in favour of a two-year period, which means the great item of news that next year we will meet again for MCMski V [or MCMskv for short], this time on the snowy slopes of the Swiss town of Lenzerheide, south of Zürich. The committees are headed by the indefatigable Antonietta Mira and Mark Girolami. The plenary speakers have already been contacted and Steve Scott (Google), Steve Fienberg (CMU), David Dunson (Duke), Krys Latuszynski (Warwick), and Tony Lelièvre (Mines, Paris), have agreed to talk. Similarly, the nine invited sessions have been selected and will include Hamiltonian Monte Carlo,  Algorithms for Intractable Problems (ABC included!), Theory of (Ultra)High-Dimensional Bayesian Computation, Bayesian NonParametrics, Bayesian Econometrics,  Quasi Monte Carlo, Statistics of Deep Learning, Uncertainty Quantification in Mathematical Models, and Biostatistics. There will be afternoon tutorials, including a practical session from the Stan team, tutorials for which call is open, poster sessions, a conference dinner at which we will be entertained by the unstoppable Imposteriors. The Richard Tweedie ski race is back as well, with a pair of Blossom skis for the winner!

As in Chamonix, there will be parallel sessions and hence the scientific committee has issued a call for proposals to organise contributed sessions, tutorials and the presentation of posters on particularly timely and exciting areas of research relevant and of current interest to Bayesian Computation. All proposals should be sent to Mark Girolami directly by May the 4th (be with him!).

intuition beyond a Beta property

Posted in Books, Kids, R, Statistics, University life with tags , , , on March 30, 2015 by xi'an

betas

A self-study question on X validated exposed an interesting property of the Beta distribution:

If x is B(n,m) and y is B(n+½,m) then √xy is B(2n,2m)

While this can presumably be established by a mere change of variables, I could not carry the derivation till the end and used instead the moment generating function E[(XY)s/2] since it naturally leads to ratios of B(a,b) functions and to nice cancellations thanks to the ½ in some Gamma functions [and this was the solution proposed on X validated]. However, I wonder at a more fundamental derivation of the property that would stem from a statistical reasoning… Trying with the ratio of Gamma random variables did not work. And the connection with order statistics does not apply because of the ½. Any idea?

off to New York

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , on March 29, 2015 by xi'an

I am off to New York City for two days, giving a seminar at Columbia tomorrow and visiting Andrew Gelman there. My talk will be about testing as mixture estimation, with slides similar to the Nice ones below if slightly upgraded and augmented during the flight to JFK. Looking at the past seminar speakers, I noticed we were three speakers from Paris in the last fortnight, with Ismael Castillo and Paul Doukhan (in the Applied Probability seminar) preceding me. Is there a significant bias there?!

likelihood-free model choice

Posted in Books, pictures, Statistics, University life, Wines with tags , , , , , , , on March 27, 2015 by xi'an

Jean-Michel Marin, Pierre Pudlo and I just arXived a short review on ABC model choice, first version of a chapter for the incoming Handbook of Approximate Bayesian computation edited by Scott Sisson, Yannan Fan, and Mark Beaumont. Except for a new analysis of a Human evolution scenario, this survey mostly argues for the proposal made in our recent paper on the use of random forests and [also argues] about the lack of reliable approximations to posterior probabilities. (Paper that was rejected by PNAS and that is about to be resubmitted. Hopefully with a more positive outcome.) The conclusion of the survey is  that

The presumably most pessimistic conclusion of this study is that the connections between (i) the true posterior probability of a model, (ii) the ABC version of this probability, and (iii) the random forest version of the above, are at best very loose. This leaves open queries for acceptable approximations of (i), since the posterior predictive error is instead an error assessment for the ABC RF model choice procedure. While a Bayesian quantity that can be computed at little extra cost, it does not necessarily compete with the posterior probability of a model.

reflecting my hope that we can eventually come up with a proper approximation to the “true” posterior probability…

importance weighting without importance weights [ABC for bandits?!]

Posted in Books, Statistics, University life with tags , , , , on March 27, 2015 by xi'an

I did not read very far in the recent arXival by Neu and Bartók, but I got the impression that it was a version of ABC for bandit problems where the probabilities behind the bandit arms are not available but can be generated. Since the stopping rule found in the “Recurrence weighting for multi-armed bandits” is the generation of an arm equal to the learner’s draw (p.5). Since there is no tolerance there, the method is exact (“unbiased”). As no reference is made to the ABC literature, this may be after all a mere analogy…

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