Archive for quantum mechanics

Maximum likelihood vs. likelihood-free quantum system identification in the atom maser

Posted in Books, Statistics, University life with tags , , , , , , on December 2, 2013 by xi'an

This paper (arXived a few days ago) compares maximum likelihood with different ABC approximations in a quantum physic setting and for an atom maser modelling that essentially bears down to a hidden Markov model. (I mostly blanked out of the physics explanations so cannot say I understand the model at all.) While the authors (from the University of Nottingham, hence Robin’s statue above…) do not consider the recent corpus of work by Ajay Jasra and coauthors (some of which was discussed on the ‘Og), they get interesting findings for an equally interesting model. First, when comparing the Fisher informations on the sole parameter of the model, the “Rabi angle” φ, for two different sets of statistics, one gets to zero at a certain value of the parameter, while the (fully informative) other is maximum (Figure 6). This is quite intriguing, esp. give the shape of the information in the former case, which reminds me of (my) inverse normal distributions. Second, the authors compare different collections of summary statistics in terms of ABC distributions against the likelihood function. While most bring much more uncertainty in the analysis, the whole collection recovers the range and shape of the likelihood function, which is nice. Third, they also use a kolmogorov-Smirnov distance to run their ABC, which is enticing, except that I cannot fathom from the paper when one would have enough of a sample (conditional on a parameter value) to rely on what is essentially an estimate of the sampling distribution. This seems to contradict the fact that they only use seven summary statistics. Or it may be that the “statistic” of waiting times happens to be a vector, in which case a Kolmogorov-Smirnov distance can indeed be adopted for the distance… The fact that the grouped seven-dimensional summary statistic provides the best ABC fit is somewhat of a surprise when considering the problem enjoys a single parameter.

“However, in practice, it is often difficult to find an s(.) which is sufficient.”

Just a point that irks me in most ABC papers is to find quotes like the above, since in most models, it is easy to show that there cannot be a non-trivial sufficient statistic! As soon as one leaves the exponential family cocoon, one is doomed in this respect!!!

On randomness

Posted in Books, pictures, Statistics, University life with tags , , , , , on February 6, 2011 by xi'an

A while ago, I posted how strangely people seem to be attracted by re- and re-explaining Bayes’ theorem when I see it as a tautological consequence of the definition of conditional probability (and hence of limited interest per se, although with immense consequences for conducting inference). Through the “spam” book mentioned earlier this week, I noticed that the same (or even worse) fatal attraction holds for randomness! (Although I had already posted on the “truly random” generators…) Having access only to one chapter, I read with a sense of growing puzzlement through Tommaso Toffoli’s chapter and came with the following comments, which are nothing but a Saturday afternoon idle thoughts!

Measure theory, and much of the axiomatic apparatus that goes into what is often called the “foundations” of probability, is just about developing more refined accounting techniques for when the outcome space becomes so large (viz., uncountably infinite) that simple minded techniques lead to paradoxes: “If a line consists of points, and a point has no length, how come a line has length?”

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Truly random [again]

Posted in Books, R, Statistics, University life with tags , , , , , , , , on December 10, 2010 by xi'an

“The measurement outputs contain at the 99% confidence level 42 new random bits. This is a much stronger statement than passing or not passing statistical tests, which merely indicate that no obvious non-random patterns are present.” arXiv:0911.3427

As often, I bought La Recherche in the station newsagent for the wrong reason! The cover of the December issue was about “God and Science” and I thought this issue would bring some interesting and deep arguments in connection with my math and realism post. The debate is very short, does not go in any depth. reproduces the Hawking’s quote that started the earlier post, and recycles the same graph about cosmology I used last summer in Vancouver! However, there are alternative interesting entries about probabilistic proof checking in Mathematics and truly random numbers… The first part is on an ACM paper on the PCP theorem by Irit Dinur, but is too terse as is (while the theory behind presumably escapes my abilities!). The second part is about a paper in Nature published by Pironio et al. and arXived as well. It is entitled “Random numbers certified by Bell’s Theorem” and also is one of the laureates of the La Recherche prize this year. I was first annoyed by the French coverage of the paper, mentioning that “a number was random with a probability of 99%” (?!) and that “a sequence of numbers is  perfectly random” (re-?!). The original paper is however stating the same thing, hence stressing the different meaning associated to randomness by those physicists, “the unpredictable character of the outcomes” and “universally-composable security”. The above “probability of randomness” is actually a p-value (associated with the null hypothesis that Bell’s inequality is not violated) that is equal to 0.00077. (So the above quote is somehow paradoxical!) The huge apparatus used to produce those random events is not very efficient: on average, 7 binary random numbers are detected per hour… A far cry from the “truly random” generator produced by Intel!

Ps-As a concidence, Julien Cornebise pointed out to me that there is a supplement in the journal about “Le Savoir du Corps” which is in fact handled by the pharmaceutical company Servier, currently under investigation for its drug Mediator… A very annoying breach of basic journalistic ethics in my opinion!


Posted in Books, University life with tags , , , , , , , , , , , on November 20, 2010 by xi'an

One colleague of mine in Dauphine gave me Anathem to read a few weeks ago. I had seen it in a bookstore once and planned to read it, so this was a perfect opportunity. I read through it slowly at first and then with more and more eagerness as the story built on, spending a fair chunk of the past evenings (and Metro rides) into finishing it. Anathem is a wonderful book, especially for mathematicians, and while it could still qualify as a science-fiction book, it blurs the frontiers between the genres of science-fiction, speculative fiction, documentary writings and epistemology… Just imagine any other sci’fi’ book being reviewed in Nature! Still, the book was awarded the 2009 Locus SF Award. So it has true sci’fi’ characteristics, including Clarke-ian bouts of space opera with a Rama-like vessel popping out of nowhere. But this is not the main feature that makes Anathem so unique and fascinating.

“The Adrakhonic theorem, which stated that the square of a right triangle hypotenuse was equal to the sum of the squares of the other two sides…” (p. 128)

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