## understanding the Hastings algorithm

Posted in Books, Statistics with tags , , , , , on August 26, 2014 by xi'an

David Minh and Paul Minh [who wrote a 2001 Applied Probability Models] have recently arXived a paper on “understanding the Hastings algorithm”. They revert to the form of the acceptance probability suggested by Hastings (1970):

$\rho(x,y) = s(x,y) \left(1+\dfrac{\pi(x) q(y|x)}{\pi(y) q(x|y)}\right)^{-1}$

where s(x,y) is a symmetric function keeping the above between 0 and 1, and q is the proposal. This obviously includes the standard Metropolis-Hastings form of the ratio, as well as Barker’s (1965):

$\rho(x,y) = \left(1+\dfrac{\pi(x) q(y|x)}{\pi(y) q(x|y)}\right)^{-1}$

which is known to be less efficient by accepting less often (see, e.g., Antonietta Mira’s PhD thesis). The authors also consider the alternative

$\rho(x,y) = \min(\pi(y)/ q(y|x),1)\,\min(q(x|y)/\pi(x),1)$

which I had not seen earlier. It is a rather intriguing quantity in that it can be interpreted as (a) a simulation of y from the cutoff target corrected by reweighing the previous x into a simulation from q(x|y); (b) a sequence of two acceptance-rejection steps, each concerned with a correspondence between target and proposal for x or y. There is an obvious caveat in this representation when the target is unnormalised since the ratio may then be arbitrarily small… Yet another alternative could be proposed in this framework, namely the delayed acceptance probability of our paper with Marco and Clara, one special case being

$\rho(x,y) = \min(\pi_1(y)q(x|y)/\pi_1(x) q(y|x),1)\,\min(\pi_2(y)/\pi_1(x),1)$

where

$\pi(x)\propto\pi_1(x)\pi_2(x)$

is an arbitrary decomposition of the target. An interesting remark in the paper is that any Hastings representation can alternatively be written as

$\rho(x,y) = \min(\pi(y)/k(x,y)q(y|x),1)\,\min(k(x,y)q(x|y)/\pi(x),1)$

where k(x,y) is a (positive) symmetric function. Hence every single Metropolis-Hastings is also a delayed acceptance in the sense that it can be interpreted as a two-stage decision.

The second part of the paper considers an extension of the accept-reject algorithm where a value y proposed from a density q(y) is accepted with probability

$\min(\pi(y)/ Mq(y),1)$

and else the current x is repeated, where M is an arbitrary constant (incl. of course the case where it is a proper constant for the original accept-reject algorithm). Curiouser and curiouser, as Alice would say! While I think I have read some similar proposal in the past, I am a wee intrigued at the appear of using only the proposed quantity y to decide about acceptance, since it does not provide the benefit of avoiding generations that are rejected. In this sense, it appears as the opposite of our vanilla Rao-Blackwellisation. (The paper however considers the symmetric version called the independent Markovian minorizing algorithm that only depends on the current x.) In the extension to proposals that depend on the current value x, the authors establish that this Markovian AR is in fine equivalent to the generic Hastings algorithm, hence providing an interpretation of the “mysterious” s(x,y) through a local maximising “constant” M(x,y). A possibly missing section in the paper is the comparison of the alternatives, albeit the authors mention Peskun’s (1973) result that exhibits the Metropolis-Hastings form as the optimum.

## the intelligent-life lottery

Posted in Books, Kids with tags , , , , , , , on August 24, 2014 by xi'an

In a theme connected with one argument in Dawkins’ The God Delusion, The New York Time just published a piece on the 20th anniversary of the debate between Carl Sagan and Ernst Mayr about the likelihood of the apparition of intelligent life. While 20 years ago, there was very little evidence if any of the existence of Earth-like planets, the current estimate is about 40 billions… The argument against the high likelihood of other inhabited planets is that the appearance of life on Earth is an accumulation of unlikely events. This is where the paper goes off-road and into the ditch, in my opinion, as it makes the comparison of the emergence of intelligent (at the level of human) life to be “as likely as if a Powerball winner kept buying tickets and — round after round — hit a bigger jackpot each time”. The later having a very clearly defined probability of occurring. Since “the chance of winning the grand prize is about one in 175 million”. The paper does not tell where the assessment of this probability can be found for the emergence of human life and I very much doubt it can be justified. Given the myriad of different species found throughout the history of evolution on Earth, some of which evolved and many more which vanished, I indeed find it hard to believe that evolution towards higher intelligence is the result of a basically zero probability event. As to conceive that similar levels of intelligence do exist on other planets, it also seems more likely than not that life took on average the same span to appear and to evolve and thus that other inhabited planets are equally missing means to communicate across galaxies. Or that the signals they managed to send earlier than us have yet to reach us. Or Earth a long time after the last form of intelligent life will have vanished…

Posted in Books, Travel with tags , , , , , , , , , , on August 23, 2014 by xi'an

I had planned my summer read long in advance to have an Amazon shipment sent to my friend Natesh out of my Amazon associate slush funds. While in Boston and Maine, I read Richard Dawkins’ The God delusion, the fourth Kelly McCullough’s Fallen Blade novel, Blade reforged, the second Ancient Blades novel, unrelated to the above, A thief in the night, by David Chandler, and also the second Tad Williams’ Bobby Dollar novel, Happy Hour in HellThe God delusion is commented on another post.

Blade reforged is not a major novel, unsurprisingly for a fourth entry, but pleasant nonetheless, especially when reading in the shade of a pavilion on Revere Beach! The characters are mostly the same as previously and it could be that the story has (hopefully) come to an end, with (spoilers!) the evil ruler replaced by the hero’s significant other and his mystical weapons returned to him. A few loose ends and a central sword fight with a more than surprising victory, but a good summer read. Checking on Kelly McCullough’s website, I notice that two more novels are in the making….

Most sadly, David Chandler’s A thief in the night had exactly the same shortcomings as another book  I had previously read and maybe reviewed, even though I cannot trace the review or even remember the title of the book (!), and somewhat those of Tad Williams’ Happy Hour in Hell as well, that is, once again a subterranean adventure in a deserted mythical mega-structure that ends up being not deserted at all and even less plausible. I really had to be stuck on a beach or in an airport lounge to finish it! The points noted about Den of Thieves apply even more forcibly here, that is, very charicaturesque characters and a weak and predictable plot. With the addition of the unbearable underground hidden world… I think I should have re-read my own review before ordering this book.

## the god delusion [statistically speaking]

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , , on August 22, 2014 by xi'an

While in Bangalore, I spotted Richard Dawkins’ The God delusion in the [fantastic if chaotic] campus bookstore and bought the Indian edition for a nominal amount.  I read most of it during my week in Boston. And finished by the lake in Maine. While I agree with most of the points made in Dawkins’ book about the irrationality of religions, and of their overall negative impact on human societies, I found the first part rather boring in that I see little appeal in dissecting so minutely the [infinitely many] incoherences of religious myths and beliefs, as this will likely miss the intended target [i.e., literal believers]. Similarly, the chapter on evolution versus intelligent design made valuable points, albeit I had already seen them before. Nothing wrong with repeating those, in particular that evolution has little to do with chance, but again unlikely to convince the [fundamentalist] masses. Overall, the book mostly focus on the Judeo-Christian-Muslim branch of religions, which may reflect on the author’s own culture and upbringing but also misses the recent attempts of Buddhism to incorporate science into their picture.

“A universe in which we are alone except for other slowly evolved intelligences is a very different universe from one with an original guiding agent whose intelligent design is responsible for its very existence.” (p.85)

What is most interesting in the book (for me) is when Dawkins tries to set the God hypothesis as a scientific hypothesis and to apply scientific methods to validate or invalidate this hypothesis. Even though there is no p-value or quantitative answer at the end. Despite the highly frequent use of “statistical” and “statistically improbable” in the corresponding chapter. What’s even more fascinating is Dawkins’ take at Bayesian arguments! Either because it is associated with a reverent or because it relies on subjective prior assessments, Bayesian statistics does not fit as a proper approach. Funny enough, Dawkins himself relies on subjective prior probabilities when discussing the likelihood of find a planet such as Earth. Now, into the details [with the Devil1] in a rather haphazard order or lack thereof: Continue reading

## on intelligent design…

Posted in Books, Kids, Travel with tags , , , , , , , on August 19, 2014 by xi'an

In connection with Dawkins’ The God delusion, which review is soon to appear on the ‘Og, a poster at an exhibit on evolution in the Harvard Museum of Natural History, which illustrates one of Dawkins’ points on scientific agosticism. Namely, that refusing to take a stand on the logical and philosophical opposition between science and religion(s) is not a scientific position. The last sentence in the poster is thus worse than unnecessary…

## STEM forums

Posted in Books, R, Statistics, University life with tags , , , , , on August 15, 2014 by xi'an

“I can calculate the movement of stars, but not the madness of men.” Isaac Newton

When visiting the exhibition hall at JSM 2014, I spoke with people from STEM forums on the Springer booth. The concept of STEM (why STEM? Nothing to do with STAN! Nor directly with Biology. It stands as the accronym for Science, Technology, Engineering, and Mathematics.) is to create a sort of peer-reviewed Cross Validated where questions would be filtered (in order to avoid the most basic questions like “How can I learn about Bayesian statistics without opening a book?” or “What is the Binomial distribution?” that often clutter the Stack Exchange boards). That’s an interesting approach which I will monitor in the future, as on the one hand, it would be nice to have a Statistics forum without “lazy undergraduate” questions as one of my interlocutors put, and on the other hand, to see how STEM forums can compete with the well-established Cross Validated and its core of dedicated moderators and editors. I left the booth with a neat tee-shirt exhibiting the above quote as well as alpha-tester on the back: STEM forums is indeed calling for entries into the Statistics section, with rewards of ebooks for the first 250 entries and a sweepstakes offering a free trip to Seattle next year!

## Bangalore workshop [ಬೆಂಗಳೂರು ಕಾರ್ಯಾಗಾರ] and new book

Posted in Books, pictures, R, Statistics, Travel, University life with tags , , , , , , , , , , , , on August 13, 2014 by xi'an

On the last day of the IFCAM workshop in Bangalore, Marc Lavielle from INRIA presented a talk on mixed effects where he illustrated his original computer language Monolix. And mentioned that his CRC Press book on Mixed Effects Models for the Population Approach was out! (Appropriately listed as out on a 14th of July on amazon!) He actually demonstrated the abilities of Monolix live and on diabets data provided by an earlier speaker from Kolkata, which was a perfect way to start initiating a collaboration! Nice cover (which is all I saw from the book at this stage!) that maybe will induce candidates to write a review for CHANCE. Estimation of those mixed effect models relies on stochastic EM algorithms developed by Marc Lavielle and Éric Moulines in the 90’s, as well as MCMC methods.