The title of this recent arXival had potential appeal, however the proposal ends up being rather straightforward and hence anti-climactic! The paper by Hu, Hendry and Heng proposes to run a mixture of proposals centred at the various modes of the target for an efficient exploration. This is a correct MCMC algorithm, granted!, but the requirement to know beforehand all the modes to be explored is self-defeating, since the major issue with MCMC is about modes that are omitted from the exploration and remain undetected throughout the simulation… As provided, this is a standard MCMC algorithm with no adaptive feature and I would rather suggest our population Monte Carlo version, given the available information. Another connection with population Monte Carlo is that I think the performances would improve by Rao-Blackwellising the acceptance rate, i.e. removing the conditioning on the (ancillary) component of the index. For PMC we proved that using the mixture proposal in the ratio led to an ideally minimal variance estimate and I do not see why randomising the acceptance ratio in the current case would bring any improvement.
Archive for the Books Category
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):
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):
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
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
is an arbitrary decomposition of the target. An interesting remark in the paper is that any Hastings representation can alternatively be written as
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
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.
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…
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 Hell. The 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….
Tad Williams’ second novel Happy Hour in Hell is much less enjoyable as the author was unable to keep up with the pace and tone of the highly imaginative first novel, full of witty and hard-boiled exchanges. The first novel introduced the (after-)life of a guardian angel in California, Doloriel (a.k.a. Bobby Dollar), with enough levels of political intrigue between Heaven and Hell and Earth and plots, pursuits, assassination attempts, etc., to make it a page-turner. This second novel sends Doloriel on a suicide mission to Hell… and the reader to a Hell of sorts where the damnation is one of eternal boredom! What made the first novel so original, namely the juxtaposition of the purpose of a guardian with his every-day terrestrial life, is lost. All we have there is a fantastic creature (from Heaven) transposed in another fantastic environment (Hell) and trying to survive without a proper guide book. The representation of Hell is not particularly enticing (!), even with acknowledged copies from Dante’s Inferno and Hieronymus Bosch’s paintings. There is a very low tolerance level to my reading of damned souls being tortured, dismembered, eaten or resuscitated, even when it gets to the hero’s turn. Add to that a continuation of the first book’s search for a particular feather. And an amazing amount of space dedicated to the characters’ meals. This makes for a very boring book. Even for a rainy day on a Maine lake! The depiction of the levels and inhabitants of Hell reminded me of another endless book by Tad Williams, Shadowmarch, where some characters end up in a subterranean semi-industrial structure, with a horde of demon-like creatures and no fun [for the reader!]. Ironically, the funniest part of reading Happy Hour in Hell was to do it after Dawkins’ as some reflections of the angel about the roles of Heaven and Hell (and religion) could have fitted well into The God delusion! (Too bad my Maine rental had Monty Python’s Holy Grail instead of The Life of Brian, as it would have made a perfect trilogy!)
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.
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
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…
“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!