Archive for exchangeability

10 great ideas about chance [book preview]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on November 13, 2017 by xi'an

[As I happened to be a reviewer of this book by Persi Diaconis and Brian Skyrms, I had the opportunity (and privilege!) to go through its earlier version. Here are the [edited] comments I sent back to PUP and the authors about this earlier version. All in  all, a terrific book!!!]

The historical introduction (“measurement”) of this book is most interesting, especially its analogy of chance with length. I would have appreciated a connection earlier than Cardano, like some of the Greek philosophers even though I gladly discovered there that Cardano was not only responsible for the closed form solutions to the third degree equation. I would also have liked to see more comments on the vexing issue of equiprobability: we all spend (if not waste) hours in the classroom explaining to (or arguing with) students why their solution is not correct. And they sometimes never get it! [And we sometimes get it wrong as well..!] Why is such a simple concept so hard to explicit? In short, but this is nothing but a personal choice, I would have made the chapter more conceptual and less chronologically historical.

“Coherence is again a question of consistent evaluations of a betting arrangement that can be implemented in alternative ways.” (p.46)

The second chapter, about Frank Ramsey, is interesting, if only because it puts this “man of genius” back under the spotlight when he has all but been forgotten. (At least in my circles.) And for joining probability and utility together. And for postulating that probability can be derived from expectations rather than the opposite. Even though betting or gambling has a (negative) stigma in many cultures. At least gambling for money, since most of our actions involve some degree of betting. But not in a rational or reasoned manner. (Of course, this is not a mathematical but rather a psychological objection.) Further, the justification through betting is somewhat tautological in that it assumes probabilities are true probabilities from the start. For instance, the Dutch book example on p.39 produces a gain of .2 only if the probabilities are correct.

> gain=rep(0,1e4)
> for (t in 1:1e4){
+ p=rexp(3);p=p/sum(p)
+ gain[t]=(p[1]*(1-.6)+p[2]*(1-.2)+p[3]*(.9-1))/sum(p)}
> hist(gain)

As I made it clear at the BFF4 conference last Spring, I now realise I have never really adhered to the Dutch book argument. This may be why I find the chapter somewhat unbalanced with not enough written on utilities and too much on Dutch books.

“The force of accumulating evidence made it less and less plausible to hold that subjective probability is, in general, approximate psychology.” (p.55)

A chapter on “psychology” may come as a surprise, but I feel a posteriori that it is appropriate. Most of it is about the Allais paradox. Plus entries on Ellesberg’s distinction between risk and uncertainty, with only the former being quantifiable by “objective” probabilities. And on Tversky’s and Kahneman’s distinction between heuristics, and the framing effect, i.e., how the way propositions are expressed impacts the choice of decision makers. However, it is leaving me unclear about the conclusion that the fact that people behave irrationally should not prevent a reliance on utility theory. Unclear because when taking actions involving other actors their potentially irrational choices should also be taken into account. (This is mostly nitpicking.)

“This is Bernoulli’s swindle. Try to make it precise and it falls apart. The conditional probabilities go in different directions, the desired intervals are of different quantities, and the desired probabilities are different probabilities.” (p.66)

The next chapter (“frequency”) is about Bernoulli’s Law of Large numbers and the stabilisation of frequencies, with von Mises making it the basis of his approach to probability. And Birkhoff’s extension which is capital for the development of stochastic processes. And later for MCMC. I like the notions of “disreputable twin” (p.63) and “Bernoulli’s swindle” about the idea that “chance is frequency”. The authors call the identification of probabilities as limits of frequencies Bernoulli‘s swindle, because it cannot handle zero probability events. With a nice link with the testing fallacy of equating rejection of the null with acceptance of the alternative. And an interesting description as to how Venn perceived the fallacy but could not overcome it: “If Venn’s theory appears to be full of holes, it is to his credit that he saw them himself.” The description of von Mises’ Kollectiven [and the welcome intervention of Abraham Wald] clarifies my previous and partial understanding of the notion, although I am unsure it is that clear for all potential readers. I also appreciate the connection with the very notion of randomness which has not yet found I fear a satisfactory definition. This chapter asks more (interesting) questions than it brings answers (to those or others). But enough, this is a brilliant chapter!

“…a random variable, the notion that Kac found mysterious in early expositions of probability theory.” (p.87)

Chapter 5 (“mathematics”) is very important [from my perspective] in that it justifies the necessity to associate measure theory with probability if one wishes to evolve further than urns and dices. To entitle Kolmogorov to posit his axioms of probability. And to define properly conditional probabilities as random variables (as my third students fail to realise). I enjoyed very much reading this chapter, but it may prove difficult to read for readers with no or little background in measure (although some advanced mathematical details have vanished from the published version). Still, this chapter constitutes a strong argument for preserving measure theory courses in graduate programs. As an aside, I find it amazing that mathematicians (even Kac!) had not at first realised the connection between measure theory and probability (p.84), but maybe not so amazing given the difficulty many still have with the notion of conditional probability. (Now, I would have liked to see some description of Borel’s paradox when it is mentioned (p.89).

“Nothing hangs on a flat prior (…) Nothing hangs on a unique quantification of ignorance.” (p.115)

The following chapter (“inverse inference”) is about Thomas Bayes and his posthumous theorem, with an introduction setting the theorem at the centre of the Hume-Price-Bayes triangle. (It is nice that the authors include a picture of the original version of the essay, as the initial title is much more explicit than the published version!) A short coverage, in tune with the fact that Bayes only contributed a twenty-plus paper to the field. And to be logically followed by a second part [formerly another chapter] on Pierre-Simon Laplace, both parts focussing on the selection of prior distributions on the probability of a Binomial (coin tossing) distribution. Emerging into a discussion of the position of statistics within or even outside mathematics. (And the assertion that Fisher was the Einstein of Statistics on p.120 may be disputed by many readers!)

“So it is perfectly legitimate to use Bayes’ mathematics even if we believe that chance does not exist.” (p.124)

The seventh chapter is about Bruno de Finetti with his astounding representation of exchangeable sequences as being mixtures of iid sequences. Defining an implicit prior on the side. While the description sticks to binary events, it gets quickly more advanced with the notion of partial and Markov exchangeability. With the most interesting connection between those exchangeabilities and sufficiency. (I would however disagree with the statement that “Bayes was the father of parametric Bayesian analysis” [p.133] as this is extrapolating too much from the Essay.) My next remark may be non-sensical, but I would have welcomed an entry at the end of the chapter on cases where the exchangeability representation fails, for instance those cases when there is no sufficiency structure to exploit in the model. A bonus to the chapter is a description of Birkhoff’s ergodic theorem “as a generalisation of de Finetti” (p..134-136), plus half a dozen pages of appendices on more technical aspects of de Finetti’s theorem.

“We want random sequences to pass all tests of randomness, with tests being computationally implemented”. (p.151)

The eighth chapter (“algorithmic randomness”) comes (again!) as a surprise as it centres on the character of Per Martin-Löf who is little known in statistics circles. (The chapter starts with a picture of him with the iconic Oberwolfach sculpture in the background.) Martin-Löf’s work concentrates on the notion of randomness, in a mathematical rather than probabilistic sense, and on the algorithmic consequences. I like very much the section on random generators. Including a mention of our old friend RANDU, the 16 planes random generator! This chapter connects with Chapter 4 since von Mises also attempted to define a random sequence. To the point it feels slightly repetitive (for instance Jean Ville is mentioned in rather similar terms in both chapters). Martin-Löf’s central notion is computability, which forces us to visit Turing’s machine. And its role in the undecidability of some logical statements. And Church’s recursive functions. (With a link not exploited here to the notion of probabilistic programming, where one language is actually named Church, after Alonzo Church.) Back to Martin-Löf, (I do not see how his test for randomness can be implemented on a real machine as the whole test requires going through the entire sequence: since this notion connects with von Mises’ Kollektivs, I am missing the point!) And then Kolmororov is brought back with his own notion of complexity (which is also Chaitin’s and Solomonov’s). Overall this is a pretty hard chapter both because of the notions it introduces and because I do not feel it is completely conclusive about the notion(s) of randomness. A side remark about casino hustlers and their “exploitation” of weak random generators: I believe Jeff Rosenthal has a similar if maybe simpler story in his book about Canadian lotteries.

“Does quantum mechanics need a different notion of probability? We think not.” (p.180)

The penultimate chapter is about Boltzmann and the notion of “physical chance”. Or statistical physics. A story that involves Zermelo and Poincaré, And Gibbs, Maxwell and the Ehrenfests. The discussion focus on the definition of probability in a thermodynamic setting, opposing time frequencies to space frequencies. Which requires ergodicity and hence Birkhoff [no surprise, this is about ergodicity!] as well as von Neumann. This reaches a point where conjectures in the theory are yet open. What I always (if presumably naïvely) find fascinating in this topic is the fact that ergodicity operates without requiring randomness. Dynamical systems can enjoy ergodic theorem, while being completely deterministic.) This chapter also discusses quantum mechanics, which main tenet requires probability. Which needs to be defined, from a frequency or a subjective perspective. And the Bernoulli shift that brings us back to random generators. The authors briefly mention the Einstein-Podolsky-Rosen paradox, which sounds more metaphysical than mathematical in my opinion, although they get to great details to explain Bell’s conclusion that quantum theory leads to a mathematical impossibility (but they lost me along the way). Except that we “are left with quantum probabilities” (p.183). And the chapter leaves me still uncertain as to why statistical mechanics carries the label statistical. As it does not seem to involve inference at all.

“If you don’t like calling these ignorance priors on the ground that they may be sharply peaked, call them nondogmatic priors or skeptical priors, because these priors are quite in the spirit of ancient skepticism.” (p.199)

And then the last chapter (“induction”) brings us back to Hume and the 18th Century, where somehow “everything” [including statistics] started! Except that Hume’s strong scepticism (or skepticism) makes induction seemingly impossible. (A perspective with which I agree to some extent, if not to Keynes’ extreme version, when considering for instance financial time series as stationary. And a reason why I do not see the criticisms contained in the Black Swan as pertinent because they savage normality while accepting stationarity.) The chapter rediscusses Bayes’ and Laplace’s contributions to inference as well, challenging Hume’s conclusion of the impossibility to finer. Even though the representation of ignorance is not unique (p.199). And the authors call again for de Finetti’s representation theorem as bypassing the issue of whether or not there is such a thing as chance. And escaping inductive scepticism. (The section about Goodman’s grue hypothesis is somewhat distracting, maybe because I have always found it quite artificial and based on a linguistic pun rather than a logical contradiction.) The part about (Richard) Jeffrey is quite new to me but ends up quite abruptly! Similarly about Popper and his exclusion of induction. From this chapter, I appreciated very much the section on skeptical priors and its analysis from a meta-probabilist perspective.

There is no conclusion to the book, but to end up with a chapter on induction seems quite appropriate. (But there is an appendix as a probability tutorial, mentioning Monte Carlo resolutions. Plus notes on all chapters. And a commented bibliography.) Definitely recommended!

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE. As appropriate for a book about Chance!]

anytime algorithm

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

Lawrence Murray, Sumeet Singh, Pierre Jacob, and Anthony Lee (Warwick) recently arXived a paper on Anytime Monte Carlo. (The earlier post on this topic is no coincidence, as Lawrence had told me about this problem when he visited Paris last Spring. Including a forced extension when his passport got stolen.) The difficulty with anytime algorithms for MCMC is the lack of exchangeability of the MCMC sequence (except for formal settings where regeneration can be used).

When accounting for duration of computation between steps of an MCMC generation, the Markov chain turns into a Markov jump process, whose stationary distribution α is biased by the average delivery time. Unless it is constant. The authors manage this difficulty by interlocking the original chain with a secondary chain so that even- and odd-index chains are independent. The secondary chain is then discarded. This provides a way to run an anytime MCMC. The principle can be extended to K+1 chains, run one after the other, since only one of those chains need be discarded. It also applies to SMC and SMC². The appeal of anytime simulation in this particle setting is that resampling is no longer a bottleneck. Hence easily distributed among processors. One aspect I do not fully understand is how the computing budget is handled, since allocating the same real time to each iteration of SMC seems to envision each target in the sequence as requiring the same amount of time. (An interesting side remark made in this paper is the lack of exchangeability resulting from elaborate resampling mechanisms, lack I had not thought of before.)

Conditional love [guest post]

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , on August 4, 2015 by xi'an

[When Dan Simpson told me he was reading Terenin’s and Draper’s latest arXival in a nice Bath pub—and not a nice bath tub!—, I asked him for a blog entry and he agreed. Here is his piece, read at your own risk! If you remember to skip the part about Céline Dion, you should enjoy it very much!!!]

Probability has traditionally been described, as per Kolmogorov and his ardent follower Katy Perry, unconditionally. This is, of course, excellent for those of us who really like measure theory, as the maths is identical. Unfortunately mathematical convenience is not necessarily enough and a large part of the applied statistical community is working with Bayesian methods. These are unavoidably conditional and, as such, it is natural to ask if there is a fundamentally conditional basis for probability.

Bruno de Finetti—and later Richard Cox and Edwin Jaynes—considered conditional bases for Bayesian probability that are, unfortunately, incomplete. The critical problem is that they mainly consider finite state spaces and construct finitely additive systems of conditional probability. For a variety of reasons, neither of these restrictions hold much truck in the modern world of statistics.

In a recently arXiv’d paper, Alexander Terenin and David Draper devise a set of axioms that make the Cox-Jaynes system of conditional probability rigorous. Furthermore, they show that the complete set of Kolmogorov axioms (including countable additivity) can be derived as theorems from their axioms by conditioning on the entire sample space.

This is a deep and fundamental paper, which unfortunately means that I most probably do not grasp it’s complexities (especially as, for some reason, I keep reading it in pubs!). However I’m going to have a shot at having some thoughts on it, because I feel like it’s the sort of paper one should have thoughts on. Continue reading

Robert’s paradox [reading in Reading]

Posted in Statistics, Travel, University life with tags , , , , , , , , , , , , on January 28, 2015 by xi'an

paradoxOn Wednesday afternoon, Richard Everitt and Dennis Prangle organised an RSS workshop in Reading on Bayesian Computation. And invited me to give a talk there, along with John Hemmings, Christophe Andrieu, Marcelo Pereyra, and themselves. Given the proximity between Oxford and Reading, this felt like a neighbourly visit, especially when I realised I could take my bike on the train! John Hemmings gave a presentation on synthetic models for climate change and their evaluation, which could have some connection with Tony O’Hagan’s recent talk in Warwick, Dennis told us about “the lazier ABC” version in connection with his “lazy ABC” paper, [from my very personal view] Marcelo expanded on the Moreau-Yoshida expansion he had presented in Bristol about six months ago, with the notion that using a Gaussian tail regularisation of a super-Gaussian target in a Langevin algorithm could produce better convergence guarantees than the competition, including Hamiltonian Monte Carlo, Luke Kelly spoke about an extension of phylogenetic trees using a notion of lateral transfer, and Richard introduced a notion of biased approximation to Metropolis-Hasting acceptance ratios, notion that I found quite attractive if not completely formalised, as there should be a Monte Carlo equivalent to the improvement brought by biased Bayes estimators over unbiased classical counterparts. (Repeating a remark by Persi Diaconis made more than 20 years ago.) Christophe Andrieu also exposed some recent developments of his on exact approximations à la Andrieu and Roberts (2009).

Since those developments are not yet finalised into an archived document, I will not delve into the details, but I found the results quite impressive and worth exploring, so I am looking forward to the incoming publication. One aspect of the talk which I can comment on is related to the exchange algorithm of Murray et al. (2006). Let me recall that this algorithm handles double intractable problems (i.e., likelihoods with intractable normalising constants like the Ising model), by introducing auxiliary variables with the same distribution as the data given the new value of the parameter and computing an augmented acceptance ratio which expectation is the targeted acceptance ratio and which conveniently removes the unknown normalising constants. This auxiliary scheme produces a random acceptance ratio and hence differs from the exact-approximation MCMC approach, which target directly the intractable likelihood. It somewhat replaces the unknown constant with the density taken at a plausible realisation, hence providing a proper scale. At least for the new value. I wonder if a comparison has been conducted between both versions, the naïve intuition being that the ratio of estimates should be more variable than the estimate of the ratio. More generally, it seemed to me [during the introductory part of Christophe’s talk] that those different methods always faced a harmonic mean danger when being phrased as expectations of ratios, since those ratios were not necessarily squared integrable. And not necessarily bounded. Hence my rather gratuitous suggestion of using other tools than the expectation, like maybe a median, thus circling back to the biased estimators of Richard. (And later cycling back, unscathed, to Reading station!)

On top of the six talks in the afternoon, there was a small poster session during the tea break, where I met Garth Holloway, working in agricultural economics, who happened to be a (unsuspected) fan of mine!, to the point of entitling his poster “Robert’s paradox”!!! The problem covered by this undeserved denomination connected to the bias in Chib’s approximation of the evidence in mixture estimation, a phenomenon that I related to the exchangeability of the component parameters in an earlier paper or set of slides. So “my” paradox is essentially label (un)switching and its consequences. For which I cannot claim any fame! Still, I am looking forward the completed version of this poster to discuss Garth’s solution, but we had a beer together after the talks, drinking to the health of our mutual friend John Deely.

improved approximate-Bayesian model-choice method for estimating shared evolutionary history [reply from the author]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , on June 3, 2014 by xi'an

[Here is a very kind and detailed reply from Jamie Oakes to the comments I made on his ABC paper a few days ago:]

First of all, many thanks for your thorough review of my pre-print! It is very helpful and much appreciated. I just wanted to comment on a few things you address in your post.

I am a little confused about how my replacement of continuous uniform probability distributions with gamma distributions for priors on several parameters introduces a potentially crippling number of hyperparameters. Both uniform and gamma distributions have two parameters. So, the new model only has one additional hyperparameter compared to the original msBayes model: the concentration parameter on the Dirichlet process prior on divergence models. Also, the new model offers a uniform prior over divergence models (though I don’t recommend it).

Your comment about there being no new ABC technique is 100% correct. The model is new, the ABC numerical machinery is not. Also, your intuition is correct, I do not use the divergence times to calculate summary statistics. I mention the divergence times in the description of the ABC algorithm with the hope of making it clear that the times are scaled (see Equation (12)) prior to the simulation of the data (from which the summary statistics are calculated). This scaling is simply to go from units proportional to time, to units that are proportional to the expected number of mutations. Clearly, my attempt at clarity only created unnecessary opacity. I’ll have to make some edits.

Regarding the reshuffling of the summary statistics calculated from different alignments of sequences, the statistics are not exchangeable. So, reshuffling them in a manner that is not conistent across all simulations and the observed data is not mathematically valid. Also, if elements are exchangeable, their order will not affect the likelihood (or the posterior, barring sampling error). Thus, if our goal is to approximate the likelihood, I would hope the reshuffling would also have little affect on the approximate posterior (otherwise my approximation is not so good?).

You are correct that my use of “bias” was not well defined in reference to the identity line of my plots of the estimated vs true probability of the one-divergence model. I think we can agree that, ideally (all assumptions are met), the estimated posterior probability of a model should estimate the probability that the model is correct. For large numbers of simulation
replicates, the proportion of the replicates for which the one-divergence model is true will approximate the probability that the one-divergence model is correct. Thus, if the method has the desirable (albeit “frequentist”) behavior such that the estimated posterior probability of the one-divergence model is an unbiased estimate of the probability that the one-divergence model is correct, the points should fall near the identity line. For example, let us say the method estimates a posterior probability of 0.90 for the one-divergence model for 1000 simulated datasets. If the method is accurately estimating the probability that the one-divergence model is the correct model, then the one-divergence model should be the true model for approximately 900 of the 1000 datasets. Any trend away from the identity line indicates the method is biased in the (frequentist) sense that it is not correctly estimating the probability that the one-divergence model is the correct model. I agree this measure of “bias” is frequentist in nature. However, it seems like a worthwhile goal for Bayesian model-choice methods to have good frequentist properties. If a method strongly deviates from the identity line, it is much more difficult to interpret the posterior probabilites that it estimates. Going back to my example of the posterior probability of 0.90 for 1000 replicates, I would be alarmed if the model was true in only 100 of the replicates.

My apologies if my citation of your PNAS paper seemed misleading. The citation was intended to be limited to the context of ABC methods that use summary statistics that are insufficient across the models under comparison (like msBayes and the method I present in the paper). I will definitely expand on this sentence to make this clearer in revisions. Thanks!

Lastly, my concluding remarks in the paper about full-likelihood methods in this domain are not as lofty as you might think. The likelihood function of the msBayes model is tractable, and, in fact, has already been derived and implemented via reversible-jump MCMC (albeit, not readily available yet). Also, there are plenty of examples of rich, Kingman-coalescent models implemented in full-likelihood Bayesian frameworks. Too many to list, but a lot of them are implemented in the BEAST software package. One noteworthy example is the work of Bryant et al. (2012, Molecular Biology and Evolution, 29(8), 1917–32) that analytically integrates over all gene trees for biallelic markers under the coalescent.