Archive for axioms of probability

What are the chances of that?

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , on May 13, 2022 by xi'an

What are the chances that I review a book with this title, a few months after reviewing a book called What is luck?! This one is written by Andrew Elliott, whose Is that a big number? I reviewed a wee bit earlier… And that the cover of this book involves a particularly unlucky sequence of die as in my much earlier review of Krysz Burdzy’s book? (About 10⁻⁶ less likely than the likeliest draw!)

The (relative) specificity of this book is to try to convey the notions of chance and uncertainty to the general public, more in demonstrating that our intuition is most often wrong by examples and simulations, than in delving into psychological reasons as in Barbara Blatchley’s book. The author advances five dualities that underly our (dysfunctional) relation to chance: individual vs. collective, randomness vs. meaning, foresight vs. insight, uniformity vs. variability, and disruption vs. opportunity.

“News programmes clearly understand that the testimonies of individuals draw better audiences than the summaries of statisticians.” (p. xvii)

Some of the nice features of the book  are (a) the description of a probabilistic problem at the beginning of each chapter, to be solved at the end, (b) the use of simulation experiments, represented by coloured pixels over a grey band crossing the page, including a section on pseudorandom generators [which is less confusing that the quote below may indicate!], (c) taking full advantage of the quincunx apparatus, and (d) very few apologies for getting into formulas. And even a relevant quote of Taleb’s Black Swan about the ludic fallacy. On the other hand, the author spends quite a large component of the book on chance games, exhibiting a ludic tendency! And contemplates biased coins, while he should know better! The historical sections may prove too much for both informed and uninformed readers. (However, I learned that the UK Government had used a form of lottery to pay interests on premium bonds.) And the later parts are less numerical and quantified, even though the author brings in the micromort measurement [invented by Ronald Howard and] favoured by David Spiegelhalter. Who actually appears to have inspired several other sections, like the one on coincidences (which remains quite light in its investigation!). I finished the book rather quickly by browsing though mostly anecdotes and a lesser feel of a unified discourse. I did not find the attempt to link with the COVID pandemic, which definitely resets our clocks on risk, particularly alluring…

“People go to a lot of trouble to generate truly random numbers—sequences that are impossible to predict.” (p.66)

The apparition of the Normal distribution is somewhat overdone and almost mystical, if the tone gets more reasonable by the end of the corresponding chapter.

“…combining random numbers from distributions that really have no business being added together (…) ends up with a statistic that actually fits the normal distribution quite well.” (p.83)

The part about Bayes and Bayesian reasoning does not include any inference, with a rather duh! criticism of prior modelling.

“If you are tempted to apply a group statistic derived from a broad analysis to a more narrow purpose, you run the risk of making an unfair judgement.” (p.263)

The section about Xenakis’ musical creations as a Markov process was most interesting (and novel to me). I also enjoyed the shared cultural entries, esp. literary ones. Like citing the recent Chernobyl TV drama. Or Philip K. Dick’s Do Androids Dream of Electric Sheep? Or yet Monty Python’s Life of Brian. Overall, there is enough trivia and engagement to keep reading the book till its end!

a new paradigm for improper priors

Posted in Books, pictures, Statistics, Travel with tags , , , , , , , , on November 6, 2017 by xi'an

Gunnar Taraldsen and co-authors have arXived a short note on using improper priors from a new perspective. Generalising an earlier 2016 paper in JSPI on the same topic. Which both relate to a concept introduced by Rényi (who himself attributes the idea to Kolmogorov). Namely that random variables measures are to be associated with arbitrary measures [not necessarily σ-finite measures, the later defining σ-finite random variables], rather than those with total mass one. Which allows for an alternate notion of conditional probability in the case of σ-finite random variables, with the perk that this conditional probability distribution is itself of mass 1 (a.e.).  Which we know happens when moving from prior to proper posterior.

I remain puzzled by the 2016 paper though as I do not follow the meaning of a random variable associated with an infinite mass probability measure. If the point is limited to construct posterior probability distributions associated with improper priors, there is little value in doing so. The argument in the 2016 paper is however that one can then define a conditional distribution in marginalisation paradoxes à la Stone, Dawid and Zidek (1973) where the marginal does not exist. Solving with this formalism the said marginalisation paradoxes as conditional distributions are only defined for σ-finite random variables. Which gives a fairly different conclusion from either Stone, Dawid and Zidek (1973) [with whom I agree, namely that there is no paradox because there is no “joint” distribution] or Jaynes (1973) [with whom I less agree!, in that the use of an invariant measure to make the discrepancy go away is not a particularly strong argument in favour of this measure]. The 2016 paper also draws an interesting connection with the study by Jim Hobert and George Casella (in Jim’s thesis) of [null recurrent or transient] Gibbs samplers with no joint [proper] distribution. Which in some situations can produce proper subchains, a phenomenon later exhibited by Alan Gelfand and Sujit Sahu (and Xiao-Li Meng as well if I correctly remember!). But I see no advantage in following this formalism, as it does not impact whether the chain is transient or null recurrent, or anything connected with its implementation. Plus a link to the approximation of improper priors by sequences of proper ones by Bioche and Druihlet I discussed a while ago.

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

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