## martingale posteriors

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , on November 7, 2022 by xi'an

A new Royal Statistical Society Read Paper featuring Edwin Fong, Chris Holmes, and Steve Walker. Starting from the predictive

$p(y_{n+1:+\infty}|y_{1:n})\ \ \ (1)$

rather than from the posterior distribution on the parameter is a fairly novel idea, also pursued by Sonia Petrone and some of her coauthors. It thus adopts a de Finetti’s perspective while adding some substance to the rather metaphysical nature of the original. It however relies on the “existence” of an infinite sample in (1) that assumes a form of underlying model à la von Mises or at least an infinite population. The representation of a parameter θ as a function of an infinite sequence comes as a shock first but starts making sense when considering it as a functional of the underlying distribution. Of course, trading (modelling) a random “opaque” parameter θ for (envisioning) an infinite sequence of random (un)observations may sound like a sure loss rather than as a great deal, but it gives substance to the epistemic uncertainty about a distributional parameter, even when a model is assumed, as in Example 1, which defines θ in the usual parametric way (i.e., the mean of the iid variables). Furthermore, the link with bootstrap and even more Bayesian bootstrap becomes clear when θ is seen this way.

Always a fan of minimal loss approaches, but (2.4) defines either a moment or a true parameter value that depends on the parametric family indexed by θ. Hence does not exist outside the primary definition of said parametric family. The following construct of the empirical cdf based on the infinite sequence as providing the θ function is elegant but what is its Bayesian justification? (I did not read Appendix C.2. in full detail but could not spot the prior on F.)

“The resemblance of the martingale posterior to a bootstrap estimator should not have gone unnoticed”

I am always fan of minimal loss approaches, but I wonder at (2.4), as it defines either a moment or a true parameter value that depends on the parametric family indexed by θ. Hence it does not exist outside the primary definition of said parametric family, which limits its appeal. The following construct of the empirical cdf based on the infinite sequence as providing the θ function is elegant and connect with bootstrap, but I wonder at its Bayesian justification. (I did not read Appendix C.2. in full detail but could not spot a prior on F.)

While I completely missed the resemblance, it is indeed the case that, if the predictive at each step is build from the earlier “sample”, the support is not going to evolve. However, this is not particularly exciting as the Bayesian non-parametric estimator is most rudimentary. This seems to bring us back to Rubin (1981) ?! A Dirichlet prior is mentioned with no further detail. And I am getting confused at the complete lack of structure, prior, &tc. It seems to contradict the next section:

“While the prescription of (3.1) remains a subjective task, we find it to be no more subjective than the selection of a likelihood function”

Copulas!!! Again, I am very glad to see copulas involved in the analysis. However, I remain unclear as to why Corollary 1 implies that any sequence of copulas could do the job. Further, why does the Gaussian copula appear as the default choice? What is the computing cost of the update (4.4) after k steps? Similarly (4.7) is using a very special form of copula, with independent-across-dimension increments. I am also missing a guided tour on the implementation, as it sounds explosive in book-keeping and multiplying, while relying on a single hyperparameter in (4.5.2)?

In the illustration section, the use of the galaxy dataset may fail to appeal to Radford Neal, in a spirit similar to Chopin’s & Ridgway’s call to leave the Pima Indians alone, since he delivered a passionate lecture on the inappropriateness of a mixture model for this dataset (at ICMS in 2001). I am unclear as to where the number of modes is extracted from the infinite predictive. What is $\theta$ in this case?

Copulas!!! Although I am unclear why Corollary 1 implies that any sequence of copulas does the job. And why the Gaussian copula appears as the default choice. What is the computing cost of the update (4.4) after k steps? Similarly (4.7) is using a very special form of copula, with independent-across-dimension increments. Missing a guided tour on the implementation, as it sounds explosive in book-keeping and multiplying. A single hyperparameter (4.5.2)?

## Finite mixture models do not reliably learn the number of components

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , on October 15, 2022 by xi'an

When preparing my talk for Padova, I found that Diana Cai, Trevor Campbell, and Tamara Broderick wrote this ICML / PLMR paper last year on the impossible estimation of the number of components in a mixture.

“A natural check on a Bayesian mixture analysis is to establish that the Bayesian posterior on the number of components increasingly concentrates near the truth as the number of data points becomes arbitrarily large.” Cai, Campbell & Broderick (2021)

Which seems to contradict [my formerly-Glaswegian friend] Agostino Nobile  who showed in his thesis that the posterior on the number of components does concentrate at the true number of components, provided the prior contains that number in its support. As well as numerous papers on the consistency of the Bayes factor, including the one against an infinite mixture alternative, as we discussed in our recent paper with Adrien and Judith. And reminded me of the rebuke I got in 2001 from the late David McKay when mentioning that I did not believe in estimating the number of components, both because of the impact of the prior modelling and of the tendency of the data to push for more clusters as the sample size increased. (This was a most lively workshop Mike Titterington and I organised at ICMS in Edinburgh, where Radford Neal also delivered an impromptu talk to argue against using the Galaxy dataset as a benchmark!)

“In principle, the Bayes factor for the MFM versus the DPM could be used as an empirical criterion for choosing between the two models, and in fact, it is quite easy to compute an approximation to the Bayes factor using importance sampling” Miller & Harrison (2018)

This is however a point made in Miller & Harrison (2018) that the estimation of k logically goes south if the data is not from the assumed mixture model. In this paper, Cai et al. demonstrate that the posterior diverges, even when it depends on the sample size. Or even the sample as in empirical Bayes solutions.

## a journal of the [less] plague and [more] pestilence year

Posted in Books, Kids, Mountains, pictures, Travel, University life with tags , , , , , , , , , , , , , , , , , , , on July 10, 2022 by xi'an

Read Rankin’s last Rebus, A song for the dark times, which takes place between Edinburgh and the Far North (of Scotland). I reasonably enjoyed it, by which I mean I was not expecting novelty, but rather reuniting with a few characters, including the Teflon villain, Big Ger Cafferty, still around at his craft. Rebus is getting older, cannot climb stairs any longer, and cannot deliver a proper punch in a fight! Still enjoyable, with a dig into Second World War internment camps for German prisoners… While not yet into the COVID era, the spirit is definitely post-Brexit, with a general resentment of what it brought (and did not bring). The character of Inspector Fox escaped me, mostly, but otherwise, an enjoyable read.

Made a light (no baking) chocolate tart, with home raspberries on top (of course) that did not last long.

Watched two Japanese shows: Any crybabies around?! by Takuma Satô which revolves around the Namahage tradition in Northern Japan (to terrify children into being obedient and no crybabies!) and the immaturity of a young father acting as such a character until disaster strikes. With a lot of cringe moments, until the utter hopelessness of this man crybaby, more straw-like than his traditional costume made me stop caring. And the mini-series Switched. Which explores a (paranormal) body switch between two teenager girls to school pressure, bullying, and depression, but in a rather perturbing manner as the girl who initiated and forced the exchange does not come out nicely, despite her overweight issues, her abusive single mother, and the attitude of the rest of the school.  The most interesting character is the other schoolgirl who has to adapt to this situation without changing her (inner) personality, but the story is slow-motioned, predictable, and heavy-handed, esp. in the sobbing department. (Plus bordering at fat-shaming at some point.)

## ICMS supports mirrors

Posted in Travel, University life with tags , , , , on December 6, 2021 by xi'an

Received an announcement from the International Centre for Mathematical Sciences (ICMS) in Edinburgh that they will support mirror meetings (albeit in the United Kindgom) as well as other inclusive initiatives:

• ICMS@: this programme allows organisers to hold ‘satellite events’, ICMS-funded activity at venues elsewhere in the UK with logistic support by the ICMS staff. The aim will be to enable more activities at a high level distributed throughout the country and to facilitate participation by those who cannot easily travel. There will be an additional emphasis on regions that have found it difficult to fund local events in the past.

• Funds to allow participants at ICMS workshops to extend their visits in the UK, especially for the purpose of visiting other institutions and engaging in extended research interaction.

• A visitor programme for researchers from low- and middle-income countries to come to the UK to attend workshops and fund their stay for up to three months.

• Workshops or schools for postgraduate students and early career researchers which can be of varying lengths and intensities.

• A fund to help people with caring responsibilities attend our events.

## northern gannets rock

Posted in Kids, pictures, Travel with tags , , , , , , , , , , , on May 9, 2021 by xi'an