dynamic mixtures and frequentist ABC

This early morning in NYC, I spotted this new arXival by Marco Bee (whom I know from the time he was writing his PhD with my late friend Bernhard Flury) and found he has been working for a while on ABC related problems. The mixture model he considers therein is a form of mixture of experts, where the weights of the mixture components are not constant but functions on (0,1) of the entry as well. This model was introduced by Frigessi, Haug and Rue in 2002 and is often used as a benchmark for ABC methods, since it is missing its normalising constant as in e.g.

even with all entries being standard pdfs and cdfs. Rather than using a (costly) numerical approximation of the “constant” (as a function of all unknown parameters involved), Marco follows the approximate maximum likelihood approach of my Warwick colleagues, Javier Rubio [now at UCL] and Adam Johansen. It is based on the [SAME] remark that under a uniform prior and using an approximation to the actual likelihood the MAP estimator is also the MLE for that approximation. The approximation is ABC-esque in that a pseudo-sample is generated from the true model (attached to a simulation of the parameter) and the pair is accepted if the pseudo-sample stands close enough to the observed sample. The paper proposes to use the Cramér-von Mises distance, which only involves ranks. Given this “posterior” sample, an approximation of the posterior density is constructed and then numerically optimised. From a frequentist view point, a direct estimate of the mode would be preferable. From my Bayesian perspective, this sounds like a step backwards, given that once a posterior sample is available, reconnecting with an approximate MLE does not sound highly compelling.

learning optimal summary statistics

“Despite the pursuit of the holy grail of sufficient statistics, most applications will have to settle for the weakest concept of optimal statistics.”Quiz #1: How does Bayes sufficiency [which preserves the posterior density] differ from sufficiency [which preserves the likelihood function]?

Quiz #2: How does Fisher-information sufficiency [which preserves the information matrix] differ from standard sufficiency [which preserves the likelihood function]?

Read a recent arXival by Till Hoffmann and Jukka-Pekka Onnela that I frankly found most puzzling… Maybe due to the Norman train where I was traveling being particularly noisy.

The argument in the paper is to find a summary statistic that minimises the [empirical] expected posterior entropy, which equivalently means minimising the expected Kullback-Leibler distance to the full posterior.  And maximizing the mutual information between parameters θ and summaries t(.). And maximizing the expected surprise. Which obviously requires breaking the sample into iid components and hence considering the gain brought by a specific transform of a single observation. The paper also contains a long comparison with other criteria for choosing summaries.

“Minimizing the posterior entropy would discard the sufficient statistic t such that the posterior is equal to the prior–we have not learned anything from the data.”

Furthermore, the expected aspect of the criterion takes us away from a proper Bayes analysis (and exhibits artifacts as the one above), which somehow makes me question the relevance of comparing entropies under different distributions. It took me a long while to realise that the collection of summaries was set by the user and quite limited. Like a neural network representation of the posterior mean. And the intractable posterior is further approximated by a closed-form function of the parameter θ and of the summary t(.). Using there a neural density estimator. Or a mixture density network.

prior elicitation

“We believe that an elicitation method should support elicitation both in the parameter and observable space, should be model-agnostic, and should be sample-efficient since human effort is costly.”

Petrus Mikkola et al. arXived a long paper on prior elicitation addressing the (most relevant) question: Why are we not widely use prior elicitation? With a massive bibliography that could be (partly) commented (and corrected as some references are incomplete, as eg my book chapter on priors!). I think the paper would make a terrific discussion paper.

The absence of a general procedure for prior elicitation is indeed hindering the adoption of Bayesian methods outside our core community and is thus eventually detrimental to their wider development. It also carries the dangers of misled or misleading prior choices. The authors put forward the absence of “software that integrates well with the current probabilistic programming tools used for other parts of the modelling workflow.” This requires setting principles that avoid “just-press-key” solutions. (Aside: This reminds me of my very first prospective PhD student, who was then working in a startup [although the name was not yet in use in the early 1990’s!] and had build such a software in a discretised, low dimension, conjugate prior, environment by returning a form of decision-theoretic impact of the chosen hyperparameters. He alas aborted his PhD attempt due to the short-term pressing matters in the under-staffed company…)

“We inspect prior elicitation from the perspectives of (1) properties of the prior distribution itself, (2) the model family and the prior elicitation method’s dependence on it, (3) the underlying elicitation space, (4) how the method interprets the information provided by the expert, (5) computation, (6) the form and quantity of interaction with the expert(s), and (7) the assumed capability of the expert (…)”

Prior elicitation is indeed a delicate balance between incorporating expert opinion(s) and avoiding over-standardisation. In my limited experience, experts tend to be over-confident about their own opinion and unwilling to attach uncertainty to their assessments. Even when being inconsistent. When several experts are involved (as, very briefly, in Section 3.6), building a common prior quickly becomes a challenge, esp. if their interests (or utility functions) diverge. As illustrated in the case of the whaling commission analysed by Adrian Raftery in the late 1990’s. (The above quote involves a single expert.) Actually, I dislike the term expert altogether, as it comes without any grading of the reliability of the person.To hit (!) at an early statement in the paper (p.5), should the prior elicitation always depend on the (sampling) model, as experts may ignore or misapprehend the model? The posterior already accounts for the likelihood and the parameter may pre-exist wrt the model, as eg cosmological constants or vaccine efficiency… In a sense, the model should be involved as little as possible in the elicitation as the expert could confuse her beliefs about the parameter with those about the accuracy of the model. (I realise this is not necessarily a mainstream position as illustrated by this paper by Andrew and friends!)

And isn’t the first stumbling block the inability of most to represent one’s prior knowledge in probabilistic terms? Innumeracy is a shared shortcoming in the general population (and since everyone’s an expert!), as repeatedly demonstrated since the start of the Covid-19 pandemic. (See also the above point about inconsistency. Accounting for such inconsistencies in a Bayesian way is a natural answer, albeit requiring the degree of expertise and reliability to be tested.)

Is prior elicitation feasible beyond a few dimensions? Even when using the constrictive tool of copulas one hits a wall after a few dimensions, assuming the expert is willing to set a prior correlation matrix.  Most of the methods described in Section 3.1 only apply to textbook examples. In their third dimension (!), the authors mention neural network parameters but later fail to cover this type of issue. (This was the example I had in mind indeed.) And they move from parameter space to observable space. Distinguishing predictive elicitation from observational elicitation, the former being what I would have suggested from scratch. Obviously, the curse of dimensionality strikes again unless one considers summary statistics (like in ABC).

While I am glad conjugate priors do not get the lion’s share, using as in Section 3.3.. non-parametric or machine learning solutions to construct the prior sounds unrealistic. (And including maximum entropy priors into that category seems wrong since they are definitely parametric.)

The proposed Bayesian treatment of the expert’s “data” (Section 4.1) is rational but requires an additional model construct to link the expert’s data with the parameter to reach a Bayes formula like (4.1). Plus a primary prior (which could then be one of the reference priors.) Reducing the expert’s input to imaginary observations may prove too narrow, though. The notion of an iterative elicitation is most appealing and its sequential aspect may not be particularly problematic in opposition to posteriors relying on using the data twice or more. I am much less buying the hierarchical construct of Section 4.3 because they imply a return to conjugate priors and hyperpriors, are not necessarily correctly understood by experts, do not always cater to observational elicitation, and are not an answer to high-dimension challenges.

Given the state of the art, it sounds like we are still far from seeing prior elicitation as a natural part of Bayesian software and probabilistic programming. Even when using a modular, model-agnostic strategy. But this is most certainly a worthy prospect!

conditioning on insufficient statistics in Bayesian regression

“…the prior distribution, the loss function, and the likelihood or sampling density (…) a healthy skepticism encourages us to question each of them”

A paper by John Lewis, Steven MacEachern, and Yoonkyung Lee has recently appeared in Bayesian Analysis. Starting with the great motivation of a misspecified model requiring the use of a (thus necessarily) insufficient statistic and moving to their central concern of simulating the posterior based on that statistic.

Model misspecification remains understudied from a B perspective and this paper is thus most welcome in addressing the issue. However, when reading through, one of my criticisms is in defining misspecification as equivalent to outliers in the sample. An outlier model is an easy case of misspecification, in the end, since the original model remains meaningful. (Why should there be “good” versus “bad” data) Furthermore, adding a non-parametric component for the unspecified part of the data would sound like a “more Bayesian” alternative. Unrelated, I also idly wondered at whether or not normalising flows could be used in this instance..

The problem in selecting a T (Darjeeling of course!) is not really discussed there, while each choice of a statistic T leads to a different signification to what misspecified means and suggests a comparison with Bayesian empirical likelihood.

“Acceptance rates of this [ABC] algorithm can be intolerably low”

Erm, this is not really the issue with ABC, is it?! Especially when the tolerance is induced by the simulations themselves.

When I reached the MCMC (Gibbs?) part of the paper, I first wondered at its relevance for the mispecification issues before realising it had become the focus of the paper. Now, simulating the observations conditional on a value of the summary statistic T is a true challenge. I remember for instance George Casella mentioning it in association with a Student’s t sample in the 1990’s and Kerrie and I having an unsuccessful attempt at it in the same period. Persi Diaconis has written several papers on the problem and I am thus surprised at the dearth of references here, like the rather recent Byrne and Girolami (2013), Florens and Simoni (2015), or Bornn et al. (2019). In the present case, the  linear model assumed as the true model has the exceptional feature that it leads to a feasible transform of an unconstrained simulation into a simulation with fixed statistics, with no measure theoretic worries if not free from considerable efforts to establish the operation is truly valid… And, while simulating (θ,y) makes perfect sense in an insufficient setting, the cost is then precisely the same as when running a vanilla ABC. Which brings us to the natural comparison with ABC. While taking ε=0 may sound as optimal for being “exact”, it is not from an ABC perspective since the convergence rate of the (summary) statistic should be roughly the one of the tolerance (Fearnhead and Liu, Frazier et al., 2018).

“[The Borel Paradox] shows that the concept of a conditional probability with regard to an isolated given hypothesis whose probability equals 0 is inadmissible.” A. Колмого́ров (1933)

As a side note for measure-theoretic purists, the derivation of the conditional of y given T(y)=T⁰ is arbitrary since the event has probability zero (ie, the conditioning set is of measure zero). See the Borel-Kolmogorov paradox. The computations in the paper are undoubtedly correct, but this is only one arbitrary choice of a transform (or conditioning σ-algebra).

Metropolis-Hastings via Classification [One World ABC seminar]

Today, Veronika Rockova is giving a webinar on her paper with Tetsuya Kaji Metropolis-Hastings via classification. at the One World ABC seminar, at 11.30am UK time. (Which was also presented at the Oxford Stats seminar last Feb.) Please register if not already a member of the 1W ABC mailing list.