Archive for curse of dimensionality

prior elicitation

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

“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!

improving bridge samplers by GANs

Posted in Books, pictures, Statistics with tags , , , , , , , on July 20, 2021 by xi'an

Hanwen Xing from Oxford recently posted a paper on arXiv about using GANs to improve the overlap bewtween the densities in bridge sampling. Bringing out new connections with noise contrastive estimation. The idea is to optimise a transform of one of the densities h() to bring it closer to the other density k(), using for instance normalising flows. (The call to transforms for bridge is not new, dating at least to Voter in 1985, the year I was starting my PhD!) Furthermore, using an f-divergence as a measure of functional distance allows for a reasonably straightforward update of the transform. That can be reformulated as a GAN target, which is somewhat natural in that the transform aims at confusing simulation from the transform of h and from k. This is quite an interesting proposal,  even though calculating the optimal transform is time-consuming and subjet to the curse of dimensionality. I also wonder at whether or not iterating the optimisation, one density after the other, would be bring further improvement.

likelihood-free and summary-free?

Posted in Books, Mountains, pictures, Statistics, Travel with tags , , , , , , , , , , , , , on March 30, 2021 by xi'an

My friends and coauthors Chris Drovandi and David Frazier have recently arXived a paper entitled A comparison of likelihood-free methods with and without summary statistics. In which they indeed compare these two perspectives on approximate Bayesian methods like ABC and Bayesian synthetic likelihoods.

“A criticism of summary statistic based approaches is that their choice is often ad hoc and there will generally be an  inherent loss of information.”

In ABC methods, the recourse to a summary statistic is often advocated as a “necessary evil” against the greater evil of the curse of dimension, paradoxically providing a faster convergence of the ABC approximation (Fearnhead & Liu, 2018). The authors propose a somewhat generic selection of summary statistics based on [my undergrad mentors!] Gouriéroux’s and Monfort’s indirect inference, using a mixture of Gaussians as their auxiliary model. Summary-free solutions, as in our Wasserstein papers, rely on distances between distributions, hence are functional distances, that can be seen as dimension-free as well (or criticised as infinite dimensional). Chris and David consider energy distances (which sound very much like standard distances, except for averaging over all permutations), maximum mean discrepancy as in Gretton et al. (2012), Cramèr-von Mises distances, and Kullback-Leibler divergences estimated via one-nearest-neighbour formulas, for a univariate sample. I am not aware of any degree of theoretical exploration of these functional approaches towards the precise speed of convergence of the ABC approximation…

“We found that at least one of the full data approaches was competitive with or outperforms ABC with summary statistics across all examples.”

The main part of the paper, besides a survey of the existing solutions, is to compare the performances of these over a few chosen (univariate) examples, with the exact posterior as the golden standard. In the g & k model, the Pima Indian benchmark of ABC studies!, Cramèr does somewhat better. While it does much worse in an M/G/1 example (where Wasserstein does better, and similarly for a stereological extremes example of Bortot et al., 2007). An ordering inversed again for a toad movement model I had not seen before. While the usual provision applies, namely that this is a simulation study on unidimensional data and a small number of parameters, the design of the four comparison experiments is very careful, eliminating versions that are either too costly or too divergence, although this could be potentially criticised for being unrealistic (i.e., when the true posterior is unknown). The computing time is roughly the same across methods, which essentially remove the call to kernel based approximations of the likelihood. Another point of interest is that the distance methods are significantly impacted by transforms on the data, which should not be so for intrinsic distances! Demonstrating the distances are not intrinsic…

the surprisingly overlooked efficiency of SMC

Posted in Books, Statistics, University life with tags , , , , , , , , , , , on December 15, 2020 by xi'an

At the Laplace demon’s seminar today (whose cool name I cannot tire of!), Nicolas Chopin gave a webinar with the above equally cool title. And the first slide debunking myths about SMC’s:

The second part of the talk is about a recent arXival Nicolas wrote with his student Hai-Dang DauI missed, about increasing the number of MCMC steps when moving the particles. Called waste-free SMC. Where only one fraction of the particles is updated, but this is enough to create a sort of independence from previous iterations of the SMC. (Hai-Dang Dau and Nicolas Chopin had to taylor their own convergence proof for this modification of the usual SMC. Producing a single-run assessment of the asymptotic variance.)

On the side, I heard about a very neat (if possibly toyish) example on estimating the number of Latin squares:

And the other item of information is that Nicolas’ and Omiros’ book, An Introduction to Sequential Monte Carlo, has now appeared! (Looking forward reading the parts I had not yet read.)

improved importance sampling via iterated moment matching

Posted in Statistics with tags , , , , on August 1, 2019 by xi'an

Topi Paananen, Juho Piironen, Paul-Christian Bürkner and Aki Vehtari have recently arXived a work on constructing an adapted importance (sampling) distribution. The beginning is more a review than a new contribution, covering the earlier work by Vehtari, Gelman  and Gabri (2017): estimating the Pareto rate for the importance weight distribution helps in assessing whether or not this distribution allows for a (necessary) second moment. In case it does not (seem to), the authors propose an affine transform of the importance distribution, using the earlier sample to match the first two moments of the distribution. Or of the targeted function. Adaptation that is controlled by the same Pareto rate technique, as in the above picture (from the paper). Predicting a natural objection as to the poor performances of the earlier samples, the paper suggests to use robust estimators of these moments, for instance via Pareto smoothing. It also suggests using multiple importance sampling as a way to regularise and robustify the estimates. While I buy the argument of fitting the target moments to achieve a better fit of the importance sampling, I remain unclear as to why an affine transform would change the (poor) tail behaviour of the importance sampler. Hence why it would apply in full generality. An alternative could consist in finding appropriate Box-Cox transforms, although the difficulty would certainly increase with the dimension.

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