My PhD student Edoardo Bandoni, along with Julien Stoehr and myself, completed a paper on validating (Bayesian) kernel quadrature with unbounded domains of integration. Which connects with probabilistic numerics, since the integrand is modelled as a Gaussian process. And RKHS methods. As opposed to Monte Carlo estimators, quadrature methods approximate integrals of smooth functions with worst-case error decaying at a minimax rate α/d for smoothness α in dimension d. Existing rate-optimal quadrature methods often depend on deterministic point sets tailored to a specific kernel, making them sensitive to misspecification and thus less robust in practice. This paper studies instead randomised quadrature methods, with a focus on robustness rather than on kernel-specific optimality. We construct an explicit, n-dependent, sampling distribution that achieves minimax rates for worst-case errors over smoothness classes without requiring knowledge of the kernel. This kernel-agnostic design does improve robustness while retaining optimal rates and extends Briol et al. (2019) to the unbounded case. Which cannot always be easily handled by a change of variables. Our result thus mostly covers unbounded sampling measures such as Gaussian and Student-t distributions, extending beyond compact domains. The results provide both theoretical guarantees and a practical recipe for robust, rate-optimal, randomised quadrature. [The above is mostly stated in the abstract.]
Archive for RKHS
optimal sampling for kernel quadrature on unbounded domains
Posted in Books, Statistics, University life with tags Bayesian quadrature, bois de Boulogne, contraction rate, ERC Synergy Grant, Gaussian processes, minimaxity, Monte Carlo integration, numerical integration, Ocean, probabilistic integration, probabilistic numerics, RKHS, sunset, uncertainty quantification, Université Paris Dauphine on May 22, 2026 by xi'annonparametric ABC [seminar]
Posted in pictures, Statistics, University life with tags ABC, AISTATS 2022, approximate Bayesian inference, Bayesian nonparametrics, g-and-k distributions, maximum mean discrepancy, misspecified model, One World ABC Seminar, RKHS, robustness, University of Warwick, WABC, Wasserstein distance, webinar on June 3, 2022 by xi'an
Puzzle: How do you run ABC when you mistrust the model?! We somewhat considered this question in our misspecified ABC paper with David and Judith. An AISTATS 2022 paper by Harita Dellaporta (Warwick), Jeremias Knoblauch, Theodoros Damoulas (Warwick), and François-Xavier Briol (formerly Warwick) is addressing this same question and Harita presented the paper at the One World ABC webinar yesterday.
It is inspired from Lyddon, Walker & Holmes (2018), who place a nonparametric prior on the generating model, in top of the assumed parametric model (with an intractable likelihood). This induces a push-forward prior on the pseudo-true parameter, that is, the value that brings the parametric family the closest possible to the true distribution of the data. Here defined as a minimum distance parameter, the maximum mean discrepancy (MMD). Choosing RKHS framework allows for a practical implementation, resorting to simulations for posterior realisations from a Dirichlet posterior and from the parametric model, and stochastic gradient for computing the pseudo-true parameter, which may prove somewhat heavy in terms of computing cost.
The paper also containts a consistency result in an ε-contaminated setting (contamination of the assumed parametric family). Comparisons like the above with a fully parametric Wasserstein-ABC approach show that this alter resists better misspecification, as could be expected since the later is not constructed for that purpose.
Next talk is on 23 June by Cosma Shalizi.
holistic framework for ABC
Posted in Books, Statistics, University life with tags ABC, AISTATS, Approximate Bayesian computation, Japan, likelihood-free methods, Okinawa, reproducing kernel Hilbert space, RKHS on April 19, 2019 by xi'anAn AISTATS 2019 paper was recently arXived by Kelvin Hsu and Fabio Ramos. Proposing an ABC method
“…consisting of (1) a consistent surrogate likelihood model that modularizes queries from simulation calls, (2) a Bayesian learning objective for hyperparameters that improves inference accuracy, and (3) a posterior surrogate density and a super-sampling inference algorithm using its closed-form posterior mean embedding.”
While this sales line sounds rather obscure to me, the authors further defend their approach against ABC-MCMC or synthetic likelihood by the points
“that (1) only one new simulation is required at each new parameter θ and (2) likelihood queries do not need to be at parameters where simulations are available.”
using a RKHS approach to approximate the likelihood or the distribution of the summary (statistic) given the parameter (value) θ. Based on the choice of a certain positive definite kernel. (As usual, I do not understand why RKHS would do better than another non-parametric approach, especially since the approach approximates the full likelihood, but I am not a non-parametrician…)
“The main advantage of using an approximate surrogate likelihood surrogate model is that it readily provides a marginal surrogate likelihood quantity that lends itself to a hyper-parameter learning algorithm”
The tolerance ε (and other cyberparameters) are estimated by maximising the approximated marginal likelihood, which happens to be available in the convenient case the prior is an anisotropic Gaussian distribution. For the simulated data in the reference table? But then missing the need for localising the simulations near the posterior? Inference is then conducting by simulating from this approximation. With the common (to RKHS) drawback that the approximation is “bounded and normalized but potentially non-positive”.
the invasion of the stochastic gradients
Posted in Statistics with tags approximate inference, arXiv, Euler discretisation, population Monte Carlo, RKHS, scalable MCMC, stochastic gradient, stochastic gradient descent, variational Bayes methods, Wales on May 10, 2017 by xi'anWithin the same day, I spotted three submissions to arXiv involving stochastic gradient descent, that I briefly browsed on my trip back from Wales:
- Stochastic Gradient Descent as Approximate Bayesian inference, by Mandt, Hoffman, and Blei, where this technique is used as a type of variational Bayes method, where the minimum Kullback-Leibler distance to the true posterior can be achieved. Rephrasing the [scalable] MCMC algorithm of Welling and Teh (2011) as such an approximation.
- Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent, by Arnak Dalalyan, which establishes a convergence of the uncorrected Langevin algorithm to the right target distribution in the sense of the Wasserstein distance. (Uncorrected in the sense that there is no Metropolis step, meaning this is a Euler approximation.) With an extension to the noisy version, when the gradient is approximated eg by subsampling. The connection with stochastic gradient descent is thus tenuous, but Arnak explains the somewhat disappointing rate of convergence as being in agreement with optimisation rates.
- Stein variational adaptive importance sampling, by Jun Han and Qiang Liu, which relates to our population Monte Carlo algorithm, but as a non-parametric version, using RKHS to represent the transforms of the particles at each iteration. The sampling method follows two threads of particles, one that is used to estimate the transform by a stochastic gradient update, and another one that is used for estimation purposes as in a regular population Monte Carlo approach. Deconstructing into those threads allows for conditional independence that makes convergence easier to establish. (A problem we also hit when working on the AMIS algorithm.)
ABC with kernelised regression
Posted in Mountains, pictures, Statistics, Travel, University life with tags 17w5025, ABC, Approximate Bayesian computation, Banff, dimension reduction, Fourier transform, ICML, reproducing kernel Hilbert space, ridge regression, RKHS, summary statistics, Wasserstein distance on February 22, 2017 by xi'an
The exact title of the paper by Jovana Metrovic, Dino Sejdinovic, and Yee Whye Teh is DR-ABC: Approximate Bayesian Computation with Kernel-Based Distribution Regression. It appeared last year in the proceedings of ICML. The idea is to build ABC summaries by way of reproducing kernel Hilbert spaces (RKHS). Regressing such embeddings to the “optimal” choice of summary statistics by kernel ridge regression. With a possibility to derive summary statistics for quantities of interest rather than for the entire parameter vector. The use of RKHS reminds me of Arthur Gretton’s approach to ABC, although I see no mention made of that work in the current paper.
In the RKHS pseudo-linear formulation, the prediction of a parameter value given a sample attached to this value looks like a ridge estimator in classical linear estimation. (I thus wonder at why one would stop at the ridge stage instead of getting the full Bayes treatment!) Things get a bit more involved in the case of parameters (and observations) of interest, as the modelling requires two RKHS, because of the conditioning on the nuisance observations. Or rather three RHKS. Since those involve a maximum mean discrepancy between probability distributions, which define in turn a sort of intrinsic norm, I also wonder at a Wasserstein version of this approach.
What I find hard to understand in the paper is how a large-dimension large-size sample can be managed by such methods with no visible loss of information and no explosion of the computing budget. The authors mention Fourier features, which never rings a bell for me, but I wonder how this operates in a general setting, i.e., outside the iid case. The examples do not seem to go into enough details for me to understand how this massive dimension reduction operates (and they remain at a moderate level in terms of numbers of parameters). I was hoping Jovana Mitrovic could present her work here at the 17w5025 workshop but she sadly could not make it to Banff for lack of funding!
