nonparametric ABC [seminar]

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

An 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

Within the same day, I spotted three submissions to arXiv involving stochastic gradient descent, that I briefly browsed on my trip back from Wales:

1. 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.
2. 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.
3. 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

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!

control functionals for Monte Carlo integration

A paper on control variates by Chris Oates, Mark Girolami (Warwick) and Nicolas Chopin (CREST) appeared in a recent issue of Series B. I had read and discussed the paper with them previously and the following is a set of comments I wrote at some stage, to be taken with enough gains of salt since Chris, Mark and Nicolas answered them either orally or in the paper. Note also that I already discussed an earlier version, with comments that are not necessarily coherent with the following ones! [Thanks to the busy softshop this week, I resorted to publish some older drafts, so mileage can vary in the coming days.]

First, it took me quite a while to get over the paper, mostly because I have never worked with reproducible kernel Hilbert spaces (RKHS) before. I looked at some proofs in the appendix and at the whole paper but could not spot anything amiss. It is obviously a major step to uncover a manageable method with a rate that is lower than √n. When I set my PhD student Anne Philippe on the approach via Riemann sums, we were quickly hindered by the dimension issue and could not find a way out. In the first versions of the nested sampling approach, John Skilling had also thought he could get higher convergence rates before realising the Monte Carlo error had not disappeared and hence was keeping the rate at the same √n speed.

The core proof in the paper leading to the 7/12 convergence rate relies on a mathematical result of Sun and Wu (2009) that a certain rate of regularisation of the function of interest leads to an average variance of order 1/6. I have no reason to mistrust the result (and anyway did not check the original paper), but I am still puzzled by the fact that it almost immediately leads to the control variate estimator having a smaller order variance (or at least variability). On average or in probability. (I am also uncertain on the possibility to interpret the boxplot figures as establishing super-√n speed.)

Another thing I cannot truly grasp is how the control functional estimator of (7) can be both a mere linear recombination of individual unbiased estimators of the target expectation and an improvement in the variance rate. I acknowledge that the coefficients of the matrices are functions of the sample simulated from the target density but still…

Another source of inner puzzlement is the choice of the kernel in the paper, which seems too simple to be able to cover all problems despite being used in every illustration there. I see the kernel as centred at zero, which means a central location must be know, decreasing to zero away from this centre, so possibly missing aspects of the integrand that are too far away, and isotonic in the reference norm, which also seems to preclude some settings where the integrand is not that compatible with the geometry.

I am equally nonplussed by the existence of a deterministic bound on the error, although it is not completely deterministic, depending on the values of the reproducible kernel at the points of the sample. Does it imply anything restrictive on the function to be integrated?

A side remark about the use of intractable in the paper is that, given the development of a whole new branch of computational statistics handling likelihoods that cannot be computed at all, intractable should possibly be reserved for such higher complexity models.