**O**ur paper with David Frazier, Gael Martin and Judith Rousseau has appeared in print in Biometrika, Volume 105, Issue 3, 1 September 2018, Pages 593–607, almost exactly two years after it was submitted. I am quite glad with the final version, though, and grateful for the editorial input, as the paper clearly characterises the connection between the tolerance level ε and the convergence rate of the summary statistic to its parameter identifying asymptotic mean. Asymptotic in the sample size, that is.

## Archive for summary statistics

## asymptotic properties of ABC now appeared

Posted in Books, Statistics, University life with tags ABC, ABC convergence, Approximate Bayesian computation, approximate Bayesian inference, Biometrika, intractable likelihood, summary statistics on September 1, 2018 by xi'an## ABC’ptotics on-line

Posted in Statistics with tags ABC, ABC convergence, Approximate Bayesian computation, approximate Bayesian inference, Biometrika, intractable likelihood, Paul Fearnhead, summary statistics on June 14, 2018 by xi'anOur paper on Asymptotic properties of ABC with David Frazier, Gael Martin, and Judith Rousseau, is now on-line on the Biometrika webpage. Coincidentally both papers by Wentao Li and Paul Fearnhead on ABC’ptotics are published in the June issue of the journal.

Approximate Bayesian computation allows for statistical analysis using models with intractable likelihoods. In this paper we consider the asymptotic behaviour of the posterior distribution obtained by this method. We give general results on the rate at which the posterior distribution concentrates on sets containing the true parameter, the limiting shape of the posterior distribution, and the asymptotic distribution of the posterior mean. These results hold under given rates for the tolerance used within the method, mild regularity conditions on the summary statistics, and a condition linked to identification of the true parameters. Implications for practitioners are discussed.

## at the Isaac Newton Institute [talks]

Posted in Statistics with tags ABC algorithm, dynamic model, empirical likelihood, INI, Isaac Newton Institute, non-i.i.d. data, summary statistics, Wasserstein distance on July 7, 2017 by xi'an**H**ere are the slides I edited this week [from previous talks by Pierre and Epstein] for the INI Workshop on scalable inference, in connection with our recently completed and submitted paper on ABC with Wasserstein distances:

## MCM 2017

Posted in Statistics with tags ABC, ABC algorithm, ABC consistency, Bayesian model choice, curse of dimensionality, Hilbert curve, MCM 2017, Montréal, population genetics, Québec, random forests, summary statistics, Wasserstein distance on July 3, 2017 by xi'an## automated ABC summary combination

Posted in Books, pictures, Statistics, University life with tags ABC, José Miguel Bernardo, Lasso, posterior distribution, semi-automatic ABC, summary statistics, University of Oxford, Wasserstein distance on March 16, 2017 by xi'an**J**onathan Harrison and Ruth Baker (Oxford University) arXived this morning a paper on the optimal combination of summaries for ABC in the sense of deriving the proper weights in an Euclidean distance involving all the available summaries. The idea is to find the weights that lead to the maximal distance between prior and posterior, in a way reminiscent of Bernardo’s (1979) maximal information principle. Plus a sparsity penalty à la Lasso. The associated algorithm is sequential in that the weights are updated at each iteration. The paper does not get into theoretical justifications but considers instead several examples with limited numbers of both parameters and summary statistics. Which may highlight the limitations of the approach in that handling (and eliminating) a large number of parameters may prove impossible this way, when compared with optimisation methods like random forests. Or summary-free distances between empirical distributions like the Wasserstein distance.

## 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**T**he 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!