## Goodness-of-fit statistics for ABC

Posted in Books, Statistics, University life with tags , , , , , on February 1, 2016 by xi'an

“Posterior predictive checks are well-suited to Approximate Bayesian Computation”

Louisiane Lemaire and her coauthors from Grenoble have just arXived a new paper on designing a goodness-of-fit statistic from ABC outputs. The statistic is constructed from a comparison between the observed (summary) statistics and replicated summary statistics generated from the posterior predictive distribution. This is a major difference with the standard ABC distance, when the replicated summary statistics are generated from the prior predictive distribution. The core of the paper is about calibrating a posterior predictive p-value derived from this distance, since it is not properly calibrated in the frequentist sense that it is not uniformly distributed “under the null”. A point I discussed in an ‘Og entry about Andrews’ book a few years ago.

The paper opposes the average distance between ABC acceptable summary statistics and the observed realisation to the average distance between ABC posterior predictive simulations of summary statistics and the observed realisation. In the simplest case (e.g., without a post-processing of the summary statistics), the main difference between both average distances is that the summary statistics are used twice in the first version: first to select the acceptable values of the parameters and a second time for the average distance. Which makes it biased downwards. The second version is more computationally demanding, especially when deriving the associated p-value. It however produces a higher power under the alternative. Obviously depending on how the alternative is defined, since goodness-of-fit is only related to the null, i.e., to a specific model.

From a general perspective, I do not completely agree with the conclusions of the paper in that (a) this is a frequentist assessment and partakes in the shortcomings of p-values and (b) the choice of summary statistics has a huge impact on the decision about the fit since hardly varying statistics are more likely to lead to a good fit than appropriately varying ones.

## a unified treatment of predictive model comparison

Posted in Books, Statistics, University life with tags , , , , , , , , , on June 16, 2015 by xi'an

“Applying various approximation strategies to the relative predictive performance derived from predictive distributions in frequentist and Bayesian inference yields many of the model comparison techniques ubiquitous in practice, from predictive log loss cross validation to the Bayesian evidence and Bayesian information criteria.”

Michael Betancourt (Warwick) just arXived a paper formalising predictive model comparison in an almost Bourbakian sense! Meaning that he adopts therein a very general representation of the issue, with minimal assumptions on the data generating process (excluding a specific metric and obviously the choice of a testing statistic). He opts for an M-open perspective, meaning that this generating process stands outside the hypothetical statistical model or, in Lindley’s terms, a small world. Within this paradigm, the only way to assess the fit of a model seems to be through the predictive performances of that model. Using for instance an f-divergence like the Kullback-Leibler divergence, based on the true generated process as the reference. I think this however puts a restriction on the choice of small worlds as the probability measure on that small world has to be absolutely continuous wrt the true data generating process for the distance to be finite. While there are arguments in favour of absolutely continuous small worlds, this assumes a knowledge about the true process that we simply cannot gather. Ignoring this difficulty, a relative Kullback-Leibler divergence can be defined in terms of an almost arbitrary reference measure. But as it still relies on the true measure, its evaluation proceeds via cross-validation “tricks” like jackknife and bootstrap. However, on the Bayesian side, using the prior predictive links the Kullback-Leibler divergence with the marginal likelihood. And Michael argues further that the posterior predictive can be seen as the unifying tool behind information criteria like DIC and WAIC (widely applicable information criterion). Which does not convince me towards the utility of those criteria as model selection tools, as there is too much freedom in the way approximations are used and a potential for using the data several times.

## Posterior predictive p-values and the convex order

Posted in Books, Statistics, University life with tags , , , , , , , , , on December 22, 2014 by xi'an

Patrick Rubin-Delanchy and Daniel Lawson [of Warhammer fame!] recently arXived a paper we had discussed with Patrick when he visited Andrew and I last summer in Paris. The topic is the evaluation of the posterior predictive probability of a larger discrepancy between data and model

$\mathbb{P}\left( f(X|\theta)\ge f(x^\text{obs}|\theta) \,|\,x^\text{obs} \right)$

which acts like a Bayesian p-value of sorts. I discussed several times the reservations I have about this notion on this blog… Including running one experiment on the uniformity of the ppp while in Duke last year. One item of those reservations being that it evaluates the posterior probability of an event that does not exist a priori. Which is somewhat connected to the issue of using the data “twice”.

“A posterior predictive p-value has a transparent Bayesian interpretation.”

Another item that was suggested [to me] in the current paper is the difficulty in defining the posterior predictive (pp), for instance by including latent variables

$\mathbb{P}\left( f(X,Z|\theta)\ge f(x^\text{obs},Z^\text{obs}|\theta) \,|\,x^\text{obs} \right)\,,$

which reminds me of the multiple possible avatars of the BIC criterion. The question addressed by Rubin-Delanchy and Lawson is how far from the uniform distribution stands this pp when the model is correct. The main result of their paper is that any sub-uniform distribution can be expressed as a particular posterior predictive. The authors also exhibit the distribution that achieves the bound produced by Xiao-Li Meng, Namely that

$\mathbb{P}(P\le \alpha) \le 2\alpha$

where P is the above (top) probability. (Hence it is uniform up to a factor 2!) Obviously, the proximity with the upper bound only occurs in a limited number of cases that do not validate the overall use of the ppp. But this is certainly a nice piece of theoretical work.

## reliable ABC model choice via random forests

Posted in pictures, R, Statistics, University life with tags , , , , , , , on October 29, 2014 by xi'an

After a somewhat prolonged labour (!), we have at last completed our paper on ABC model choice with random forests and submitted it to PNAS for possible publication. While the paper is entirely methodological, the primary domain of application of ABC model choice methods remains population genetics and the diffusion of this new methodology to the users is thus more likely via a media like PNAS than via a machine learning or statistics journal.

When compared with our recent update of the arXived paper, there is not much different in contents, as it is mostly an issue of fitting the PNAS publication canons. (Which makes the paper less readable in the posted version [in my opinion!] as it needs to fit the main document within the compulsory six pages, relegated part of the experiments and of the explanations to the Supplementary Information section.)

## posterior predictive distributions of Bayes factors

Posted in Books, Kids, Statistics with tags , , , on October 8, 2014 by xi'an

Once a Bayes factor B(y)  is computed, one needs to assess its strength. As repeated many times here, Jeffreys’ scale has no validation whatsoever, it is simply a division of the (1,∞) range into regions of convenience. Following earlier proposals in the literature (Box, 1980; García-Donato and Chen, 2005; Geweke and Amisano, 2008), an evaluation of this strength within the issue at stake, i.e. the comparison of two models, can be based on the predictive distribution. While most authors (like García-Donato and Chen) consider the prior predictive, I think using the posterior predictive distribution is more relevant since

1. it exploits the information contained in the data y, thus concentrates on a region of relevance in the parameter space(s), which is especially interesting in weakly informative settings (even though we should abstain from testing in those cases, dixit Andrew);
2. it reproduces the behaviour of the Bayes factor B(x) for values x of the observation similar to the original observation y;
3. it does not hide issues of indeterminacy linked with improper priors: the Bayes factor B(x) remains indeterminate, even with a well-defined predictive;
4. it does not separate between errors of type I and errors of type II but instead uses the natural summary provided by the Bayesian analysis, namely the predictive distribution π(x|y);
5. as long as the evaluation is not used to reach a decision, there is no issue of “using the data twice”, we are simply producing an estimator of the posterior loss, for instance the (posterior) probability of selecting the wrong model. The Bayes factor B(x) is thus functionally  independent of y, while x is probabilistically dependent on y.

Note that, even though probabilities of errors of type I and errors of type II can be computed, they fail to account for the posterior probabilities of both models. (This is the delicate issue with the solution of García-Donato and Chen.) Another nice feature is that the predictive distribution of the Bayes factor can be computed even in complex settings where ABC needs to be used.

## ABC model choice via random forests [expanded]

Posted in Statistics, University life with tags , , , , , , , , , , , on October 1, 2014 by xi'an

Today, we arXived a second version of our paper on ABC model choice with random forests. Or maybe [A]BC model choice with random forests. Since the random forest is built on a simulation from the prior predictive and no further approximation is used in the process. Except for the computation of the posterior [predictive] error rate. The update wrt the earlier version is that we ran massive simulations throughout the summer, on existing and new datasets. In particular, we have included a Human dataset extracted from the 1000 Genomes Project. Made of 51,250 SNP loci. While this dataset is not used to test new evolution scenarios, we compared six out-of-Africa scenarios, with a possible admixture for Americans of African ancestry. The scenario selected by a random forest procedure posits a single out-of-Africa colonization event with a secondary split into a European and an East Asian population lineages, and a recent genetic admixture between African and European lineages, for Americans of African origin. The procedure reported a high level of confidence since the estimated posterior error rate is equal to zero. The SNP loci were carefully selected using the following criteria: (i) all individuals have a genotype characterized by a quality score (GQ)>10, (ii) polymorphism is present in at least one of the individuals in order to fit the SNP simulation algorithm of Hudson (2002) used in DIYABC V2 (Cornuet et al., 2014), (iii) the minimum distance between two consecutive SNPs is 1 kb in order to minimize linkage disequilibrium between SNP, and (iv) SNP loci showing significant deviation from Hardy-Weinberg equilibrium at a 1% threshold in at least one of the four populations have been removed.

In terms of random forests, we optimised the size of the bootstrap subsamples for all of our datasets. While this optimisation requires extra computing time, it is negligible when compared with the enormous time taken by a logistic regression, which is [yet] the standard ABC model choice approach. Now the data has been gathered, it is only a matter of days before we can send the paper to a journal

## ABC model choice by random forests [guest post]

Posted in pictures, R, Statistics, University life with tags , , , , , , , , , , on August 11, 2014 by xi'an

[Dennis Prangle sent me his comments on our ABC model choice by random forests paper. Here they are! And I appreciate very much contributors commenting on my paper or others, so please feel free to join.]

This paper proposes a new approach to likelihood-free model choice based on random forest classifiers. These are fit to simulated model/data pairs and then run on the observed data to produce a predicted model. A novel “posterior predictive error rate” is proposed to quantify the degree of uncertainty placed on this prediction. Another interesting use of this is to tune the threshold of the standard ABC rejection approach, which is outperformed by random forests.

The paper has lots of thought-provoking new ideas and was an enjoyable read, as well as giving me the encouragement I needed to read another chapter of the indispensable Elements of Statistical Learning However I’m not fully convinced by the approach yet for a few reasons which are below along with other comments.

Alternative schemes

The paper shows that random forests outperform rejection based ABC. I’d like to see a comparison to more efficient ABC model choice algorithms such as that of Toni et al 2009. Also I’d like to see if the output of random forests could be used as summary statistics within ABC rather than as a separate inference method.

Posterior predictive error rate (PPER)

This is proposed to quantify the performance of a classifier given a particular data set. The PPER is the proportion of times the classifier’s most favoured model is incorrect for simulated model/data pairs drawn from an approximation to the posterior predictive. The approximation is produced by a standard ABC analysis.

Misclassification could be due to (a) a poor classifier or (b) uninformative data, so the PPER aggregrates these two sources of uncertainty. I think it is still very desirable to have an estimate of the uncertainty due to (b) only i.e. a posterior weight estimate. However the PPER is useful. Firstly end users may sometimes only care about the aggregated uncertainty. Secondly relative PPER values for a fixed dataset are a useful measure of uncertainty due to (a), for example in tuning the ABC threshold. Finally, one drawback of the PPER is the dependence on an ABC estimate of the posterior: how robust are the results to the details of how this is obtained?

Classification

This paper illustrates an important link between ABC and machine learning classification methods: model choice can be viewed as a classification problem. There are some other links: some classifiers make good model choice summary statistics (Prangle et al 2014) or good estimates of ABC-MCMC acceptance ratios for parameter inference problems (Pham et al 2014). So the good performance random forests makes them seem a generally useful tool for ABC (indeed they are used in the Pham et al al paper).