**E**dwin Fong and Chris Holmes (Oxford) just wrote a paper on Bayesian scalable methods from a M-open perspective. Borrowing from the conformal prediction framework of Vovk et al. (2005) to achieve frequentist coverage for prediction intervals. The method starts with the choice of a conformity measure that measures how well each observation in the sample agrees with the sample. Which is exchangeable and hence leads to a rank statistic from which a p-value can be derived. Which is the empirical cdf associated with the observed conformities. Following Vovk et al. (2005) and Wasserman (2011) Edwin and Chris note that the Bayesian predictive itself acts like a conformity measure. Predictive that can itself be approximated by MCMC and importance sampling (possibly smoothed by Pareto). The paper also expands the setting to partial exchangeable models, renamed group conformal predictions. While reluctant to engage into turning Bayesian solutions into frequentist ones, I can see some worth in deriving both in order to expose discrepancies and hence signal possible issues with models and priors.

## Archive for Bayesian predictive

## Conformal Bayesian Computation

Posted in Books, pictures, Statistics, University life with tags Bayesian predictive, conformal prediction, frequentist coverage, p-value, Pareto smoothed importance sampling, University of Oxford on July 8, 2021 by xi'an## focused Bayesian prediction

Posted in Books, pictures, Statistics, Travel, University life with tags Australia, Bayesian non-parametrics, Bayesian predictive, Casa Matemática Oaxaca, econometrics, likelihood-free inference, Mexico, misspecification, Monash University, One World ABC Seminar, prediction, pseudo-marginal MCMC, score function, webinar on June 3, 2020 by xi'an**I**n this fourth session of our One World ABC Seminar, my friend and coauthor Gael Martin, gave an after-dinner talk on focused Bayesian prediction, more in the spirit of Bissiri et al. than following a traditional ABC approach. because along with Ruben Loaiza-Maya and [my friend and coauthor] David Frazier, they consider the possibility of a (mild?) misspecification of the model. Using thus scoring rules à la Gneiting and Raftery. Gael had in fact presented an earlier version at our workshop in Oaxaca, in November 2018. As in other solutions of that kind, difficulty in weighting the score into a distribution. Although asymptotic irrelevance, direct impact on the current predictions, at least for the early dates in the time series… Further calibration of the set of interest A. Or the focus of the prediction. As a side note the talk perfectly fits the One World likelihood-free seminar as it does not use the likelihood function!

“The very premise of this paper is that, in reality, any choice of predictive class is such that the truth is not contained therein, at which point there is no reason to presume that the expectation of any particular scoring rule will be maximized at the truth or, indeed, maximized by the same predictive distribution that maximizes a different (expected) score.”

This approach requires the proxy class to be close enough to the true data generating model. Or in the word of the authors to be *plausible predictive* models. And to produce the true distribution via the score as it is proper. Or the closest to the true model in the misspecified family. I thus wonder at a possible extension with a non-parametric version, the prior being thus on functionals rather than parameters, if I understand properly the meaning of Π(P_{θ}). (Could the score function be misspecified itself?!) Since the score is replaced with its empirical version, the implementation is resorting to off-the-shelf MCMC. (I wonder for a few seconds if the approach could be seen as a pseudo-marginal MCMC but the estimation is always based on the same observed sample, hence does not directly fit the pseudo-marginal MCMC framework.)

*[Notice: Next talk in the series is tomorrow, 11:30am GMT+1.]*

## a unified treatment of predictive model comparison

Posted in Books, Statistics, University life with tags AIC, Bayesian model comparison, Bayesian predictive, Bourbaki, DIC, Kullback-Leibler divergence, M-open inference, marginal likelihood, posterior predictive, small worlds on June 16, 2015 by xi'an

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

**M**ichael 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 distributions of Bayes factors

Posted in Books, Kids, Statistics with tags Bayes factor, Bayesian predictive, Bayesian tests, posterior predictive on October 8, 2014 by xi'an**O**nce 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

- 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);
- it reproduces the behaviour of the Bayes factor B(x) for values x of the observation similar to the original observation y;
- it does not hide issues of indeterminacy linked with improper priors: the Bayes factor B(x) remains indeterminate, even with a well-defined predictive;
- 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);
- 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.