## posterior predictive checks for admixture models

**I**n a posting coincidence, just a few days after we arXived our paper on ABC model choice with random forests, where we use posterior predictive errors for assessing the variability of the random forest procedure, David Mimno, David Blei, and Barbara Engelhardt arXived a paper on posterior predictive checks to quantify lack-of-fit in admixture models of latent population structure, which deals with similar data and models, while also using the posterior predictive as a central tool. (*Marginalia:* the paper is a wee bit difficult to read [esp. with France-Germany playing in the airport bar!] as the modelling is only clearly described at the very end. I suspect this arXived version was put together out of a submission to a journal like Nature or PNAS, with mentions of a Methods section that does not appear here and of Supplementary Material that turned into subsections of the Discussion section.)

**T**he dataset are genomic datasets made of SNPs (single nucleotide polymorphisms). For instance, the first (HapMap) dataset corresponds to 1,043 individuals and 468,167 SNPs. The model is simpler than Kingman’s coalescent, hence its likelihood does not require ABC steps to run inference. The admixture model in the paper is essentially a mixture model over ancestry indices with individual dependent weights with Bernoulli observations, hence resulting into a completed likelihood of the form

(which looks more formidable than it truly is!). Regular Bayesian inference is thus possible in this setting, implementing e.g. Gibbs sampling. The authors chose instead to rely on EM and thus derived the maximum likelihood estimators of the (many) parameters of the admixture. And of the latent variables **z**. Their posterior predictive check is based on the simulation of pseudo-observations (as in ABC!) from the above likelihood, with parameters and latent variables replaced with their EM estimates (unlike ABC). There is obviously some computational reason in doing this instead of simulating from the posterior, albeit implicit in the paper. I am however slightly puzzled by the conditioning on the latent variable estimate **ẑ**, as its simulation is straightforward and as a latent variable is more a missing observation than a parameter. Given those 30 to 100 replications of the data, an empirical distribution of a discrepancy function is used to assess whether or not the equivalent discrepancy for the observation is an outlier. If so, the model is not appropriate for the data. (Interestingly, the discrepancy is measured via the Bayes factor of z-scores.)

**T**he connection with our own work is that the construction of discrepancy measures proposed in this paper could be added to our already large collection of summary statistics to check to potential impact in model comparison, i.e. for a significant contribution to the random forest nodes. Conversely, the most significant summary statistics could then be tested as discrepancy measures. Or, more in tune with our Series B paper on the proper selection of summary variables, the distribution of those discrepancy measures could be compared across potential models. Assuming this does not take too much computing power…

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