Archive for consistence

improved approximate-Bayesian model-choice method for estimating shared evolutionary history [reply from the author]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , on June 3, 2014 by xi'an

[Here is a very kind and detailed reply from Jamie Oakes to the comments I made on his ABC paper a few days ago:]

First of all, many thanks for your thorough review of my pre-print! It is very helpful and much appreciated. I just wanted to comment on a few things you address in your post.

I am a little confused about how my replacement of continuous uniform probability distributions with gamma distributions for priors on several parameters introduces a potentially crippling number of hyperparameters. Both uniform and gamma distributions have two parameters. So, the new model only has one additional hyperparameter compared to the original msBayes model: the concentration parameter on the Dirichlet process prior on divergence models. Also, the new model offers a uniform prior over divergence models (though I don’t recommend it).

Your comment about there being no new ABC technique is 100% correct. The model is new, the ABC numerical machinery is not. Also, your intuition is correct, I do not use the divergence times to calculate summary statistics. I mention the divergence times in the description of the ABC algorithm with the hope of making it clear that the times are scaled (see Equation (12)) prior to the simulation of the data (from which the summary statistics are calculated). This scaling is simply to go from units proportional to time, to units that are proportional to the expected number of mutations. Clearly, my attempt at clarity only created unnecessary opacity. I’ll have to make some edits.

Regarding the reshuffling of the summary statistics calculated from different alignments of sequences, the statistics are not exchangeable. So, reshuffling them in a manner that is not conistent across all simulations and the observed data is not mathematically valid. Also, if elements are exchangeable, their order will not affect the likelihood (or the posterior, barring sampling error). Thus, if our goal is to approximate the likelihood, I would hope the reshuffling would also have little affect on the approximate posterior (otherwise my approximation is not so good?).

You are correct that my use of “bias” was not well defined in reference to the identity line of my plots of the estimated vs true probability of the one-divergence model. I think we can agree that, ideally (all assumptions are met), the estimated posterior probability of a model should estimate the probability that the model is correct. For large numbers of simulation
replicates, the proportion of the replicates for which the one-divergence model is true will approximate the probability that the one-divergence model is correct. Thus, if the method has the desirable (albeit “frequentist”) behavior such that the estimated posterior probability of the one-divergence model is an unbiased estimate of the probability that the one-divergence model is correct, the points should fall near the identity line. For example, let us say the method estimates a posterior probability of 0.90 for the one-divergence model for 1000 simulated datasets. If the method is accurately estimating the probability that the one-divergence model is the correct model, then the one-divergence model should be the true model for approximately 900 of the 1000 datasets. Any trend away from the identity line indicates the method is biased in the (frequentist) sense that it is not correctly estimating the probability that the one-divergence model is the correct model. I agree this measure of “bias” is frequentist in nature. However, it seems like a worthwhile goal for Bayesian model-choice methods to have good frequentist properties. If a method strongly deviates from the identity line, it is much more difficult to interpret the posterior probabilites that it estimates. Going back to my example of the posterior probability of 0.90 for 1000 replicates, I would be alarmed if the model was true in only 100 of the replicates.

My apologies if my citation of your PNAS paper seemed misleading. The citation was intended to be limited to the context of ABC methods that use summary statistics that are insufficient across the models under comparison (like msBayes and the method I present in the paper). I will definitely expand on this sentence to make this clearer in revisions. Thanks!

Lastly, my concluding remarks in the paper about full-likelihood methods in this domain are not as lofty as you might think. The likelihood function of the msBayes model is tractable, and, in fact, has already been derived and implemented via reversible-jump MCMC (albeit, not readily available yet). Also, there are plenty of examples of rich, Kingman-coalescent models implemented in full-likelihood Bayesian frameworks. Too many to list, but a lot of them are implemented in the BEAST software package. One noteworthy example is the work of Bryant et al. (2012, Molecular Biology and Evolution, 29(8), 1917–32) that analytically integrates over all gene trees for biallelic markers under the coalescent.

Estimating the number of species

Posted in Statistics with tags , , , , , on November 20, 2009 by xi'an

Bayesian Analysis just published on-line a paper by Hongmei Zhang and Hal Stern on a (new) Bayesian analysis of the problem of estimating the number of unseen species within a population. This problem has always fascinated me, as it seems at first sight to be an impossible problem, how can you estimate the number of species you do not know?! The approach relates to capture-recapture models, with an extra hierarchical layer for the species. The Bayesian analysis of the model obviously makes a lot of sense, with the prior modelling being quite influential. Zhang and Stern use a hierarchical Dirichlet prior on the capture probabilities, \theta_i, when the captures follow a multinomial model

y|\theta,S \sim \mathcal{M}(N, \theta_1,\ldots,\theta_S)

where N=\sum_i y_i the total number of observed individuals,

\mathbf{\theta}|S \sim \mathcal{D}(\alpha,\ldots,\alpha)

and

\pi(\alpha,S) = f(1-f)^{S-S_\text{min}} \alpha^{-3/2}

forcing the coefficients of the Dirichlet prior towards zero. The paper also covers predictive design, analysing the capture effort corresponding to a given recovery rate of species. The overall approach is not immensely innovative in its methodology, the MCMC part being rather straightforward, but the predictive abilities of the model are nonetheless interesting.

The previously accepted paper in Bayesian Analysis is a note by Ron Christensen about an inconsistent Bayes estimator that you may want to use in an advanced Bayesian class. For all practical purposes, it should not overly worry you, since the example involves a sampling distribution that is normal when its parameter is irrational and is Cauchy otherwise. (The prior is assumed to be absolutely continuous wrt the Lebesgue measure and it thus gives mass zero to the set of rational numbers \mathbb{Q}. The fact that \mathbb{Q} is dense in \mathbb{R} is irrelevant from a measure-theoretic viewpoint.)