Archive for Bayes factors

Bayesian parameter estimation versus model comparison

Posted in Books, pictures, Statistics with tags , , , , , , on December 5, 2016 by xi'an

John Kruschke [of puppies’ fame!] wrote a paper in Perspectives in Psychological Science a few years ago on the comparison between two Bayesian approaches to null hypotheses. Of which I became aware through a X validated question that seemed to confuse Bayesian parameter estimation with Bayesian hypothesis testing.

“Regardless of the decision rule, however, the primary attraction of using parameter estimation to assess null values is that the an explicit posterior distribution reveals the relative credibility of all the parameter values.” (p.302)

After reading this paper, I realised that Kruschke meant something completely different, namely that a Bayesian approach to null hypothesis testing could operate from the posterior on the corresponding parameter, rather than to engage into formal Bayesian model comparison (null versus the rest of the World). The notion is to check whether or not the null value stands within the 95% [why 95?] HPD region [modulo a buffer zone], which offers the pluses of avoiding a Dirac mass at the null value and a long-term impact of the prior tails on the decision, with the minus of replacing the null with a tolerance region around the null and calibrating the rejection level. This opposition is thus a Bayesian counterpart of running tests on point null hypotheses either by Neyman-Pearson procedures or by confidence intervals. Note that in problems with nuisance parameters this solution requires a determination of the 95% HPD region associated with the marginal on the parameter of interest, which may prove a challenge.

“…the measure provides a natural penalty for vague priors that allow a broad range of parameter values, because a vague prior dilutes credibility across a broad range of parameter values, and therefore the weighted average is also attenuated.” (p. 306)

While I agree with most of the critical assessment of Bayesian model comparison, including Kruschke’s version of Occam’s razor [and Lindley’s paradox] above, I do not understand how Bayesian model comparison fails to return a full posterior on both the model indices [for model comparison] and the model parameters [for estimation]. To state that it does not because the Bayes factor only depends on marginal likelihoods (p.307) sounds unfair if only because most numerical techniques to approximate the Bayes factors rely on preliminary simulations of the posterior. The point that the Bayes factor strongly depends on the modelling of the alternative model is well-taken, albeit the selection of the null in the “estimation” approach does depend as well on this alternative modelling. Which is an issue if one ends up accepting the null value and running a Bayesian analysis based on this null value.

“The two Bayesian approaches to assessing null values can be unified in a single hierarchical model.” (p.308)

Incidentally, the paper briefly considers a unified modelling that can be interpreted as a mixture across both models, but this mixture representation completely differs from ours [where we also advocate estimation to replace testing] since the mixture is at the likelihood x prior level, as in O’Neill and Kypriaos.

ISBA 2016 [#4]

Posted in pictures, Running, Statistics, Travel with tags , , , , , , , , , , on June 17, 2016 by xi'an

As an organiser of the ABC session (along with Paul Fearnhead), I was already aware of most results behind the talks, but nonetheless got some new perspectives from the presentations. Ewan Cameron discussed a two-stage ABC where the first step is actually an indirect inference inference, which leads to a more efficient ABC step. With applications to epidemiology. Lu presented extensions of his work with Paul Fearnhead, incorporating regression correction à la Beaumont to demonstrate consistency and using defensive sampling to control importance sampling variance. (While we are working on a similar approach, I do not want to comment on the consistency part, but I missed how defensive sampling can operate in complex ABC settings, as it requires advanced knowledge on the target to be effective.) And Ted Meeds spoke about two directions for automatising ABC (as in the ABcruise), from incorporating the pseudo-random generator into the representation of the ABC target, to calling for deep learning advances. The inclusion of random generators in the transform is great, provided they can remain black boxes as otherwise they require recoding. (This differs from quasi-Monte Carlo ABC, which aims at reducing the variability due to sheer noise.) It took me a little while, but I eventually understood why Jan Haning saw this inclusion as a return to fiducial inference!

Merlise Clyde gave a wide-ranging plenary talk on (linear) model selection that looked at a large range of priors under the hat of generalised confluent hypergeometric priors over the mixing scale in Zellner’s g-prior. Some were consistent under one or both models, maybe even for misspecified models. Some parts paralleled my own talk on the foundations of Bayesian tests, no wonder since I mostly give a review before launching into a criticism of the Bayes factor. Since I think this may be a more productive perspective than trying to over-come the shortcomings of Bayes factors in weakly informative settings. Some comments at the end of Merlise’s talk were loosely connected to this view in that they called for an unitarian perspective [rather than adapting a prior to a specific inference problem] with decision-theoretic backup. Conveniently the next session was about priors and testing, obviously connected!, with Leo Knorr-Held considering g-priors for the Cox model, Kerrie Mengersen discussing priors for over-fitted mixtures and HMMs, and Dan Simpson entertaining us with his quest of a prior for a point process, eventually reaching PC priors.

Using MCMC output to efficiently estimate Bayes factors

Posted in Books, R, Statistics, University life with tags , , , , on May 19, 2016 by xi'an

As I was checking for software to answer a query on X validated about generic Bayes factor derivation, I came across an R software called BayesFactor, which only applies in regression settings and relies on the Savage-Dickey representation of the Bayes factor


when the null hypothesis writes as θ=θ⁰ (and possibly additional nuisance parameters with [roughly speaking] an independent prior). As we discussed in our paper with Jean-Michel Marin [which got ignored by large!], this representation of the Bayes factor is based on picking a very specific version of the prior, or more exactly of three prior densities. Assuming such versions are selected, I wonder at the performances of this approximation, given that it involves approximating the marginal posterior at θ⁰….

“To ensure that the Bayes factor we compute using the Savage–Dickey ratio is the the ratio of marginal densities that we intend, the condition (…) is easily met by models which specify priors in which the nuisance parameters are independent of the parameters of interest.” Morey et al. (2011)

First, when reading Morey at al. (2011), I realised (a wee bit late!) that Chib’s method is nothing but a version of the Savage-Dickey representation when the marginal posterior can be estimated in a parametric (Rao-Blackwellised) way. However, outside hierarchical models based on conjugate priors such parametric approximations are intractable and non-parametric versions must be invoked instead, which necessarily degrades the quality of the method. A degradation that escalates with the dimension of the parameter θ. In addition, I am somewhat perplexed by the use of a Rao-Blackwell argument in the setting of the Dickey-Savage representation. Indeed this representation assumes that

\pi_1(\psi|\theta_0)=\pi_0(\psi) \  \ \text{or}\quad \pi_1(\theta_0,\psi)=\pi_1(\theta_0)\pi_0(\psi)

which means that [the specific version of] the conditional density of θ⁰ given ψ should not depend on the nuisance parameter. But relying on a Rao-Blackwellisation leads to estimate the marginal posterior via full conditionals. Of course, θ given ψ and y may depend on ψ, but still… Morey at al. (2011) advocate the recourse to Chib’s formula as optimal but this obviously requires the full conditional to be available. They acknowledge this point as moot, since it is sufficient from their perspective to specify a conjugate prior. They consider this to be a slight modification of the model (p.377). However, I see the evaluation of an estimated density at a single (I repeat, single!) point as being the direst part of the method as it is clearly more sensitive to approximations that the evaluation of a whole integral, since the later incorporates an averaging effect by definition. Hence, even if this method was truly available for all models, I would be uncertain of its worth when compared with other methods, except the harmonic mean estimator of course!

On the side, Morey at al. (2011) study a simple one-sample t test where they use an improper prior on the nuisance parameter σ, under both models. While the Savage-Dickey representation is correct in this special case, I fail to see why the identity would apply in every case under an improper prior. In particular, independence does not make sense with improper priors. The authors also indicate the possible use of this Bayes factor approximation for encompassing models. At first, I thought this could be most useful in our testing by mixture framework where we define an encompassing model as a mixture. However, I quickly realised that using a Beta Be(a,a) prior on the weight α with a<1 leads to an infinite density value at both zero and one, hence cannot be compatible with a Savage-Dickey representation of the Bayes factor.

reversible chain[saw] massacre

Posted in Books, pictures, R, Statistics, University life with tags , , , , , , , , on May 16, 2016 by xi'an

A paper in Nature this week that uses reversible-jump MCMC, phylogenetic trees, and Bayes factors. And that looks at institutionalised or ritual murders in Austronesian cultures. How better can it get?!

“by applying Bayesian phylogenetic methods (…) we find strong support for models in which human sacrifice stabilizes social stratification once stratification has arisen, and promotes a shift to strictly inherited class systems.” Joseph Watts et al.

The aim of the paper is to establish that societies with human sacrifices are more likely to have become stratified and stable than societies without such niceties. The hypothesis to be tested is then about the evolution towards more stratified societies rather the existence of a high level of stratification.

“The social control hypothesis predicts that human sacrifice (i) co-evolves with social stratification, (ii) increases the chance of a culture gaining social stratification, and (iii) reduces the chance of a culture losing social stratification once stratification has arisen.” Joseph Watts et al.

The methodological question is then how can this be tested when considering those are extinct societies about which little is known. Grouping together moderate and high stratification societies against egalitarian societies, the authors tested independence of both traits versus dependence, with a resulting Bayes factor of 3.78 in favour of the latest. Other hypotheses of a similar flavour led to Bayes factors in the same range. Which is thus not overwhelming. Actually, given that the models are quite simplistic, I do not agree that those Bayes factors prove anything of the magnitude of such anthropological conjectures. Even if the presence/absence of human sacrifices is confirmed in all of the 93 societies, and if the stratification of the cultures is free from uncertainties, the evolutionary part is rather involved, from my neophyte point of view: the evolutionary structure (reproduced above) is based on a sample of 4,200 trees based on Bayesian analysis of Austronesian basic vocabulary items, followed by a call to the BayesTrait software to infer about evolution patterns between stratification levels, concluding (with p-values!) at a phylogenetic structure of the data. BayesTrait was also instrumental in deriving MLEs for the various transition rates, “in order to inform our choice of priors” (!). BayesTrait has an MCMC function used by the authors “to test for correlated evolution between traits” and derive the above Bayes factors. Using a stepping-stone method I am unaware of. And 10⁹ iterations (repeated 3 times for checking consistency)… Reversible jump is apparently used to move between constrained and unconstrained models, leading to the pie charts at the inner nodes of the above picture. Again a by-product of BayesTrait. The trees on the left and the right are completely identical, the difference being in the inference about stratification evolution (right) and sacrifice evolution (left). While the overall hypothesis makes sense at my layman level (as a culture has to be stratified enough to impose sacrifices from its members), I am not convinced that this involved statistical analysis brings that strong a support. (But it would make a fantastic topic for an undergraduate or a Master thesis!)

contemporary issues in hypothesis testing

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , on May 3, 2016 by xi'an

hipocontemptNext Fall, on 15-16 September, I will take part in a CRiSM workshop on hypothesis testing. In our department in Warwick. The registration is now open [until Sept 2] with a moderate registration free of £40 and a call for posters. Jim Berger and Joris Mulder will both deliver a plenary talk there, while Andrew Gelman will alas give a remote talk from New York. (A terrific poster by the way!)

seminar in Harvard

Posted in Statistics, Travel with tags , , , , , , , , , , on March 16, 2016 by xi'an

harvard2103Next week, I will be in Harvard Monday and Tuesday, visiting friends in the Department of Statistics and giving a seminar. The slides for the talk will be quite similar to those of my talk in Bristol, a few weeks ago. Hopefully, there will not be too much overlap between both audiences! And hopefully I’ll manage to get to my conclusion before all hell breaks loose (which is why I strategically set my conclusion in the early slides!)

read paper [in Bristol]

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on January 29, 2016 by xi'an

Clifton & Durdham Downs, Bristol, Sept. 25, 2012I went to give a seminar in Bristol last Friday and I chose to present the testing with mixture paper. As we are busy working on the revision, I was eagerly looking for comments and criticisms that could strengthen this new version. As it happened, the (Bristol) Bayesian Cake (Reading) Club had chosen our paper for discussion, two weeks in a row!, hence the title!, and I got invited to join the group the morning prior to the seminar! This was, of course, most enjoyable and relaxed, including an home-made cake!, but also quite helpful in assessing our arguments in the paper. One point of contention or at least of discussion was the common parametrisation between the components of the mixture. Although all parametrisations are equivalent from a single component point of view, I can [almost] see why using a mixture with the same parameter value on all components may impose some unsuspected constraint on that parameter. Even when the parameter is the same moment for both components. This still sounds like a minor counterpoint in that the weight should converge to either zero or one and hence eventually favour the posterior on the parameter corresponding to the “true” model.

Another point that was raised during the discussion is the behaviour of the method under misspecification or for an M-open framework: when neither model is correct does the weight still converge to the boundary associated with the closest model (as I believe) or does a convexity argument produce a non-zero weight as it limit (as hinted by one example in the paper)? I had thought very little about this and hence had just as little to argue though as this does not sound to me like the primary reason for conducting tests. Especially in a Bayesian framework. If one is uncertain about both models to be compared, one should have an alternative at the ready! Or use a non-parametric version, which is a direction we need to explore deeper before deciding it is coherent and convergent!

A third point of discussion was my argument that mixtures allow us to rely on the same parameter and hence the same prior, whether proper or not, while Bayes factors are less clearly open to this interpretation. This was not uniformly accepted!

Thinking afresh about this approach also led me to broaden my perspective on the use of the posterior distribution of the weight(s) α: while previously I had taken those weights mostly as a proxy to the posterior probabilities, to be calibrated by pseudo-data experiments, as for instance in Figure 9, I now perceive them primarily as the portion of the data in agreement with the corresponding model [or hypothesis] and more importantly as a solution for staying away from a Neyman-Pearson-like decision. Or error evaluation. Usually, when asked about the interpretation of the output, my answer is to compare the behaviour of the posterior on the weight(s) with a posterior associated with a sample from each model. Which does sound somewhat similar to posterior predictives if the samples are simulated from the associated predictives. But the issue was not raised during the visit to Bristol, which possibly reflects on how unfrequentist the audience was [the Statistics group is], as it apparently accepted with no further ado the use of a posterior distribution as a soft assessment of the comparative fits of the different models. If not necessarily agreeing the need of conducting hypothesis testing (especially in the case of the Pima Indian dataset!).