Archive for Bayesian model comparison

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.

ABC by subset simulation

Posted in Books, Statistics, Travel with tags , , , , , , , , , on August 25, 2016 by xi'an

Last week, Vakilzadeh, Beck and Abrahamsson arXived a paper entitled “Using Approximate Bayesian Computation by Subset Simulation for Efficient Posterior Assessment of Dynamic State-Space Model Classes”. It follows an earlier paper by Beck and co-authors on ABC by subset simulation, paper that I did not read. The model of interest is a hidden Markov model with continuous components and covariates (input), e.g. a stochastic volatility model. There is however a catch in the definition of the model, namely that the observable part of the HMM includes an extra measurement error term linked with the tolerance level of the ABC algorithm. Error term that is dependent across time, the vector of errors being within a ball of radius ε. This reminds me of noisy ABC, obviously (and as acknowledged by the authors), but also of some ABC developments of Ajay Jasra and co-authors. Indeed, as in those papers, Vakilzadeh et al. use the raw data sequence to compute their tolerance neighbourhoods, which obviously bypasses the selection of a summary statistic [vector] but also may drown signal under noise for long enough series.

“In this study, we show that formulating a dynamical system as a general hierarchical state-space model enables us to independently estimate the model evidence for each model class.”

Subset simulation is a nested technique that produces a sequence of nested balls (and related tolerances) such that the conditional probability to be in the next ball given the previous one remains large enough. Requiring a new round of simulation each time. This is somewhat reminding me of nested sampling, even though the two methods differ. For subset simulation, estimating the level probabilities means that there also exists a converging (and even unbiased!) estimator for the evidence associated with different tolerance levels. Which is not a particularly natural object unless one wants to turn it into a tolerance selection principle, which would be quite a novel perspective. But not one adopted in the paper, seemingly. Given that the application section truly compares models I must have missed something there. (Blame the long flight from San Francisco to Sydney!) Interestingly, the different models as in Table 4 relate to different tolerance levels, which may be an hindrance for the overall validation of the method.

I find the subsequent part on getting rid of uncertain prediction error model parameters of lesser [personal] interest as it essentially replaces the marginal posterior on the parameters of interest by a BIC approximation, with the unsurprising conclusion that “the prior distribution of the nuisance parameter cancels out”.

ABC for repulsive point processes

Posted in Books, pictures, Statistics, University life with tags , , , , , , , on May 5, 2016 by xi'an

garden tree, Jan. 12, 2012Shinichiro Shirota and Alan Gelfand arXived a paper on the use of ABC for analysing some repulsive point processes, more exactly the Gibbs point processes, for which ABC requires a perfect sampler to operate, unless one is okay with stopping an MCMC chain before it converges, and determinantal point processes studied by Lavancier et al. (2015) [a paper I wanted to review and could not find time to!]. Detrimental point processes have an intensity function that is the determinant of a covariance kernel, hence repulsive. Simulation of a determinantal process itself is not straightforward and involves approximations. But the likelihood itself is unavailable and Lavancier et al. (2015) use approximate versions by fast Fourier transforms, which means MCMC is challenging even with those approximate steps.

“The main computational cost of our algorithm is simulation of x for each iteration of the ABC-MCMC.”

The authors propose here to use ABC instead. With an extra approximative step for simulating the determinantal process itself. Interestingly, the Gibbs point process allows for a sufficient statistic, the number of R-closed points, although I fail to see how the radius R is determined by the model, while the determinantal process does not. The summary statistics end up being a collection of frequencies within various spheres of different radii. However, these statistics are then processed by Fearnhead’s and Prangle’s proposal, namely to come up as an approximation of E[θ|y] as the natural summary. Obtained by regression over the original summaries. Another layer of complexity stems from using an ABC-MCMC approach. And including a Lasso step in the regression towards excluding less relevant radii. The paper also considers Bayesian model validation for such point processes, implementing prior predictive tests with a ranked probability score, rather than a Bayes factor.

As point processes have always been somewhat mysterious to me, I do not have any intuition about the strength of the distributional assumptions there and the relevance of picking a determinantal process against, say, a Strauss process. The model comparisons operated in the paper are not strongly supporting one repulsive model versus the others, with the authors concluding at the need for many points towards a discrimination between models. I also wonder at the possibility of including other summaries than Ripley’s K-functions, which somewhat imply a discretisation of the space, by concentric rings. Maybe using other point processes for deriving summary statistics as MLEs or Bayes estimators for those models would help. (Or maybe not.)

approximating evidence with missing data

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , on December 23, 2015 by xi'an

University of Warwick, May 31 2010Panayiota Touloupou (Warwick), Naif Alzahrani, Peter Neal, Simon Spencer (Warwick) and Trevelyan McKinley arXived a paper yesterday on Model comparison with missing data using MCMC and importance sampling, where they proposed an importance sampling strategy based on an early MCMC run to approximate the marginal likelihood a.k.a. the evidence. Another instance of estimating a constant. It is thus similar to our Frontier paper with Jean-Michel, as well as to the recent Pima Indian survey of James and Nicolas. The authors give the difficulty to calibrate reversible jump MCMC as the starting point to their research. The importance sampler they use is the natural choice of a Gaussian or t distribution centred at some estimate of θ and with covariance matrix associated with Fisher’s information. Or derived from the warmup MCMC run. The comparison between the different approximations to the evidence are done first over longitudinal epidemiological models. Involving 11 parameters in the example processed therein. The competitors to the 9 versions of importance samplers investigated in the paper are the raw harmonic mean [rather than our HPD truncated version], Chib’s, path sampling and RJMCMC [which does not make much sense when comparing two models]. But neither bridge sampling, nor nested sampling. Without any surprise (!) harmonic means do not converge to the right value, but more surprisingly Chib’s method happens to be less accurate than most importance solutions studied therein. It may be due to the fact that Chib’s approximation requires three MCMC runs and hence is quite costly. The fact that the mixture (or defensive) importance sampling [with 5% weight on the prior] did best begs for a comparison with bridge sampling, no? The difficulty with such study is obviously that the results only apply in the setting of the simulation, hence that e.g. another mixture importance sampler or Chib’s solution would behave differently in another model. In particular, it is hard to judge of the impact of the dimensions of the parameter and of the missing data.

never mind the big data here’s the big models [workshop]

Posted in Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , on December 22, 2015 by xi'an

Maybe the last occurrence this year of the pastiche of the iconic LP of the Sex Pistols!, made by Tamara Polajnar. The last workshop as well of the big data year in Warwick, organised by the Warwick Data Science Institute. I appreciated the different talks this afternoon, but enjoyed particularly Dan Simpson’s and Rob Scheichl’s. The presentation by Dan was so hilarious that I could not resist asking him for permission to post the slides here:

Not only hilarious [and I have certainly missed 67% of the jokes], but quite deep about the meaning(s) of modelling and his views about getting around the most blatant issues. Ron presented a more computational talk on the ways to reach petaflops on current supercomputers, in connection with weather prediction models used (or soon to be used) by the Met office. For a prediction area of 1 km². Along with significant improvements resulting from multiscale Monte Carlo and quasi-Monte Carlo. Definitely impressive! And a brilliant conclusion to the Year of Big Data (and big models).

never mind the big data here’s the big models [workshop]

Posted in Kids, pictures, Statistics with tags , , , , , , , on December 10, 2015 by xi'an

A perfect opportunity to recycle the pastiche of the iconic LP of the Sex Pistols!, that Mark Girolami posted for the ATI Scoping workshop  last month in Warwick. There is an open workshop on the theme of big data/big models next week in Warwick, organised by the Warwick Data Science Institute. It will take place on December 15, from noon till 5:30pm in the Zeeman Building. Invited speakers are

• Robert Scheichl (University of Bath, Dept of Mathematical Sciences)
• Shiwei Lan (University of Warwick, Dept of Statistics)
• Konstantinos Zygalakis (University of Southampton, Dept of Mathematical Sciences)
Dan Simpson (University of Bath, Dept of Mathematical Sciences), with the enticing title of “To avoid fainting, keep repeating ‘It’s only a model’…”

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.