estimating the marginal likelihood (or an information criterion)

Tory Imai (from Kyoto University) arXived a paper last summer on what first looked like a novel approximation of the marginal likelihood. Based on the variance of thermodynamic integration. The starting argument is that there exists a power 0<t⁰<1 such that the expectation of the logarithm of the product of the prior by the likelihood to the power t⁰ or t⁰-powered likelihood  is equal to the standard log-marginal

when the expectation is under the posterior corresponding to the t⁰-powered likelihood (rather than the full likelihood). By an application of the mean value theorem. Watanabe’s (2013) WBIC replaces the optimum t⁰ with 1/log(n), n being the sample size. The issue in terms of computational statistics is of course that the error of WBIC (against the true log m(x)) is only characterised as an order of n.

The second part of the paper is rather obscure to me, as the motivation for the real log canonical threshold is missing, even though the quantity is connected with the power likelihood. And the DIC effective dimension. It then goes on to propose a new approximation of sBIC, where s stands for singular, of Drton and Plummer (2017) which I had missed (and may ask my colleague Martin later today at Warwick!). Quickly reading through the later however brings explanations about the real log canonical threshold being simply the effective dimension in Schwarwz’s BIC approximation to the log marginal,

(as derived by Watanabe), where m is called the multiplicity of the real log canonical threshold. Both λ and m being unknown, Drton and Plummer (2017) estimate the above approximation in a Bayesian fashion, which leads to a double indexed marginal approximation for a collection of models. Since this thread leads me further and further from a numerical resolution of the marginal estimation, but brings in a different perspective on mixture Bayesian estimation, I will return to this highly  in a later post. The paper of Imai discusses a different numerical approximation to sBIC, With a potential improvement in computing sBIC. (The paper was proposed as a poster to BayesComp 2020, so I am looking forward discussing it with the author.)

 

thermodynamic integration plus temperings

Biljana Stojkova and David Campbel recently arXived a paper on the used of parallel simulated tempering for thermodynamic integration towards producing estimates of marginal likelihoods. Resulting into a rather unwieldy acronym of PT-STWNC for “Parallel Tempering – Simulated Tempering Without Normalizing Constants”. Remember that parallel tempering runs T chains in parallel for T different powers of the likelihood (from 0 to 1), potentially swapping chain values at each iteration. Simulated tempering monitors a single chain that explores both the parameter space and the temperature range. Requiring a prior on the temperature. Whose optimal if unrealistic choice was found by Geyer and Thomson (1995) to be proportional to the inverse (and unknown) normalising constant (albeit over a finite set of temperatures). Proposing the new temperature instead via a random walk, the Metropolis within Gibbs update of the temperature τ then involves normalising constants.

“This approach is explored as proof of concept and not in a general sense because the precision of the approximation depends on the quality of the interpolator which in turn will be impacted by smoothness and continuity of the manifold, properties which are difficult to characterize or guarantee given the multi-modal nature of the likelihoods.”

To bypass this issue, the authors pick for their (formal) prior on the temperature τ, a prior such that the profile posterior distribution on τ is constant, i.e. the joint distribution at τ and at the mode [of the conditional posterior distribution of the parameter] is constant. This choice makes for a closed form prior, provided this mode of the tempered posterior can de facto be computed for each value of τ. (However it is unclear to me why the exact mode would need to be used.) The resulting Metropolis ratio becomes independent of the normalising constants. The final version of the algorithm runs an extra exchange step on both this simulated tempering version and the untempered version, i.e., the original unnormalised posterior. For the marginal likelihood, thermodynamic integration is invoked, following Friel and Pettitt (2008), using simulated tempering samples of (θ,τ) pairs (associated instead with the above constant profile posterior) and simple Riemann integration of the expected log posterior. The paper stresses the gain due to a continuous temperature scale, as it “removes the need for optimal temperature discretization schedule.” The method is applied to the Glaxy (mixture) dataset in order to compare it with the earlier approach of Friel and Pettitt (2008), resulting in (a) a selection of the mixture with five components and (b) much more variability between the estimated marginal  likelihoods for different numbers of components than in the earlier approach (where the estimates hardly move with k). And (c) a trimodal distribution on the means [and unimodal on the variances]. This example is however hard to interpret, since there are many contradicting interpretations for the various numbers of components in the model. (I recall Radford Neal giving an impromptu talks at an ICMS workshop in Edinburgh in 2001 to warn us we should not use the dataset without a clear(er) understanding of the astrophysics behind. If I remember well he was excluded all low values for the number of components as being inappropriate…. I also remember taking two days off with Peter Green to go climbing Craigh Meagaidh, as the only authorised climbing place around during the foot-and-mouth epidemics.) In conclusion, after presumably too light a read (I did not referee the paper!), it remains unclear to me why the combination of the various tempering schemes is bringing a noticeable improvement over the existing. At a given computational cost. As the temperature distribution does not seem to favour spending time in the regions where the target is most quickly changing. As such the algorithm rather appears as a special form of exchange algorithm.

postdoctoral position in computational statistical physics and machine learning

Bayesian goodness of fit

 

Persi Diaconis and Guanyang Wang have just arXived an interesting reflection on the notion of Bayesian goodness of fit tests. Which is a notion that has always bothered me, in a rather positive sense (!), as

“I also have to confess at the outset to the zeal of a convert, a born again believer in stochastic methods. Last week, Dave Wright reminded me of the advice I had given a graduate student during my algebraic geometry days in the 70’s :`Good Grief, don’t waste your time studying statistics. It’s all cookbook nonsense.’ I take it back! …” David Mumford

The paper starts with a reference to David Mumford, whose paper with Wu and Zhou on exponential “maximum entropy” synthetic distributions is at the source (?) of this paper, and whose name appears in its very title: “A conversation for David Mumford”…, about his conversion from pure (algebraic) maths to applied maths. The issue of (Bayesian) goodness of fit is addressed, with card shuffling examples, the null hypothesis being that the permutation resulting from the shuffling is uniformly distributed if shuffling takes enough time. Interestingly, while the parameter space is compact as a distribution on a finite set, Lindley’s paradox still occurs, namely that the null (the permutation comes from a Uniform) is always accepted provided there is no repetition under a “flat prior”, which is the Dirichlet D(1,…,1) over all permutations. (In this finite setting an improper prior is definitely improper as it does not get proper after accounting for observations. Although I do not understand why the Jeffreys prior is not the Dirichlet(½,…,½) in this case…) When resorting to the exponential family of distributions entertained by Zhou, Wu and Mumford, including the uniform distribution as one of its members, Diaconis and Wang advocate the use of a conjugate prior (exponential family, right?!) to compute a Bayes factor that simplifies into a ratio of two intractable normalising constants. For which the authors suggest using importance sampling, thermodynamic integration, or the exchange algorithm. Except that they rely on the (dreaded) harmonic mean estimator for computing the Bayes factor in the following illustrative section! Due to the finite nature of the space, I presume this estimator still has a finite variance. (Remark 1 calls for convergence results on exchange algorithms, which can be found I think in the just as recent arXival by Christophe Andrieu and co-authors.) An interesting if rare feature of the example processed in the paper is that the sufficient statistic used for the permutation model can be directly simulated from a Multinomial distribution. This is rare as seen when considering the benchmark of Ising models, for which the summary and sufficient statistic cannot be directly simulated. (If only…!) In fine, while I enjoyed the paper a lot, I remain uncertain as to its bearings, since defining an objective alternative for the goodness-of-fit test becomes quickly challenging outside simple enough models.

WBIC, practically

“Thus far, WBIC has received no more than a cursory mention by Gelman et al. (2013)”

I had missed this 2015  paper by Nial Friel and co-authors on a practical investigation of Watanabe’s WBIC. Where WBIC stands for widely applicable Bayesian information criterion. The thermodynamic integration approach explored by Nial and some co-authors for the approximation of the evidence, thermodynamic integration that produces the log-evidence as an integral between temperatures t=0 and t=1 of a powered evidence, is eminently suited for WBIC, as the widely applicable Bayesian information criterion is associated with the specific temperature t⁰ that makes the power posterior equidistant, Kullback-Leibler-wise, from the prior and posterior distributions. And the expectation of the log-likelihood under this very power posterior equal to the (genuine) evidence. In fact, WBIC is often associated with the sub-optimal temperature 1/log(n), where n is the (effective?) sample size. (By comparison, if my minimalist description is unclear!, thermodynamic integration requires a whole range of temperatures and associated MCMC runs.) In an ideal Gaussian setting, WBIC improves considerably over thermodynamic integration, the larger the sample the better. In more realistic settings, though, including a simple regression and a logistic [Pima Indians!] model comparison, thermodynamic integration may do better for a given computational cost although the paper is unclear about these costs. The paper also runs a comparison with harmonic mean and nested sampling approximations. Since the integral of interest involves a power of the likelihood, I wonder if a safe version of the harmonic mean resolution can be derived from simulations of the genuine posterior. Provided the exact temperature t⁰ is known…