Colin Wei and Iain Murray arXived a new version of their paper on doubly-intractable distributions, which is to be presented at AISTATS. It builds upon the Russian roulette estimator of Lyne et al. (2015), which itself exploits the debiasing technique of McLeish et al. (2011) [found earlier in the physics literature as in Carter and Cashwell, 1975, according to the current paper]. Such an unbiased estimator of the inverse of the normalising constant can be used for pseudo-marginal MCMC, except that the estimator is sometimes negative and has to be so as proved by Pierre Jacob and co-authors. As I discussed in my post on the Russian roulette estimator, replacing the negative estimate with its absolute value does not seem right because a negative value indicates that the quantity is close to zero, hence replacing it with zero would sound more appropriate. Wei and Murray start from the property that, while the expectation of the importance weight is equal to the normalising constant, the expectation of the inverse of the importance weight converges to the inverse of the weight for an MCMC chain. This however sounds like an harmonic mean estimate because the property would also stand for any substitute to the importance density, as it only requires the density to integrate to one… As noted in the paper, the variance of the resulting Roulette estimator “will be high” or even infinite. Following Glynn et al. (2014), the authors build a coupled version of that solution, which key feature is to cut the higher order terms in the debiasing estimator. This does not guarantee finite variance or positivity of the estimate, though. In order to decrease the variance (assuming it is finite), backward coupling is introduced, with a Rao-Blackwellisation step using our 1996 Biometrika derivation. Which happens to be of lower cost than the standard Rao-Blackwellisation in that special case, O(N) versus O(N²), N being the stopping rule used in the debiasing estimator. Under the assumption that the inverse importance weight has finite expectation [wrt the importance density], the resulting backward-coupling Russian roulette estimator can be proven to be unbiased, as it enjoys a finite expectation. (As in the generalised harmonic mean case, the constraint imposes thinner tails on the importance function, which then hampers the convergence of the MCMC chain.) No mention is made of achieving finite variance for those estimators, which again is a serious concern due to the similarity with harmonic means…
Archive for normalising constant
“Methods to estimate the marginal likelihood should be sensitive to the prior choice. Non-informative priors should increase the contribution of low-likelihood regions of parameter space in the estimated marginal likelihood. Consequently, the prior choice should affect the estimated evidence.”
In a most recent arXival, Maturana, Brewer, and Klaere discuss of the appeal of nested sampling for conducting model choice in philogenetic models. In comparison with the “generalized steppingstone sampling” method, which represents the evidence as a product of ratios of evidences (Fan et al., 2011). And which I do not think I have previously met, with all references provided therein relating to Bayesian philogenetics, apparently. The stepping stone approach relies on a sequence of tempered targets, moving from a reference distribution to the real target as a temperature β goes from zero to one. (The paper also mentions thermodynamic integration as too costly.) Nested sampling—much discussed on this blog!—is presented in this paper as having the ability to deal with partly convex likelihoods, although I do not really get how or why. (As there is nothing new in the fairly pedagogical pretentation of nested sampling therein.) Nothing appears to be mentioned about the difficulty to handle multimodal as high likelihood isolated regions are unlikely to be sampled from poorly weighted priors (by which I mean that a region with significant likelihood mass is unlikely to get sampled if the prior distribution gives little prior weight to that region). The novelty in the paper is to compare nested sampling with generalized steppingstone sampling and path sampling on several phylogenetic examples. I did not spot computing time mentioned there. As usual with examples, my reservation is that the conclusions drawn for one particular analysis of one (three) particular example(s) does not convey a general method about the power and generality of a method. Even though I acknowledge that nested sampling is on principle operational in wide generality.
A new survey on Bayesian inference with intractable normalising constants was posted on arXiv yesterday by Jaewoo Park and Murali Haran. A rather massive work of 58 pages, almost handy for a short course on the topic! In particular, it goes through the most common MCMC methods with a detailed description, followed by comments on components to be calibrated and the potential theoretical backup. This includes for instance the method of Liang et al. (2016) that I reviewed a few months ago. As well as the Wang-Landau technique we proposed with Yves Atchadé and Nicolas Lartillot. And the noisy MCMC of Alquier et al. (2016), also reviewed a few months ago. (The Russian Roulette solution is only mentioned very briefly as” computationally very expensive”. But still used in some illustrations. The whole area of pseudo-marginal MCMC is also missing from the picture.)
“…auxiliary variable approaches tend to be more efficient than likelihood approximation approaches, though efficiencies vary quite a bit…”
The authors distinguish between MCMC methods where the normalizing constant is approximated and those where it is omitted by an auxiliary representation. The survey also distinguishes between asymptotically exact and asymptotically inexact solutions. For instance, using a finite number of MCMC steps instead of the associated target results in an asymptotically inexact method. The question that remains open is what to do with the output, i.e., whether or not there is a way to correct for this error. In the illustration for the Ising model, the double Metropolis-Hastings version of Liang et al. (2010) achieves for instance massive computational gains, but also exhibits a persistent bias that would go undetected were it the sole method implemented. This aspect of approximate inference is not really explored in the paper, but constitutes a major issue for modern statistics (and machine learning as well, when inference is taken into account.)
In conclusion, this survey provides a serious exploration of recent MCMC methods. It begs for a second part involving particle filters, which have often proven to be faster and more efficient than MCMC methods, at least in state space models. In that regard, Nicolas Chopin and James Ridgway examined further techniques when calling to leave the Pima Indians [dataset] alone.
In the March 2016 issue of JASA that currently sits on my desk, there is a paper by Liang, Jim, Song and Liu on the adaptive exchange algorithm, which aims at handling posteriors for sampling distributions with intractable normalising constants. The concept behind the algorithm is the exchange principle initiated by Jesper Møller and co-authors in 2006, where an auxiliary pseudo-observation is simulated for the missing constants to vanish in a Metropolis-Hastings ratio. (The name exchangeable was introduced in a subsequent paper by Iain Murray, Zoubin Ghahramani and David MacKay, also in 2006.)
The crux of the method is to run an iteration as [where y denotes the observation]
- Proposing a new value θ’ of the parameter from a proposal q(θ’|θ);
- Generate a pseudo-observation z~ƒ(z|θ’);
- Accept with probability
which has the appeal to cancel all normalising constants. And the repeal of requiring an exact simulation from the very distribution with the missing constant, ƒ(.|θ). Which means that in practice a finite number of MCMC steps will be used and will bias the outcome. The algorithm is unusual in that it replaces the exact proposal q(θ’|θ) with an unbiased random version q(θ’|θ)ƒ(z|θ’), z being just an augmentation of the proposal. (The current JASA paper by Liang et al. seems to confuse augment and argument, see p.378.)
To avoid the difficulty in simulating from ƒ(.|θ), the authors draw pseudo-observations from sampling distributions with a finite number m of parameter values under the [unrealistic] assumption (A⁰) that this collection of values provides an almost complete cover of the posterior support. One of the tricks stands with an auxiliary [time-heterogeneous] chain of pseudo-observations generated by single Metropolis steps from one of these m fixed targets. These pseudo-observations are then used in the main (or target) chain to define the above exchange probability. The auxiliary chain is Markov but time-heterogeneous since the probabilities of accepting a move are evolving with time according to a simulated annealing schedule. Which produces a convergent estimate of the m normalising constants. The main chain is not Markov in that it depends on the whole history of the auxiliary chain [see Step 5, p.380]. Even jointly the collection of both chains is not Markov. The paper prefers to consider the process as an adaptive Markov chain. I did not check the rather intricate in details, so cannot judge of the validity of the overall algorithm; I simply note that one condition (A², p.383) is incredibly strong in that it assumes the Markov transition kernel to be Doeblin uniformly on any compact set of the calibration parameters. However, the major difficulty with this approach seems to be in its delicate calibration. From providing a reference set of m parameter values scanning the posterior support to picking transition kernels on both the parameter and the sample spaces, to properly cooling the annealing schedule [always a fun part!], there seems to be [from my armchair expert’s perspective, of course!] a wide range of opportunities for missing the target or running into zero acceptance problems. Both examples analysed in the paper, the auto-logistic and the auto-normal models, are actually of limited complexity in that they depend on a few parameters, 2 and 4 resp., and enjoy sufficient statistics, of dimensions 2 and 4 as well. Hence simulating (pseudo-)realisations of those sufficient statistics should be less challenging than the original approach replicating an entire vector of thousands of dimensions.
“By employing a modification to existing naïve subsampling techniques we can obtain an algorithm which is still exact but has sub-linear iterative cost as a function of data size.”
A few weeks ago Murray Pollock, Paul Fearnhead, Adam Johansen and Gareth Roberts (all from Warwick except for Paul) arXived a paper The Scalable Langevin Exact Algorithm: Bayesian Inference for Big Data. (This was also the topic of Murray’s talk last year at JSM in Seattle.) One major advance found in the paper is the derivation of an “exact” algorithm that is sub-linear in the data size. As discussed in the introduction, the current approaches to large data problems either suffer from being approximate (like divide-and-conquer methods) or do not achieve significant reduction in the computing time, being of order O(n). The authors mention Teh and Welling (2011) sand their tochastic gradient approximation to the Langevin diffusion, when the gradient is based on a subsample. Without the Metropolis correction that would ensure an exact target but at a cost of order O(n). (Which makes the technique rather difficult to recommend.)
A novel [for me] notion at the core of this paper is the concept of quasi-stationary distribution, which is the limiting distribution of a Markov chain X[t] conditional on a Markov stopping time [being larger than t]. The approach is based on diffusions with appropriate stationary distributions like the Langevin diffusion. (Actually, as in most papers I have read and remember, the current paper only considers the Langevin diffusion.) In order to avoid the issues with unadjusted and Metropolis-adjusted Langevin schemes, a killed Brownian motion is created, which means a Brownian motion conditional of being alive till time T when the instantaneous killing rate is a given function of the chain, Φ(X[t]), related with the stationary measure of the Langevin diffusion ν. Under appropriate conditions, the density of this killed Brownian motion converges [in T] to √ν. Which immediately hints at creating a new Langevin diffusion targeting ν² instead of ν. And killing it with the proper rate, which can be done by thinning a Poisson process. Simulating the original path can be done by path-space rejection sampling, following the technique set by Gareth Roberts and co-authors more than ten years ago. Based on finite dimensional realisations of the path on [0,T]. And including the killing part can be done by importance sampling and checking that the simulated killing time is larger than the current (exponentially simulated) time.
One practical difficulty in the implementation of this neat principle is the derivation of the normalising constant, which evaluation degrades with the time horizon T. The solution adopted in the paper is through a sequential Monte Carlo method, using another discretisation of the time interval [0,T] (relying on the original one would get too costly?). As for subsampling, since the survival probability for the Brownian motion is based on an unbiased estimator, subsampling does not hurt if conducted in a random manner. Although this increases the variance on principle, the use of a control variate computed just once helps in reducing the complexity to O(1).
This is a tough paper and I have not gone through the effort of trying to implement it, but this is an original and innovative construct I would like to monitor in further details on a toy example, maybe next week while in Warwick. Or at least to discuss it with the authors.
“The new method circumvents the challenges associated with accurate evidence calculations by computing posterior odds ratios using Bayesian parameter estimation”
One paper leading to another, I had a look at Hee et al. 2015 paper on Bayes factor estimation. The “novelty” stands in introducing the model index as an extra parameter in a single model encompassing all models under comparison, the “new” parameterisation being in (θ,n) rather than in θ. With the distinction that the parameter θ is now made of the union of all parameters across all models. Which reminds us very much of Carlin and Chib (1995) approach to the problem. (Peter Green in his Biometrika (1995) paper on reversible jump MCMC uses instead a direct sum of parameter spaces.) The authors indeed suggest simulating jointly (θ,n) in an MCMC or nested sampling scheme. Rather than being updated by arbitrary transforms as in Carlin and Chib (1995) the useless parameters from the other models are kept constant… The goal being to estimate P(n|D) the marginal posterior on the model index, aka the posterior probability of model n.
Now, I am quite not certain keeping the other parameter constants is a valid move: given a uniform prior on n and an equally uniform proposal, the acceptance probability simplifies into the regular Metropolis-Hastings ratio for model n. Hence the move is valid within model n. If not, I presume the previous pair (θ⁰,n⁰) is repeated. Wait!, actually, this is slightly more elaborate: if a new value of n, m, is proposed, then the acceptance ratio involves the posteriors for both n⁰ and m, possibly only the likelihoods when the proposal is the prior. So the move will directly depend on the likelihood ratio in this simplified case, which indicates the scheme could be correct after all. Except that this neglects the measure theoretic subtleties that led to reversible jump symmetry and hence makes me wonder. In other words, it follows exactly the same pattern as reversible jump without the constraints of the latter… Free lunch, anyone?!
A short announcement that the slides of almost all talks at the CRiSM workshop on estimating constants last April 20-22 are now available. Enjoy (and dicuss)!