Archive for doubly intractable problems

the HMC algorithm meets the exchange algorithm

Posted in Mountains, pictures, Statistics, Travel, University life with tags , , , , , , , , on July 26, 2017 by xi'an

Julien Stoehr (now in Dublin, soon to join us as a new faculty in Paris-Dauphine!), Alan Benson and Nial Friel (both at UCD) arXived last week a paper entitled Noisy HMC for doubly-intractable distributions. Which considers solutions for adapting Hamiltonian Monte Carlo to target densities that involve a missing constant. In the sense of our workshop last year in Warwick. And in the theme pursued by Nial in the past years. The notion is thus to tackle a density π(θ)∞exp(V(X|θ)/Z(θ) when Z(θ) is intractable. In that case the gradient of log Z(θ) can be estimated as the expectation of the gradient of V(X|θ) [as a standard exponential family identity]. And the ratio of the Z(θ)’s appearing in the Metropolis ratio can be derived by Iain Murray’s exchange algorithm, based on simulations from the sampling distribution attached to the parameter in the denominator.

The resulting algorithm proposed by the authors thus uses N simulations of auxiliary variables at each step þ of the leapfrog part, towards an approximation of the gradient term, plus another N simulations for approximating the ratio of the normalising constants Z(θ)/Z(θ’). While justified from an importance sampling perspective, this approximation is quite poor when θ and θ’ differ. A better solution [as shown in the paper] is to take advantage of all leapfrog steps and of associated auxiliary simulations to build a telescopic product of ratios where the parameter values θ and θ’ are much closer. The main difficulty is in drawing a comparison with the exchange algorithm, since the noisy HMC version is computationally more demanding. (A secondary difficulty is in having an approximate algorithm that no longer leaves the target density stationary.)

oxwasp@amazon.de

Posted in Books, Kids, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , on April 12, 2017 by xi'an

The reason for my short visit to Berlin last week was an OxWaSP (Oxford and Warwick Statistics Program) workshop hosted by Amazon Berlin with talks between statistics and machine learning, plus posters from our second year students. While the workshop was quite intense, I enjoyed very much the atmosphere and the variety of talks there. (Just sorry that I left too early to enjoy the social programme at a local brewery, Brauhaus Lemke, and the natural history museum. But still managed nice runs east and west!) One thing I found most interesting (if obvious in retrospect) was the different focus of academic and production talks, where the later do not aim at a full generality or at a guaranteed improvement over the existing, provided the new methodology provides a gain in efficiency over the existing.

This connected nicely with me reading several Nature articles on quantum computing during that trip,  where researchers from Google predict commercial products appearing in the coming five years, even though the technology is far from perfect and the outcome qubit error prone. Among the examples they provided, quantum simulation (not meaning what I consider to be simulation!), quantum optimisation (as a way to overcome multimodality), and quantum sampling (targeting given probability distributions). I find the inclusion of the latest puzzling in that simulation (in that sense) shows very little tolerance for errors, especially systematic bias. It may be that specific quantum architectures can be designed for specific probability distributions, just like some are already conceived for optimisation. (It may even be the case that quantum solutions are (just next to) available for intractable constants as in Ising or Potts models!)

Russian roulette still rolling

Posted in Statistics with tags , , , , , , , , , , , , on March 22, 2017 by xi'an

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…

adaptive exchange

Posted in Books, Statistics, University life with tags , , , , , , , , , , on October 27, 2016 by xi'an

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]

  1. Proposing a new value θ’ of the parameter from a proposal q(θ’|θ);
  2. Generate a pseudo-observation z~ƒ(z|θ’);
  3. Accept with probability

\dfrac{\pi(\theta')f(y|\theta')}{\pi(\theta)f(y|\theta)}\dfrac{q(\theta|\theta')f(z|\theta)}{q(\theta'|\theta)f(z|\theta')}

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.

auxiliary variable methods as ABC

Posted in Books, pictures, Statistics, University life with tags , , , , , on May 9, 2016 by xi'an

ruins of the abbey at Tynemouth, Sept. 03, 2013Dennis Prangle and Richard Everitt arXived a note today where they point out the identity between the auxiliary variable approach of Møller et al. (2006) [or rather its multiple or annealed version à la Murray] and [exact] ABC (as in our 2009 paper) in the case of Markov random fields. The connection between the two appears when using an importance sampling step in the ABC algorithm and running a Markov chain forward and backward the same number of steps as there are levels in the annealing scheme of MAV. Maybe more a curiosity than an indicator of a large phenomenon, since it is so rare that ABC can be use in its exact form.