invertible flow non equilibrium sampling (InFiNE)

With Achille Thin and a few other coauthors [and friends], we just arXived a paper on a new form of importance sampling, motivated by a recent paper of Rotskoff and Vanden-Eijnden (2019) on non-equilibrium importance sampling. The central ideas of this earlier paper are the introduction of conformal Hamiltonian dynamics, where a dissipative term is added to the ODE found in HMC, namely

which means that all orbits converge to fixed points that satisfy ∇U(q) = 0 as the energy eventually vanishes. And the property that, were T be a conformal Hamiltonian integrator associated with H, i.e. perserving the invariant measure, averaging over orbits of T would improve the precision of Monte Carlo unbiased estimators, while remaining unbiased. The fact that Rotskoff and Vanden-Eijnden (2019) considered only continuous time makes their proposal hard to implement without adding approximation error, while our approach is directly set in discrete-time and preserves unbiasedness. And since measure preserving transforms are too difficult to come by, a change of variable correction, as in normalising flows, allows for an arbitrary choice of T, while keeping the estimator unbiased. The use of conformal maps makes for a natural choice of T in this context.

The resulting InFiNE algorithm is an MCMC particular algorithm which can be represented as a  partially collapsed Gibbs sampler when using the right auxiliary variables. As in Andrieu, Doucet and Hollenstein (2010) and their ISIR algorithm. The algorithm can be used for estimating normalising constants, comparing favourably with AIS, sampling from complex targets, and optimising variational autoencoders and their ELBO.

I really appreciated working on this project, with links to earlier notions like multiple importance sampling à la Owen and Zhou (2000), nested sampling, non-homogeneous normalising flows, measure estimation à la Kong et al. (2002), on which I worked in a more or less distant past.

non-reversible jump MCMC

Philippe Gagnon and et Arnaud Doucet have recently arXived a paper on a non-reversible version of reversible jump MCMC, the methodology introduced by Peter Green in 1995 to tackle Bayesian model choice/comparison/exploration. Whom Philippe presented at BayesComp20.

“The objective of this paper is to propose sampling schemes which do not suffer from such a diffusive behaviour by exploiting the lifting idea (…)”

The idea is related to lifting, creating non-reversible behaviour by adding a direction index (a spin) to the exploration of the models, assumed to be totally ordered, as with nested models (mixtures, changepoints, &tc.).  As with earlier versions of lifting, the chain proceeds along one (spin) direction until the proposal is rejected in which case the spin spins. The acceptance probability in the event of a change of model (upwards or downwards) is essentially the same as the reversible one (meaning it includes the dreaded Jacobian!). The original difficulty with reversible jump remains active with non-reversible jump in that the move from one model to the next must produce plausible values. The paper recalls two methods proposed by Christophe Andrieu and his co-authors. One consists in buffering a tempering sequence, but this proves costly.  Pursuing the interesting underlying theme that both reversible and non-reversible versions are noisy approximations of the marginal ratio, the other one consists in marginalising out the parameter to approximate the marginal probability of moving between nearby models. Combined with multiple choice to preserve stationarity and select more likely moves at the same time. Still requiring a multiplication of the number of simulations but parallelisable. The paper contains an exact comparison result that non-reversible jump leads to a smaller asymptotic variance than reversible jump, but it is unclear to me whether or not this accounts for the extra computing time resulting from the multiple paths in the proposed algorithms. (Even though the numerical illustration shows an improvement brought by the non-reversible side for the same computational budget.)

Bernoulli race particle filters

Sebastian Schmon, Arnaud Doucet and George Deligiannidis have recently arXived an AISTATS paper with the above nice title. The motivation for the extension is facing intractable particle weights for state space models, as for instance in discretised diffusions.  In most cases, actually, the weight associated with the optimal forward proposal involves an intractable integral which is the predictive of the current observed variate given the past hidden states. And in some cases, there exist unbiased and non-negative estimators of the targets,  which can thus be substituted, volens nolens,  to the original filter. As in many pseudo-marginal derivations, this new algorithm can be interpreted as targeting an augmented distribution that involves the auxiliary random variates behind the unbiased estimators of the particle weights. A worthwhile remark since it allows for the preservation of the original target as in (8) provided the auxiliary random variates are simulated from the right conditionals. (At least ideally as I have no clue when this is feasible.)

“if Bernoulli resampling is per-formed, the variance for any Monte Carlo estimate will be the same as if the true weights were known and one applies standard multinomial resampling.”

The Bernoulli race in the title stands for a version of the Bernoulli factory problem, where an intractable and bounded component of the weight can be turned into a probability, for which a Bernoulli draw is available, hence providing a Multinomial sampling with the intractable weights since replacing the exact probability with an estimate does not modify the Bernoulli distribution, amazingly so! Even with intractable normalising constants in particle filters. The practicality of the approach may however be restricted by the possibility of some intractable terms being very small and requiring many rejections for one acceptance, as the number of attempts is a compound geometric. The intractability may add to the time request the drawback of keeping this feature hidden as well. Or force some premature interruption in the settings of a parallel implementation.

Bayesian inference with intractable normalizing functions

In the latest September issue of JASA I received a few days ago, I spotted a review paper by Jaewoo Park & Murali Haran on intractable normalising constants Z(θ). There have been many proposals for solving this problem as well as several surveys, some conferences and even a book. The current survey focus on MCMC solutions, from auxiliary variable approaches to likelihood approximation algorithms (albeit without ABC entries, even though the 2006 auxiliary variable solutions of Møller et al. et of Murray et al. do simulate pseudo-observations and hence…). This includes the MCMC approximations to auxiliary sampling proposed by Faming Liang and co-authors across several papers. And the paper Yves Atchadé, Nicolas Lartillot and I wrote ten years ago on an adaptive MCMC targeting Z(θ) and using stochastic approximation à la Wang-Landau. Park & Haran stress the relevance of using sufficient statistics in this approach towards fighting computational costs, which makes me wonder if an ABC version could be envisioned.  The paper also includes pseudo-marginal techniques like Russian Roulette (once spelled Roullette) and noisy MCMC as proposed in Alquier et al.  (2016). These methods are compared on three examples: (1) the Ising model, (2) a social network model, the Florentine business dataset used in our original paper, and a larger one where most methods prove too costly, and (3) an attraction-repulsion point process model. In conclusion, an interesting survey, taking care to spell out the calibration requirements and the theoretical validation, if of course depending on the chosen benchmarks.

unbiased consistent nested sampling via sequential Monte Carlo [a reply]

Rob Salomone sent me the following reply on my comments of yesterday about their recently arXived paper.

Our main goal in the paper was to show that Nested Sampling (when interpreted a certain way) is really just a member of a larger class of SMC algorithms, and exploring the consequences of that. We should point out that the section regarding calibration applies generally to SMC samplers, and hope that people give those techniques a try regardless of their chosen SMC approach.

Regarding your question about “whether or not it makes more sense to get completely SMC and forego any nested sampling flavour!”, this is an interesting point. After all, if Nested Sampling is just a special form of SMC, why not just use more standard SMC approaches? It seems that the Nested Sampling’s main advantage is its ability to cope with problems that have “phase transition’’ like behaviour, and thus is robust to a wider range of difficult problems than annealing approaches. Nevertheless, we hope this way of looking at NS (and showing that there may be variations of SMC with certain advantages) leads to improved NS and SMC methods down the line.  

Regarding your post, I should clarify a point regarding unbiasedness. The largest likelihood bound is actually set to infinity. Thus, for the fixed version of NS—SMC, one has an unbiased estimator of the “final” band. Choosing a final band prematurely will of course result in very high variance. However, the estimator is unbiased. For example, consider NS—SMC with only one strata. Then, the method reduces to simply using the prior as an importance sampling distribution for the posterior (unbiased, but often high variance).

Comments related to two specific parts of your post are below (your comments in italicised bold):

“Which never occurred as the number one difficulty there, as the simplest implementation runs a Markov chain from the last removed entry, independently from the remaining entries. Even stationarity is not an issue since I believe that the first occurrence within the level set is distributed from the constrained prior.”

This is an interesting point that we had not considered! In practice, and in many papers that apply Nested Sampling with MCMC, the common approach is to start the MCMC at one of the randomly selected “live points”, so the discussion related to independence was in regard to these common implementations.

Regarding starting the chain from outside of the level set. This is likely not done in practice as it introduces an additional difficulty of needing to propose a sample inside the required region (Metropolis–Hastings will have non—zero probability of returning a sample that is still outside the constrained region for any fixed number of iterations). Forcing the continuation of MCMC until a valid point is proposed I believe will be a subtle violation of detailed balance. Of course, the bias of such a modification may be small in practice, but it is an additional awkwardness introduced by the requirement of sample independence!

“And then, in a twist that is not clearly explained in the paper, the focus moves to an improved nested sampler that moves one likelihood value at a time, with a particle step replacing a single  particle. (Things get complicated when several particles may take the very same likelihood value, but randomisation helps.) At this stage the algorithm is quite similar to the original nested sampler. Except for the unbiased estimation of the constants, the  final constant, and the replacement of exponential weights exp(-t/N) by powers of (N-1/N)”

Thanks for pointing out that this isn’t clear, we will try to do better in the next revision! The goal of this part of the paper wasn’t necessarily to propose a new version of nested sampling. Our focus here was to demonstrate that NS–SMC is not simply the Nested Sampling idea with an SMC twist, but that the original NS algorithm with MCMC (and restarting the MCMC sampling at one of the “live points’” as people do in practice) actually is a special case of SMC (with the weights replaced with a suboptimal choice).

The most curious thing is that, as you note, the estimates of remaining prior mass in the SMC context come out as powers of (N-1)/N and not exp(-t/N). In the paper by Walter (2017), he shows that the former choice is actually superior in terms of bias and variance. It was a nice touch that the superior choice of weights came out naturally in the SMC interpretation! 

That said, as the fixed version of NS-SMC is the one with the unbiasedness and consistency properties, this was the version we used in the main statistical examples.