neural importance sampling

Dennis Prangle signaled this paper during his talk of last week, first of our ABC ‘minars now rechristened as The One World ABC Seminar to join the “One World xxx Seminar” franchise! The paper is written by Thomas Müller and co-authors, all from Disney research [hence the illustration], and we discussed it in our internal reading seminar at Dauphine. The authors propose to parameterise the importance sampling density via neural networks, just like Dennis is using auto-encoders. Starting with the goal of approximating

(where they should assume f to be non-negative for the following), the authors aim at simulating from an approximation of f(x)/ℑ since this “ideal” pdf would give zero variance.

“Unfortunately, the above integral is often not solvable in closed form, necessitating its estimation with another Monte Carlo estimator.”

Among the discussed solutions, the Latent-Variable Model one is based on a pdf represented as a marginal. A mostly intractable integral, which the authors surprisingly seem to deem an issue as they do not mention the standard solution of simulating from the joint and using the conditional in the importance weight. (Or even more surprisingly and obviously wrongly see the latter as a biased approximation to the weight.)

“These “autoregressive flows” offer the desired exact evaluation of q(x;θ). Unfortunately, they generally only permit either efficient sample generation or efficient evaluation of q(x;θ), which makes them prohibitively expensive for our application to Mont Carlo integration.”

When presenting normalizing flows, namely the representation of the simulation output as the result of an invertible mapping of a standard (e.g., Gaussian or Uniform) random variable, x=h(u,θ), which can itself be decomposed into a composition of suchwise functions. And I am thus surprised this cannot be done in an efficient manner if transforms are well chosen…

“The key proposition of Dinh et al. (2014) is to focus on a specific class of mappings—referred to as coupling layers—that admit Jacobian matrices where determinants reduce to the product of diagonal terms.

Using a transform with a triangular Jacobian at each stage has the appeal of keeping the change of variable simple and allowing for non-linear transforms. Namely piecewise polynomials. When reading the one-blob (!) encoding , I am however uncertain the approach is more than the choice of a particular functional basis, as for instance wavelets (which may prove more costly to handle, granted!)

“Given that NICE scales well to high-dimensional problems…”

It is always unclear to me why almost every ML paper feels the urge to redefine & motivate the KL divergence. And to recall that it avoids bothering about the normalising constant. Looking at the variance of the MC estimator & seeking minimal values is praiseworthy, but only when the variance exists. What are the guarantees on the density estimate for this to happen? And where are the arguments for NICE scaling nicely to high dimensions? Interesting intrusion of path sampling, but is it of any use outside image analysis—I had forgotten Eric Veach’s original work was on light transport—?

g-and-k [or -h] distributions

Dennis Prangle released last week an R package called gk and an associated arXived paper for running inference on the g-and-k and g-and-h quantile distributions. As should be clear from an earlier review on Karian’s and Dudewicz’s book quantile distributions, I am not particularly fond of those distributions which construction seems very artificial to me, as mostly based on the production of a closed-form quantile function. But I agree they provide a neat benchmark for ABC methods, if nothing else. However, as recently pointed out in our Wasserstein paper with Espen Bernton, Pierre Jacob and Mathieu Gerber, and explained in a post of Pierre’s on Statisfaction, the pdf can be easily constructed by numerical means, hence allows for an MCMC resolution, which is also a point made by Dennis in his paper. Using the closed-form derivation of the Normal form of the distribution [i.e., applied to Φ(x)] so that numerical derivation is not necessary.

Bayesian optimization for likelihood-free inference of simulator-based statistical models [guest post]

[The following comments are from Dennis Prangle, about the second half of the paper by Gutmann and Corander I commented last week.]

Here are some comments on the paper of Gutmann and Corander. My brief skim read through this concentrated on the second half of the paper, the applied methodology. So my comments should be quite complementary to Christian’s on the theoretical part!

ABC algorithms generally follow the template of proposing parameter values, simulating datasets and accepting/rejecting/weighting the results based on similarity to the observations. The output is a Monte Carlo sample from a target distribution, an approximation to the posterior. The most naive proposal distribution for the parameters is simply the prior, but this is inefficient if the prior is highly diffuse compared to the posterior. MCMC and SMC methods can be used to provide better proposal distributions. Nevertheless they often still seem quite inefficient, requiring repeated simulations in parts of parameter space which have already been well explored.

The strategy of this paper is to instead attempt to fit a non-parametric model to the target distribution (or in fact to a slight variation of it). Hopefully this will require many fewer simulations. This approach is quite similar to Richard Wilkinson’s recent paper. Richard fitted a Gaussian process to the ABC analogue of the log-likelihood. Gutmann and Corander introduce two main novelties:

1. They model the expected discrepancy (i.e. distance) Δ_θ between the simulated and observed summary statistics. This is then transformed to estimate the likelihood. This is in contrast to Richard who transformed the discrepancy before modelling. This is the standard ABC approach of weighting the discrepancy depending on how close to 0 it is. The drawback of the latter approach is it requires picking a tuning parameter (the ABC acceptance threshold or bandwidth) in advance of the algorithm. The new approach still requires a tuning parameter but its choice can be delayed until the transformation is performed.
2. They generate the θ values on-line using “Bayesian optimisation”. The idea is to pick θ to concentrate on the region near the minimum of the objective function, and also to reduce uncertainty in the Gaussian process. Thus well explored regions can usually be neglected. This is in contrast to Richard who chose θs using space filling design prior to performing any simulations.

I didn’t read the paper’s theory closely enough to decide whether (1) is a good idea. Certainly the results for the paper’s examples look convincing. Also, one issue with Richard‘s approach was that because the log-likelihood varied over such a wide variety of magnitudes, he needed to fit several “waves” of GPs. It would be nice to know if the approach of modelling the discrepancy has removed this problem, or if a single GP is still sometimes an insufficiently flexible model.

Novelty (2) is a very nice and natural approach to take here. I did wonder why the particular criterion in Equation (45) was used to decide on the next θ. Does this correspond to optimising some information theoretic quantity? Other practical questions were whether it’s possible to parallelise the method (I seem to remember talking to Michael Gutmann about this at NIPS but can’t remember his answer!), and how well the approach scales up with the dimension of the parameters.

ABC model choice by random forests [guest post]

[Dennis Prangle sent me his comments on our ABC model choice by random forests paper. Here they are! And I appreciate very much contributors commenting on my paper or others, so please feel free to join.]

This paper proposes a new approach to likelihood-free model choice based on random forest classifiers. These are fit to simulated model/data pairs and then run on the observed data to produce a predicted model. A novel “posterior predictive error rate” is proposed to quantify the degree of uncertainty placed on this prediction. Another interesting use of this is to tune the threshold of the standard ABC rejection approach, which is outperformed by random forests.

The paper has lots of thought-provoking new ideas and was an enjoyable read, as well as giving me the encouragement I needed to read another chapter of the indispensable Elements of Statistical Learning However I’m not fully convinced by the approach yet for a few reasons which are below along with other comments.

Alternative schemes

The paper shows that random forests outperform rejection based ABC. I’d like to see a comparison to more efficient ABC model choice algorithms such as that of Toni et al 2009. Also I’d like to see if the output of random forests could be used as summary statistics within ABC rather than as a separate inference method.

Posterior predictive error rate (PPER)

This is proposed to quantify the performance of a classifier given a particular data set. The PPER is the proportion of times the classifier’s most favoured model is incorrect for simulated model/data pairs drawn from an approximation to the posterior predictive. The approximation is produced by a standard ABC analysis.

Misclassification could be due to (a) a poor classifier or (b) uninformative data, so the PPER aggregrates these two sources of uncertainty. I think it is still very desirable to have an estimate of the uncertainty due to (b) only i.e. a posterior weight estimate. However the PPER is useful. Firstly end users may sometimes only care about the aggregated uncertainty. Secondly relative PPER values for a fixed dataset are a useful measure of uncertainty due to (a), for example in tuning the ABC threshold. Finally, one drawback of the PPER is the dependence on an ABC estimate of the posterior: how robust are the results to the details of how this is obtained?

Classification

This paper illustrates an important link between ABC and machine learning classification methods: model choice can be viewed as a classification problem. There are some other links: some classifiers make good model choice summary statistics (Prangle et al 2014) or good estimates of ABC-MCMC acceptance ratios for parameter inference problems (Pham et al 2014). So the good performance random forests makes them seem a generally useful tool for ABC (indeed they are used in the Pham et al al paper).