Xichen Huang, Jin Wang and Feng Liang have recently arXived a paper where they rely on variational Bayes in conjunction with a spike-and-slab prior modelling. This actually stems from an earlier paper by Carbonetto and Stephens (2012), the difference being in the implementation of the method, which is less Gibbs-like for the current paper. The approach is not fully Bayesian in that, not only an approximate (variational) representation is used for the parameters of interest (regression coefficient and presence-absence indicators) but also the nuisance parameters are replaced with MAPs. The variational approximation on the regression parameters is an independent product of spike-and-slab distributions. The authors show the approximate approach is consistent in both frequentist and Bayesian terms (under identifiability assumptions). The method is undoubtedly faster than MCMC since it shares many features with EM but I still wonder at the Bayesian interpretability of the outcome, which writes out as a product of estimated spike-and-slab mixtures. First, the weights in the mixtures are estimated by EM, hence fixed. Second, the fact that the variational approximation is a product is confusing in that the posterior distribution on the regression coefficients is unlikely to produce posterior independence.
Archive for variational Bayes methods
Elaine Angelino, Matthew Johnson and Ryan Adams just arXived a massive survey of 118 pages on scalable Bayesian inference, which could have been entitled Bayes for Big Data, as this monograph covers state-of-the-art computational approaches to large and complex data structures. I did not read each and every line of it, but I have already recommended it to my PhD students. Some of its material unsurprisingly draws from the recent survey by Rémi Bardenet et al. (2015) I discussed a while ago. It also relates rather frequently to the somewhat parallel ICML paper of Korattikara et al. (2014). And to the firefly Monte Carlo procedure also discussed previously here.
Chapter 2 provides some standard background on computational techniques, Chapter 3 covers MCMC with data subsets, Chapter 4 gives some entries on MCMC with parallel and distributed architectures, Chapter 5 focus on variational solutions, and Chapter 6 is about open questions and challenges.
“Insisting on zero asymptotic bias from Monte Carlo estimates of expectations may leave us swamped in errors from high variance or transient bias.”
One central theme of the paper is the need for approximate solutions, MCMC being perceived as the exact solution. (Somewhat wrongly in the sense that the product of an MCMC is at best an empirical version of the true posterior, hence endowed with a residual and incompressible variation for a given computing budget.) While Chapter 3 stresses the issue of assessing the distance to the true posterior, it does not dwell at all on computing times and budget, which is arguably a much harder problem. Chapter 4 seems to be more aware of this issue since arguing that “a way to use parallel computing resources is to run multiple sequential MCMC algorithms at once [but that this] does not reduce the transient bias in MCMC estimates of posterior expectations” (p.54). The alternatives are to use either prefetching (which was the central theme of Elaine Angelino’s thesis), asynchronous Gibbs with the new to me (?) Hogwild Gibbs algorithms (connected in Terenin et al.’s recent paper, not quoted in the paper), some versions of consensus Monte Carlo covered in earlier posts, the missing links being in my humble opinion an assessment of the worth of those solutions (in the spirit of “here’s the solution, what was the problem again?”) and once again the computing time issue. Chapter 5 briefly discusses some recent developments in variational mean field approximations, which is farther from my interests and (limited) competence, but which appears as a particular class of approximate models and thus could (and should?) relate to likelihood-free methods. Chapter 6 about the current challenges of the field is presumably the most interesting in this monograph in that it produces open questions and suggests directions for future research. For instance, opposing the long term MCMC error with the short term transient part. Or the issue of comparing different implementations in a practical and timely perspective.
“Unfortunately, the factorization does not make it immediately clear how to aggregate on the level of samples without first having to obtain an estimate of the densities themselves.” (p.2)
The recently arXived variational consensus Monte Carlo is a paper by Maxim Rabinovich, Elaine Angelino, and Michael Jordan that approaches the consensus Monte Carlo principle from a variational perspective. As in the embarrassingly parallel version, the target is split into a product of K terms, each being interpreted as an unnormalised density and being fed to a different parallel processor. The most natural partition is to break the data into K subsamples and to raise the prior to the power 1/K in each term. While this decomposition makes sense from a storage perspective, since each bit corresponds to a different subsample of the data, it raises the question of the statistical pertinence of splitting the prior and my feelings about it are now more lukewarm than when I commented on the embarrassingly parallel version, mainly for the reason that it is not reparameterisation invariant—getting different targets if one does the reparameterisation before or after the partition—and hence does not treat the prior as the reference measure it should be. I therefore prefer the version where the same original prior is attached to each part of the partitioned likelihood (and even more the random subsampling approaches discussed in the recent paper of Bardenet, Doucet, and Holmes). Another difficulty with the decomposition is that a product of densities is not a density in most cases (it may even be of infinite mass) and does not offer a natural path to the analysis of samples generated from each term in the product. Nor an explanation as to why those samples should be relevant to construct a sample for the original target.
“The performance of our algorithm depends critically on the choice of aggregation function family.” (p.5)
Since the variational Bayes approach is a common answer to complex products models, Rabinovich et al. explore the use of variational Bayes techniques to build the consensus distribution out of the separate samples. As in Scott et al., and Neiswanger et al., the simulation from the consensus distribution is a transform of simulations from each of the terms in the product, e.g., a weighted average. Which determines the consensus distribution as a member of an aggregation family defined loosely by a Dirac mass. When the transform is a sum of individual terms, variational Bayes solutions get much easier to find and the authors work under this restriction… In the empirical evaluation of this variational Bayes approach as opposed to the uniform and Gaussian averaging options in Scott et al., it improves upon those, except in a mixture example with a large enough common variance.
In fine, despite the relevance of variational Bayes to improve the consensus approximation, I still remain unconvinced about the use of the product of (pseudo-)densities and the subsequent mix of simulations from those components, for the reason mentioned above and also because the tail behaviour of those components is not related with the tail behaviour of the target. Still, this is a working solution to a real problem and as such is a reference for future works.
With my friends Peter Green (Bristol), Krzysztof Łatuszyński (Warwick) and Marcello Pereyra (Bristol), we just arXived the first version of “Bayesian computation: a perspective on the current state, and sampling backwards and forwards”, which first title was the title of this post. This is a survey of our own perspective on Bayesian computation, from what occurred in the last 25 years [a lot!] to what could occur in the near future [a lot as well!]. Submitted to Statistics and Computing towards the special 25th anniversary issue, as announced in an earlier post.. Pulling strength and breadth from each other’s opinion, we have certainly attained more than the sum of our initial respective contributions, but we are welcoming comments about bits and pieces of importance that we miss and even more about promising new directions that are not posted in this survey. (A warning that is should go with most of my surveys is that my input in this paper will not differ by a large margin from ideas expressed here or in previous surveys.)
Second and last day of the NIPS workshops! The collection of topics was quite broad and would have made my choosing an ordeal, except that I was invited to give a talk at the probabilistic programming workshop, solving my dilemma… The first talk by Kathleen Fisher was quite enjoyable in that it gave a conceptual discussion of the motivations for probabilistic languages, drawing an analogy with the early days of computer programming that saw a separation between higher level computer languages and machine programming, with a compiler interface. And calling for a similar separation between the models faced by statistical inference and machine-learning and the corresponding code, if I understood her correctly. This was connected with Frank Wood’s talk of the previous day where he illustrated the concept through a generation of computer codes to approximately generate from standard distributions like Normal or Poisson. Approximately as in ABC, which is why the organisers invited me to talk in this session. However, I was a wee bit lost in the following talks and presumably lost part of my audience during my talk, as I realised later to my dismay when someone told me he had not perceived the distinction between the trees in the random forest procedure and the phylogenetic trees in the population genetic application. Still, while it had for me a sort of Twilight Zone feeling of having stepped in another dimension, attending this workshop was an worthwhile experiment as an eye-opener into a highly different albeit connected field, where code and simulator may take the place of a likelihood function… To the point of defining Hamiltonian Monte Carlo directly on the former, as Vikash Mansinghka showed me at the break.
I completed the day with the final talks in the variational inference workshop, if only to get back on firmer ground! Apart from attending my third talk by Vikash in the conference (but on a completely different topic on variational approximations for discrete particle-ar distributions), a talk by Tim Salimans linked MCMC and variational approximations, using MCMC and HMC to derive variational bounds. (He did not expand on the opposite use of variational approximations to build better proposals.) Overall, I found these two days and my first NIPS conference quite exciting, if somewhat overpowering, with a different atmosphere and a different pace compared with (small or large) statistical meetings. (And a staggering gender imbalance!)
On Thursday, I will travel to Montréal for the two days of NIPS workshop there. On Friday, there is the ABC in Montréal workshop that I cannot but attend! (First occurrence of an “ABC in…” in North America! Sponsored by ISBA as well.) And on Saturday, there is the 3rd NIPS Workshop on Probabilistic Programming where I am invited to give a talk on… ABC! And maybe will manage to get a sneak at the nearby workshop on Advances in variational inference… (0n a very personal side, I wonder if the weather will remain warm enough to go running in the early morning.)
reflections on the probability space induced by moment conditions with implications for Bayesian Inference [refleXions]Posted in Statistics, University life with tags ABC, compatible conditional distributions, empirical likelihood, expectation-propagation, harmonic mean estimator, INLA, latent variable, MCMC, prior distributions, structural model, variational Bayes methods on November 26, 2014 by xi'an
“The main finding is that if the moment functions have one of the properties of a pivotal, then the assertion of a distribution on moment functions coupled with a proper prior does permit Bayesian inference. Without the semi-pivotal condition, the assertion of a distribution for moment functions either partially or completely specifies the prior.” (p.1)
Ron Gallant will present this paper at the Conference in honour of Christian Gouréroux held next week at Dauphine and I have been asked to discuss it. What follows is a collection of notes I made while reading the paper , rather than a coherent discussion, to come later. Hopefully prior to the conference.
The difficulty I have with the approach presented therein stands as much with the presentation as with the contents. I find it difficult to grasp the assumptions behind the model(s) and the motivations for only considering a moment and its distribution. Does it all come down to linking fiducial distributions with Bayesian approaches? In which case I am as usual sceptical about the ability to impose an arbitrary distribution on an arbitrary transform of the pair (x,θ), where x denotes the data. Rather than a genuine prior x likelihood construct. But I bet this is mostly linked with my lack of understanding of the notion of structural models.
“We are concerned with situations where the structural model does not imply exogeneity of θ, or one prefers not to rely on an assumption of exogeneity, or one cannot construct a likelihood at all due to the complexity of the model, or one does not trust the numerical approximations needed to construct a likelihood.” (p.4)
As often with econometrics papers, this notion of structural model sets me astray: does this mean any latent variable model or an incompletely defined model, and if so why is it incompletely defined? From a frequentist perspective anything random is not a parameter. The term exogeneity also hints at this notion of the parameter being not truly a parameter, but including latent variables and maybe random effects. Reading further (p.7) drives me to understand the structural model as defined by a moment condition, in the sense that
has a unique solution in θ under the true model. However the focus then seems to make a major switch as Gallant considers the distribution of a pivotal quantity like
as induced by the joint distribution on (x,θ), hence conversely inducing constraints on this joint, as well as an associated conditional. Which is something I have trouble understanding, First, where does this assumed distribution on Z stem from? And, second, exchanging randomness of terms in a random variable as if it was a linear equation is a pretty sure way to produce paradoxes and measure theoretic difficulties.
The purely mathematical problem itself is puzzling: if one knows the distribution of the transform Z=Z(X,Λ), what does that imply on the joint distribution of (X,Λ)? It seems unlikely this will induce a single prior and/or a single likelihood… It is actually more probable that the distribution one arbitrarily selects on m(x,θ) is incompatible with a joint on (x,θ), isn’t it?
“The usual computational method is MCMC (Markov chain Monte Carlo) for which the best known reference in econometrics is Chernozhukov and Hong (2003).” (p.6)
While I never heard of this reference before, it looks like a 50 page survey and may be sufficient for an introduction to MCMC methods for econometricians. What I do not get though is the connection between this reference to MCMC and the overall discussion of constructing priors (or not) out of fiducial distributions. The author also suggests using MCMC to produce the MAP estimate but this always stroke me as inefficient (unless one uses our SAME algorithm of course).
“One can also compute the marginal likelihood from the chain (Newton and Raftery (1994)), which is used for Bayesian model comparison.” (p.22)
Not the best solution to rely on harmonic means for marginal likelihoods…. Definitely not. While the author actually uses the stabilised version (15) of Newton and Raftery (1994) estimator, which in retrospect looks much like a bridge sampling estimator of sorts, it remains dangerously close to the original [harmonic mean solution] especially for a vague prior. And it only works when the likelihood is available in closed form.
“The MCMC chains were comprised of 100,000 draws well past the point where transients died off.” (p.22)
I wonder if the second statement (with a very nice image of those dying transients!) is intended as a consequence of the first one or independently.
“A common situation that requires consideration of the notions that follow is that deriving the likelihood from a structural model is analytically intractable and one cannot verify that the numerical approximations one would have to make to circumvent the intractability are sufficiently accurate.” (p.7)
This then is a completely different business, namely that defining a joint distribution by mean of moment equations prevents regular Bayesian inference because the likelihood is not available. This is more exciting because (i) there are alternative available! From ABC to INLA (maybe) to EP to variational Bayes (maybe). And beyond. In particular, the moment equations are strongly and even insistently suggesting that empirical likelihood techniques could be well-suited to this setting. And (ii) it is no longer a mathematical worry: there exist a joint distribution on m(x,θ), induced by a (or many) joint distribution on (x,θ). So the question of finding whether or not it induces a single proper prior on θ becomes relevant. But, if I want to use ABC, being given the distribution of m(x,θ) seems to mean I can only generate new values of this transform while missing a natural distance between observations and pseudo-observations. Still, I entertain lingering doubts that this is the meaning of the study. Where does the joint distribution come from..?!
“Typically C is coarse in the sense that it does not contain all the Borel sets (…) The probability space cannot be used for Bayesian inference”
My understanding of that part is that defining a joint on m(x,θ) is not always enough to deduce a (unique) posterior on θ, which is fine and correct, but rather anticlimactic. This sounds to be what Gallant calls a “partial specification of the prior” (p.9).
Overall, after this linear read, I remain very much puzzled by the statistical (or Bayesian) implications of the paper . The fact that the moment conditions are central to the approach would once again induce me to check the properties of an alternative approach like empirical likelihood.