Archive for Monash University
[I wrote this set of comments right after MCqMC 2016 on a preliminary version of the paper so mileage may vary in terms of the adequation to the current version!]
In warp-U bridge sampling, newly arXived and first presented at MCqMC 16, Xiao-Li Meng continues (in collaboration with Lahzi Wang) his exploration of bridge sampling techniques towards improving the estimation of normalising constants and ratios thereof. The bridge sampling estimator of Meng and Wong (1996) is an harmonic mean importance sampler that requires iterations as it depends on the ratio of interest. Given that the normalising constant of a density does not depend on the chosen parameterisation in the sense that the Jacobian transform preserves this constant, a degree of freedom is in the choice of the parameterisation. This is the idea behind warp transformations. The initial version of Meng and Schilling (2002) used location-scale transforms, while the warp-U solution goes for a multiple location-scale transform that can be seen as based on a location-scale mixture representation of the target. With K components. This approach can also be seen as a sort of artificial reversible jump algorithm when one model is fully known. A strategy Nicolas and I also proposed in our nested sampling Biometrika paper.
Once such a mixture approximation is obtained. each and every component of the mixture can be turned into the standard version of the location-scale family by the appropriate location-scale transform. Since the component index k is unknown for a given X, they call this transform a random transform, which I find somewhat more confusing that helpful. The conditional distribution of the index given the observable x is well-known for mixtures and it is used here to weight the component-wise location-scale transforms of the original distribution p into something that looks rather similar to the standard version of the location-scale family. If no mode has been forgotten by the mixture. The simulations from the original p are then rescaled by one of those transforms, which index k is picked according to the conditional distribution. As explained later to me by XL, the random[ness] in the picture is due to the inclusion of a random ± sign. Still, in the notation introduced in (13), I do not get how the distribution Þ [sorry for using different symbols, I cannot render a tilde on a p] is defined since both ψ and W are random. Is it the marginal? In which case it would read as a weighted average of rescaled versions of p. I have the same problem with Theorem 1 in that I do not understand how one equates Þ with the joint distribution.
Equation (21) is much more illuminating (I find) than the previous explanation in that it exposes the fact that the principle is one of aiming at a new distribution for both the target and the importance function, with hopes that the fit will get better. It could have been better to avoid the notion of random transform, then, but this is mostly a matter of conveying the notion.
On more specifics points (or minutiae), the unboundedness of the likelihood is rarely if ever a problem when using EM. An alternative to the multiple start EM proposal would then be to get sequential and estimate the mixture in a sequential manner, only adding a component when it seems worth it. See eg Chopin and Pelgrin (2004) and Chopin (2007). This could also help with the bias mentioned therein since only a (tiny?) fraction of the data would be used. And the number of components K has an impact on the accuracy of the approximation, as in not missing a mode, and on the computing time. However my suggestion would be to avoid estimating K as this must be immensely costly.
Section 6 obviously relates to my folded Markov interests. If I understand correctly, the paper argues that the transformed density Þ does not need to be computed when considering the folding-move-unfolding step as a single step rather than three steps. I fear the description between equations (30) and (31) is missing the move step over the transformed space. Also on a personal basis I still do not see how to add this approach to our folding methodology, even though the different transforms act as as many replicas of the original Markov chain.
Scandinavian picaresque, in the spirit of the novels of Paasilinna, and following another book by Jonas Jonasson already commented on the ‘Og, The Girl who saved the King of Sweden, but not as funny, because of the heavy recourse to World history, the main (100 year old) character meeting a large collection of major historical figures. And crossing the Himalayas when escaping from a Russian Gulag, which reminded me of this fantastic if possibly apocryphal The Long Walk where a group of Polish prisoners was making it through the Gobi desert to reach India and freedom (or death). The story here is funny but not that funny and once it is over, there is not much to say about it, which is why I left it on a bookshare table in Monash. The current events are somewhat dull, in opposition to the 100 year life of Allan, and the police enquiry a tad too predictable. Plus the themes are somewhat comparable to The Girl who …, with atom bombs, cold war, brothers hating one another…
“This formulation reveals an interesting connection between multiple hypothesis testing and mixture modelling with the class labels corresponding to the accepted hypotheses in each test.”
After my seminar at Monash University last Friday, David Dowe pointed out to me the recent work by Enes Makalic and Daniel Schmidt on minimum description length (MDL) methods for multiple testing as somewhat related to our testing by mixture paper. Work which appeared in the proceedings of the 4th Workshop on Information Theoretic Methods in Science and Engineering (WITMSE-11), that took place in Helsinki, Finland, in 2011. Minimal encoding length approaches lead to choosing the model that enjoys the smallest coding length. Connected with, e.g., Rissannen‘s approach. The extension in this paper consists in considering K hypotheses at once on a collection of m datasets (the multiple then bears on the datasets rather than on the hypotheses). And to associate an hypothesis index to each dataset. When the objective function is the sum of (generalised) penalised likelihoods [as in BIC], it leads to selecting the “minimal length” model for each dataset. But the authors introduce weights or probabilities for each of the K hypotheses, which indeed then amounts to a mixture-like representation on the exponentiated codelengths. Which estimation by optimal coding was first proposed by Chris Wallace in his book. This approach eliminates the model parameters at an earlier stage, e.g. by maximum likelihood estimation, to return a quantity that only depends on the model index and the data. In fine, the purpose of the method differs from ours in that the former aims at identifying an appropriate hypothesis for each group of observations, rather than ranking those hypotheses for the entire dataset by considering the posterior distribution of the weights in the later. The mixture has somehow more of a substance in the first case, where separating the datasets into groups is part of the inference.
Taking advantage of being in San Francisco, I flew yesterday to Australia over the Pacific, crossing for the first time the day line. The 15 hour Qantas flight to Sydney was remarkably smooth and quiet, with most passengers sleeping for most of the way, and it gave me a great opportunity to go over several papers I wanted to read and review. Over the next week or so, I will work with my friends and co-authors David Frazier and Gael Martin at Monash University (and undoubtedly enjoy the great food and wine scene!). Before flying back to Paris (alas via San Francisco rather than direct).
With David Frazier and Gael Martin from Monash University, and with Judith Rousseau (Paris-Dauphine), we have now completed and arXived a paper entitled Asymptotic Properties of Approximate Bayesian Computation. This paper undertakes a fairly complete study of the large sample properties of ABC under weak regularity conditions. We produce therein sufficient conditions for posterior concentration, asymptotic normality of the ABC posterior estimate, and asymptotic normality of the ABC posterior mean. Moreover, those (theoretical) results are of significant import for practitioners of ABC as they pertain to the choice of tolerance ε used within ABC for selecting parameter draws. In particular, they [the results] contradict the conventional ABC wisdom that this tolerance should always be taken as small as the computing budget allows.
Now, this paper bears some similarities with our earlier paper on the consistency of ABC, written with David and Gael. As it happens, the paper was rejected after submission and I then discussed it in an internal seminar in Paris-Dauphine, with Judith taking part in the discussion and quickly suggesting some alternative approach that is now central to the current paper. The previous version analysed Bayesian consistency of ABC under specific uniformity conditions on the summary statistics used within ABC. But conditions for consistency are now much weaker conditions than earlier, thanks to Judith’s input!
- Li and Fearnhead (2015) considers an ABC algorithm based on kernel smoothing, whereas our interest is the original ABC accept-reject and its many derivatives
- our theoretical approach permits a complete study of the asymptotic properties of ABC, posterior concentration, asymptotic normality of ABC posteriors, and asymptotic normality of the ABC posterior mean, whereas Li and Fearnhead (2015) is only concerned with asymptotic normality of the ABC posterior mean estimator (and various related point estimators);
- the results of Li and Fearnhead (2015) are derived under very strict uniformity and continuity/differentiability conditions, which bear a strong resemblance to those conditions in Yuan and Clark (2004) and Creel et al. (2015), while the result herein do not rely on such conditions and only assume very weak regularity conditions on the summaries statistics themselves; this difference allows us to characterise the behaviour of ABC in situations not covered by the approach taken in Li and Fearnhead (2015);
With Gael Martin, Brendan McCabe, David T. Frazier, and Worapree Maneesoonthorn, we arXived (and submitted) a strongly revised version of our earlier paper. We begin by demonstrating that reduction to a set of sufficient statistics of reduced dimension relative to the sample size is infeasible for most state-space models, hence calling for the use of partial posteriors in such settings. Then we give conditions [like parameter identification] under which ABC methods are Bayesian consistent, when using an auxiliary model to produce summaries, either as MLEs or [more efficiently] scores. Indeed, for the order of accuracy required by the ABC perspective, scores are equivalent to MLEs but are computed much faster than MLEs. Those conditions happen to to be weaker than those found in the recent papers of Li and Fearnhead (2016) and Creel et al. (2015). In particular as we make no assumption about the limiting distributions of the summary statistics. We also tackle the dimensionality curse that plagues ABC techniques by numerically exhibiting the improved accuracy brought by looking at marginal rather than joint modes. That is, by matching individual parameters via the corresponding scalar score of the integrated auxiliary likelihood rather than matching on the multi-dimensional score statistics. The approach is illustrated on realistically complex models, namely a (latent) Ornstein-Ulenbeck process with a discrete time linear Gaussian approximation is adopted and a Kalman filter auxiliary likelihood. And a square root volatility process with an auxiliary likelihood associated with a Euler discretisation and the augmented unscented Kalman filter. In our experiments, we compared our auxiliary based technique to the two-step approach of Fearnhead and Prangle (in the Read Paper of 2012), exhibiting improvement for the examples analysed therein. Somewhat predictably, an important challenge in this approach that is common with the related techniques of indirect inference and efficient methods of moments, is the choice of a computationally efficient and accurate auxiliary model. But most of the current ABC literature discusses the role and choice of the summary statistics, which amounts to the same challenge, while missing the regularity provided by score functions of our auxiliary models.