Archive for University of Warwick
“By accepting of having obtained a poor approximation to the posterior, except for the location of its main mode, we switch to maximum likelihood estimation.”
Presumably the first paper ever quoting from the ‘Og! Indeed, Umberto Picchini arXived a paper about a technique merging ABC with prior feedback (rechristened data cloning by S. Lele), where a maximum likelihood estimate is produced by an ABC-MCMC algorithm. For state-space models. This relates to an earlier paper by Fabio Rubio and Adam Johansen (Warwick), who also suggested using ABC to approximate the maximum likelihood estimate. Here, the idea is to use an increasing number of replicates of the latent variables, as in our SAME algorithm, to spike the posterior around the maximum of the (observed) likelihood. An ABC version of this posterior returns a mean value as an approximate maximum likelihood estimate.
“This is a so-called “likelihood-free” approach [Sisson and Fan, 2011], meaning that knowledge of the complete expression for the likelihood function is not required.”
The above remark is sort of inappropriate in that it applies to a non-ABC setting where the latent variables are simulated from the exact marginal distributions, that is, unconditional on the data, and hence their density cancels in the Metropolis-Hastings ratio. This pre-dates ABC by a few years, since this was an early version of particle filter.
“In this work we are explicitly avoiding the most typical usage of ABC, where the posterior is conditional on summary statistics of data S(y), rather than y.”
Another point I find rather negative in that, for state-space models, using the entire time-series as a “summary statistic” is unlikely to produce a good approximation.
The discussion on the respective choices of the ABC tolerance δ and on the prior feedback number of copies K is quite interesting, in that Umberto Picchini suggests setting δ first before increasing the number of copies. However, since the posterior gets more and more peaked as K increases, the consequences on the acceptance rate of the related ABC algorithm are unclear. Another interesting feature is that the underlying MCMC proposal on the parameter θ is an independent proposal, tuned during the warm-up stage of the algorithm. Since the tuning is repeated at each temperature, there are some loose ends as to whether or not it is a genuine Markov chain method. The same question arises when considering that additional past replicas need to be simulated when K increases. (Although they can be considered as virtual components of a vector made of an infinite number of replicas, to be used when needed.)
The simulation study involves a regular regression with 101 observations, a stochastic Gompertz model studied by Sophie Donnet, Jean-Louis Foulley, and Adeline Samson in 2010. With 12 points. And a simple Markov model. Again with 12 points. While the ABC-DC solutions are close enough to the true MLEs whenever available, a comparison with the cheaper ABC Bayes estimates would have been of interest as well.
As a coincidence, I received my copy of JRSS Series B with the Read Paper by Mathieu Gerber and Nicolas Chopin on sequential quasi Monte Carlo just as I was preparing an arXival of a few discussions on the paper! Among the [numerous and diverse] discussions, a few were of particular interest to me [I highlighted members of the University of Warwick and of Université Paris-Dauphine to suggest potential biases!]:
- Mike Pitt (Warwick), Murray Pollock et al. (Warwick) and Finke et al. (Warwick) all suggested combining quasi Monte Carlo with pseudomarginal Metropolis-Hastings, pMCMC (Pitt) and Rao-Bklackwellisation (Finke et al.);
- Arnaud Doucet pointed out that John Skilling had used the Hilbert (ordering) curve in a 2004 paper;
- Chris Oates, Dan Simpson and Mark Girolami (Warwick) suggested combining quasi Monte Carlo with their functional control variate idea;
- Richard Everitt wondered about the dimension barrier of d=6 and about possible slice extensions;
- Zhijian He and Art Owen pointed out simple solutions to handle a random number of uniforms (for simulating each step in sequential Monte Carlo), namely to start with quasi Monte Carlo and end up with regular Monte Carlo, in an hybrid manner;
- Hans Künsch points out the connection with systematic resampling à la Carpenter, Clifford and Fearnhead (1999) and wonders about separating the impact of quasi Monte Carlo between resampling and propagating [which vaguely links to one of my comments];
- Pierre L’Ecuyer points out a possible improvement over the Hilbert curve by a preliminary sorting;
- Frederik Lindsten and Sumeet Singh propose using ABC to extend the backward smoother to intractable cases [but still with a fixed number of uniforms to use at each step], as well as Mateu and Ryder (Paris-Dauphine) for a more general class of intractable models;
- Omiros Papaspiliopoulos wonders at the possibility of a quasi Markov chain with “low discrepancy paths”;
- Daniel Rudolf suggest linking the error rate of sequential quasi Monte Carlo with the bounds of Vapnik and Ĉervonenkis (1977).
The arXiv document also includes the discussions by Julyan Arbel and Igor Prünster (Turino) on the Bayesian nonparametric side of sqMC and by Robin Ryder (Dauphine) on the potential of sqMC for ABC.
While visiting Dauphine, Natesh Pillai and Aaron Smith pointed out this interesting paper of Joris Bierkens (Warwick) that had escaped my arXiv watch/monitoring. The paper is about turning Metropolis-Hastings algorithms into non-reversible versions, towards improving mixing.
In a discrete setting, a way to produce a non-reversible move is to mix the proposal kernel Q with its time-reversed version Q’ and use an acceptance probability of the form
where ε is any weight. This construction is generalised in the paper to any vorticity (skew-symmetric with zero sum rows) matrix Γ, with the acceptance probability
where ε is small enough to ensure all numerator values are non-negative. This is a rather annoying assumption in that, except for the special case derived from the time-reversed kernel, it has to be checked over all pairs (x,y). (I first thought it also implied the normalising constant of π but everything can be set in terms of the unormalised version of π, Γ or ε included.) The paper establishes that the new acceptance probability preserves π as its stationary distribution. An alternative construction is to make the proposal change from Q in H such that H(x,y)=Q(x,y)+εΓ(x,y)/π(x). Which seems more pertinent as not changing the proposal cannot improve that much the mixing behaviour of the chain. Still, the move to the non-reversible versions has the noticeable plus of decreasing the asymptotic variance of the Monte Carlo estimate for any integrable function. Any. (Those results are found in the physics literature of the 2000’s.)
The extension to the continuous case is a wee bit more delicate. One needs to find an anti-symmetric vortex function g with zero integral [equivalent to the row sums being zero] such that g(x,y)+π(y)q(y,x)>0 and with same support as π(x)q(x,y) so that the acceptance probability of g(x,y)+π(y)q(y,x)/π(x)q(x,y) leads to π being the stationary distribution. Once again g(x,y)=ε(π(y)q(y,x)-π(x)q(x,y)) is a natural candidate but it is unclear to me why it should work. As the paper only contains one illustration for the discretised Ornstein-Uhlenbeck model, with the above choice of g for a small enough ε (a point I fail to understand since any ε<1 should provide a positive g(x,y)+π(y)q(y,x)), it is also unclear to me that this modification (i) is widely applicable and (ii) is relevant for genuine MCMC settings.
This week in Warwick was one of the busiest ones ever as I had to juggle between two workshops, including one in Oxford, a departmental meeting, two paper revisions, two pre-vivas, and a seminar in Leeds. Not to mention a broken toe (!), a flat tire (!!), and a diner at the X. Hardly anytime for writing blog entries..! Fortunately, I managed to squeeze time for working with Kerrie Mengersen who was visiting Warwick this fortnight. Finding new directions for the (A)BCel approach we developed a few years ago with Pierre Pudlo. The workshop in Oxford was quite informal with talks from PhD students [I fear I cannot discuss here as the papers are not online yet]. And one talk by François Caron about estimating sparse networks with not exactly exchangeable priors and completely random measures. And one talk by Kerrie Mengersen on a new and in-progress approach to handling Big Data that I found quite convincing (if again cannot discuss here). The probabilistic numerics workshop was discussed in yesterday’s post and I managed to discuss it a wee bit further with the organisers at The X restaurant in Kenilworth. (As a superfluous aside, and after a second sampling this year, I concluded that the Michelin star somewhat undeserved in that the dishes at The X are not particularly imaginative or tasty, the excellent sourdough bread being the best part of the meal!) I was expecting the train ride to Leeds to be highly bucolic as it went through the sunny countryside of South Yorkshire, with newly born lambs running in the bright green fields surrounded by old stone walls…, but instead went through endless villages with their rows of brick houses. Not that I have anything against brick houses, mind! Only, I had not realised how dense this part of England was, this presumably getting back all the way to the Industrial Revolution with the Manchester-Leeds-Birmingham triangle.
My seminar in Leeds was as exciting as in Amsterdam last week and with a large audience, so I got many and only interesting questions, from the issue of turning the output (i.e., the posterior on α) into a decision rule, to making decision in the event of a non-conclusive posterior, to links with earlier frequentist resolutions, to whether or not we were able to solve the Lindley-Jeffreys paradox (we are not!, which makes a lot of sense), to the possibility of running a subjective or a sequential version. After the seminar I enjoyed a perfect Indian dinner at Aagrah, apparently a Yorkshire institution, with the right balance between too hot and too mild, i.e., enough spices to break a good sweat but not too many to loose any sense of taste!
I attended an highly unusual workshop while in Warwick last week. Unusual for me, obviously. It was about probabilistic numerics, i.e., the use of probabilistic or stochastic arguments in the numerical resolution of (possibly) deterministic problems. The notion in this approach is fairly Bayesian in that it makes use to prior information or belief about the quantity of interest, e.g., a function, to construct an usually Gaussian process prior and derive both an estimator that is identical to a numerical method (e.g., Runge-Kutta or trapezoidal integration) and uncertainty or variability around this estimator. While I did not grasp much more than the classy introduction talk by Philipp Hennig, this concept sounds fairly interesting, if only because of the Bayesian connection, and I wonder if we will soon see a probability numerics section at ISBA! More seriously, placing priors on functions or functionals is a highly formal perspective (as in Bayesian non-parametrics) and it makes me wonder how much of the data (evaluation of a function at a given set of points) and how much of the prior is reflected in the output [variability]. (Obviously, one could also ask a similar question for statistical analyses!) For instance, issues of singularity arise among those stochastic process priors.
Another question that stemmed from this talk is whether or not more efficient numerical methods can derived that way, in addition to recovering the most classical ones. Somewhat, somehow, given the idealised nature of the prior, it feels like priors could be more easily compared or ranked than in classical statistical problems. Since the aim is to figure out the value of an integral or the solution to an ODE. (Or maybe not, since again almost the same could be said about estimating a normal mean.)
Cristiano Varin, Manuela Cattelan and David Firth (Warwick) have written a paper on the statistical analysis of citations and index factors, paper that is going to be Read at the Royal Statistical Society next May the 13th. And hence is completely open to contributed discussions. Now, I have written several entries on the ‘Og about the limited trust I set to citation indicators, as well as about the abuse made of those. However I do not think I will contribute to the discussion as my reservations are about the whole bibliometrics excesses and not about the methodology used in the paper.
The paper builds several models on the citation data provided by the “Web of Science” compiled by Thompson Reuters. The focus is on 47 Statistics journals, with a citation horizon of ten years, which is much more reasonable than the two years in the regular impact factor. A first feature of interest in the descriptive analysis of the data is that all journals have a majority of citations from and to journals outside statistics or at least outside the list. Which I find quite surprising. The authors also build a cluster based on the exchange of citations, resulting in rather predictable clusters, even though JCGS and Statistics and Computing escape the computational cluster to end up in theory and methods along Annals of Statistics and JRSS Series B.
In addition to the unsavoury impact factor, a ranking method discussed in the paper is the eigenfactor score that starts with a Markov exploration of articles by going at random to one of the papers in the reference list and so on. (Which shares drawbacks with the impact factor, e.g., in that it does not account for the good or bad reason the paper is cited.) Most methods produce the Big Four at the top, with Series B ranked #1, and Communications in Statistics A and B at the bottom, along with Journal of Applied Statistics. Again, rather anticlimactic.
The major modelling input is based on Stephen Stigler’s model, a generalised linear model on the log-odds of cross citations. The Big Four once again receive high scores, with Series B still much ahead. (The authors later question the bias due to the Read Paper effect, but cannot easily evaluate this impact. While some Read Papers like Spiegelhalter et al. 2002 DIC do generate enormous citation traffic, to the point of getting re-read!, other journals also contain discussion papers. And are free to include an on-line contributed discussion section if they wish.) Using an extra ranking lasso step does not change things.
In order to check the relevance of such rankings, the authors also look at the connection with the conclusions of the (UK) 2008 Research Assessment Exercise. They conclude that the normalised eigenfactor score and Stigler model are more correlated with the RAE ranking than the other indicators. Which means either that the scores are good predictors or that the RAE panel relied too heavily on bibliometrics! The more global conclusion is that clusters of journals or researchers have very close indicators, hence that ranking should be conducted with more caution that it is currently. And, more importantly, that reverting the indices from journals to researchers has no validation and little information.