Archive for Kalman filter

Computational Bayesian Statistics [book review]

Posted in Books, Statistics with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , on February 1, 2019 by xi'an

This Cambridge University Press book by M. Antónia Amaral Turkman, Carlos Daniel Paulino, and Peter Müller is an enlarged translation of a set of lecture notes in Portuguese. (Warning: I have known Peter Müller from his PhD years in Purdue University and cannot pretend to perfect objectivity. For one thing, Peter once brought me frozen-solid beer: revenge can also be served cold!) Which reminds me of my 1994 French edition of Méthodes de Monte Carlo par chaînes de Markov, considerably upgraded into Monte Carlo Statistical Methods (1998) thanks to the input of George Casella. (Re-warning: As an author of books on the same topic(s), I can even less pretend to objectivity.)

“The “great idea” behind the development of computational Bayesian statistics is the recognition that Bayesian inference can be implemented by way of simulation from the posterior distribution.”

The book is written from a strong, almost militant, subjective Bayesian perspective (as, e.g., when half-Bayesians are mentioned!). Subjective (and militant) as in Dennis Lindley‘s writings, eminently quoted therein. As well as in Tony O’Hagan‘s. Arguing that the sole notion of a Bayesian estimator is the entire posterior distribution. Unless one brings in a loss function. The book also discusses the Bayes factor in a critical manner, which is fine from my perspective.  (Although the ban on improper priors makes its appearance in a very indirect way at the end of the last exercise of the first chapter.)

Somewhat at odds with the subjectivist stance of the previous chapter, the chapter on prior construction only considers non-informative and conjugate priors. Which, while understandable in an introductory book, is a wee bit disappointing. (When mentioning Jeffreys’ prior in multidimensional settings, the authors allude to using univariate Jeffreys’ rules for the marginal prior distributions, which is not a well-defined concept or else Bernardo’s and Berger’s reference priors would not have been considered.) The chapter also mentions the likelihood principle at the end of the last exercise, without a mention of the debate about its derivation by Birnbaum. Or Deborah Mayo’s recent reassessment of the strong likelihood principle. The following chapter is a sequence of illustrations in classical exponential family models, classical in that it is found in many Bayesian textbooks. (Except for the Poison model found in Exercise 3.3!)

Nothing to complain (!) about the introduction of Monte Carlo methods in the next chapter, especially about the notion of inference by Monte Carlo methods. And the illustration by Bayesian design. The chapter also introduces Rao-Blackwellisation [prior to introducing Gibbs sampling!]. And the simplest form of bridge sampling. (Resuscitating the weighted bootstrap of Gelfand and Smith (1990) may not be particularly urgent for an introduction to the topic.) There is furthermore a section on sequential Monte Carlo, including the Kalman filter and particle filters, in the spirit of Pitt and Shephard (1999). This chapter is thus rather ambitious in the amount of material covered with a mere 25 pages. Consensus Monte Carlo is even mentioned in the exercise section.

“This and other aspects that could be criticized should not prevent one from using this [Bayes factor] method in some contexts, with due caution.”

Chapter 5 turns back to inference with model assessment. Using Bayesian p-values for model assessment. (With an harmonic mean spotted in Example 5.1!, with no warning about the risks, except later in 5.3.2.) And model comparison. Presenting the whole collection of xIC information criteria. from AIC to WAIC, including a criticism of DIC. The chapter feels somewhat inconclusive but methinks this is the right feeling on the current state of the methodology for running inference about the model itself.

“Hint: There is a very easy answer.”

Chapter 6 is also a mostly standard introduction to Metropolis-Hastings algorithms and the Gibbs sampler. (The argument given later of a Metropolis-Hastings algorithm with acceptance probability one does not work.) The Gibbs section also mentions demarginalization as a [latent or auxiliary variable] way to simulate from complex distributions [as we do], but without defining the notion. It also references the precursor paper of Tanner and Wong (1987). The chapter further covers slice sampling and Hamiltonian Monte Carlo, the later with sufficient details to lead to reproducible implementations. Followed by another standard section on convergence assessment, returning to the 1990’s feud of single versus multiple chain(s). The exercise section gets much larger than in earlier chapters with several pages dedicated to most problems. Including one on ABC, maybe not very helpful in this context!

“…dimension padding (…) is essentially all that is to be said about the reversible jump. The rest are details.”

The next chapter is (somewhat logically) the follow-up for trans-dimensional problems and marginal likelihood approximations. Including Chib’s (1995) method [with no warning about potential biases], the spike & slab approach of George and McCulloch (1993) that I remember reading in a café at the University of Wyoming!, the somewhat antiquated MC³ of Madigan and York (1995). And then the much more recent array of Bayesian lasso techniques. The trans-dimensional issues are covered by the pseudo-priors of Carlin and Chib (1995) and the reversible jump MCMC approach of Green (1995), the later being much more widely employed in the literature, albeit difficult to tune [and even to comprehensively describe, as shown by the algorithmic representation in the book] and only recommended for a large number of models under comparison. Once again the exercise section is most detailed, with recent entries like the EM-like variable selection algorithm of Ročková and George (2014).

The book also includes a chapter on analytical approximations, which is also the case in ours [with George Casella] despite my reluctance to bring them next to exact (simulation) methods. The central object is the INLA methodology of Rue et al. (2009) [absent from our book for obvious calendar reasons, although Laplace and saddlepoint approximations are found there as well]. With a reasonable amount of details, although stopping short of implementable reproducibility. Variational Bayes also makes an appearance, mostly following the very recent Blei et al. (2017).

The gem and originality of the book are primarily to be found in the final and ninth chapter where four software are described, all with interfaces to R: OpenBUGS, JAGS, BayesX, and Stan, plus R-INLA which is processed in the second half of the chapter (because this is not a simulation method). As in the remainder of the book, the illustrations are related to medical applications. Worth mentioning is the reminder that BUGS came in parallel with Gelfand and Smith (1990) Gibbs sampler rather than as a consequence. Even though the formalisation of the Markov chain Monte Carlo principle by the later helped in boosting the power of this software. (I also appreciated the mention made of Sylvia Richardson’s role in this story.) Since every software is illustrated in depth with relevant code and output, and even with the shortest possible description of its principle and modus vivendi, the chapter is 60 pages long [and missing a comparative conclusion]. Given my total ignorance of the very existence of the BayesX software, I am wondering at the relevance of its inclusion in this description rather than, say, other general R packages developed by authors of books such as Peter Rossi. The chapter also includes a description of CODA, with an R version developed by Martin Plummer [now a Warwick colleague].

In conclusion, this is a high-quality and all-inclusive introduction to Bayesian statistics and its computational aspects. By comparison, I find it much more ambitious and informative than Albert’s. If somehow less pedagogical than the thicker book of Richard McElreath. (The repeated references to Paulino et al.  (2018) in the text do not strike me as particularly useful given that this other book is written in Portuguese. Unless an English translation is in preparation.)

Disclaimer: this book was sent to me by CUP for endorsement and here is what I wrote in reply for a back-cover entry:

An introduction to computational Bayesian statistics cooked to perfection, with the right mix of ingredients, from the spirited defense of the Bayesian approach, to the description of the tools of the Bayesian trade, to a definitely broad and very much up-to-date presentation of Monte Carlo and Laplace approximation methods, to an helpful description of the most common software. And spiced up with critical perspectives on some common practices and an healthy focus on model assessment and model selection. Highly recommended on the menu of Bayesian textbooks!

And this review is likely to appear in CHANCE, in my book reviews column.

from least squares to signal processing and particle filtering

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , on June 6, 2017 by xi'an

Nozer Singpurwalla, Nick. Polson, and Refik Soyer have just arXived a remarkable survey on the history of signal processing, from Gauß, Yule, Kolmogorov and Wiener, to Ragazzini, Shanon, Kálmán [who, I was surprised to learn, died in Gainesville last year!], Gibbs sampling, and the particle filters of the 1990’s.

auxiliary likelihood-based approximate Bayesian computation in state-space models

Posted in Books, pictures, Statistics, University life with tags , , , , , , , on May 2, 2016 by xi'an

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.

Bayesian filtering and smoothing [book review]

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , , , , on February 25, 2015 by xi'an

When in Warwick last October, I met Simo Särkkä, who told me he had published an IMS monograph on Bayesian filtering and smoothing the year before. I thought it would be an appropriate book to review for CHANCE and tried to get a copy from Oxford University Press, unsuccessfully. I thus bought my own book that I received two weeks ago and took the opportunity of my Czech vacations to read it… [A warning pre-empting accusations of self-plagiarism: this is a preliminary draft for a review to appear in CHANCE under my true name!]

“From the Bayesian estimation point of view both the states and the static parameters are unknown (random) parameters of the system.” (p.20)

 Bayesian filtering and smoothing is an introduction to the topic that essentially starts from ground zero. Chapter 1 motivates the use of filtering and smoothing through examples and highlights the naturally Bayesian approach to the problem(s). Two graphs illustrate the difference between filtering and smoothing by plotting for the same series of observations the successive confidence bands. The performances are obviously poorer with filtering but the fact that those intervals are point-wise rather than joint, i.e., that the graphs do not provide a confidence band. (The exercise section of that chapter is superfluous in that it suggests re-reading Kalman’s original paper and rephrases the Monty Hall paradox in a story unconnected with filtering!) Chapter 2 gives an introduction to Bayesian statistics in general, with a few pages on Bayesian computational methods. A first remark is that the above quote is both correct and mildly confusing in that the parameters can be consistently estimated, while the latent states cannot. A second remark is that justifying the MAP as associated with the 0-1 loss is incorrect in continuous settings.  The third chapter deals with the batch updating of the posterior distribution, i.e., that the posterior at time t is the prior at time t+1. With applications to state-space systems including the Kalman filter. The fourth to sixth chapters concentrate on this Kalman filter and its extension, and I find it somewhat unsatisfactory in that the collection of such filters is overwhelming for a neophyte. And no assessment of the estimation error when the model is misspecified appears at this stage. And, as usual, I find the unscented Kalman filter hard to fathom! The same feeling applies to the smoothing chapters, from Chapter 8 to Chapter 10. Which mimic the earlier ones. Continue reading

Approximate Bayesian Computation in state space models

Posted in Statistics, Travel, University life with tags , , , , , , , on October 2, 2014 by xi'an

While it took quite a while (!), with several visits by three of us to our respective antipodes, incl. my exciting trip to Melbourne and Monash University two years ago, our paper on ABC for state space models was arXived yesterday! Thanks to my coauthors, Gael Martin, Brendan McCabe, and  Worapree Maneesoonthorn,  I am very glad of this outcome and of the new perspective on ABC it produces.  For one thing, it concentrates on the selection of summary statistics from a more econometrics than usual point of view, defining asymptotic sufficiency in this context and demonstrated that both asymptotic sufficiency and Bayes consistency can be achieved when using maximum likelihood estimators of the parameters of an auxiliary model as summary statistics. In addition, the proximity to (asymptotic) sufficiency yielded by the MLE is replicated by the score vector. Using the score instead of the MLE as a summary statistics allows for huge gains in terms of speed. The method is then applied to a continuous time state space model, using as auxiliary model an augmented unscented Kalman filter. We also found in the various state space models tested therein that the ABC approach based on the marginal [likelihood] score was performing quite well, including wrt Fearnhead’s and Prangle’s (2012) approach… I like the idea of using such a generic object as the unscented Kalman filter for state space models, even when it is not a particularly accurate representation of the true model. Another appealing feature of the paper is in the connections made with indirect inference.