Archive for maximum likelihood estimation

efficient approximate Bayesian inference for models with intractable likleihood

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , on July 6, 2015 by xi'an

Awalé board on my garden table, March 15, 2013Dalhin, Villani [Mattias, not Cédric] and Schön arXived a paper this week with the above title. The type of intractable likelihood they consider is a non-linear state-space (HMM) model and the SMC-ABC they propose is based on an optimised Laplace approximation. That is, replacing the posterior distribution on the parameter θ with a normal distribution obtained by a Taylor expansion of the log-likelihood. There is no obvious solution for deriving this approximation in the case of intractable likelihood functions and the authors make use of a Bayesian optimisation technique called Gaussian process optimisation (GPO). Meaning that the Laplace approximation is the Laplace approximation of a surrogate log-posterior. GPO is a Bayesian numerical method in the spirit of the probabilistic numerics discussed on the ‘Og a few weeks ago. In the current setting, this means iterating three steps

  1. derive an approximation of the log-posterior ξ at the current θ using SMC-ABC
  2. construct a surrogate log-posterior by a Gaussian process using the past (ξ,θ)’s
  3. determine the next value of θ

In the first step, a standard particle filter cannot be used to approximate the observed log-posterior at θ because the conditional density of observed given latent is intractable. The solution is to use ABC for the HMM model, in the spirit of many papers by Ajay Jasra and co-authors. However, I find the construction of the substitute model allowing for a particle filter very obscure… (A side effect of the heat wave?!) I can spot a noisy ABC feature in equation (7), but am at a loss as to how the reparameterisation by the transform τ is compatible with the observed-given-latent conditional being unavailable: if the pair (x,v) at time t has a closed form expression, so does (x,y), at least on principle, since y is a deterministic transform of (x,v). Another thing I do not catch is why having a particle filter available prevent the use of a pMCMC approximation.

The second step constructs a Gaussian process posterior on the log-likelihood, with Gaussian errors on the ξ’s. The Gaussian process mean is chosen as zero, while the covariance function is a Matérn function. With hyperparameters that are estimated by maximum likelihood estimators (based on the argument that the marginal likelihood is available in closed form). Turning the approach into an empirical Bayes version.

The next design point in the sequence of θ’s is the argument of the maximum of a certain acquisition function, which is chosen here as a sort of maximum regret associated with the posterior predictive associated with the Gaussian process. With possible jittering. At this stage, it reminded me of the Gaussian process approach proposed by Michael Gutmann in his NIPS poster last year.

Overall, the method is just too convoluted for me to assess its worth and efficiency without a practical implementation to… practice upon, for which I do not have time! Hence I would welcome any comment from readers having attempted such implementations. I also wonder at the lack of link with Simon Wood‘s Gaussian approximation that appeared in Nature (2010) and was well-discussed in the Read Paper of Fearnhead and Prangle (2012).

hierarchical models are not Bayesian models

Posted in Books, Kids, Statistics, University life with tags , , , , , , , on February 18, 2015 by xi'an

When preparing my OxWaSP projects a few weeks ago, I came perchance on a set of slides, entitled “Hierarchical models are not Bayesian“, written by Brian Dennis (University of Idaho), where the author argues against Bayesian inference in hierarchical models in ecology, much in relation with the previously discussed paper of Subhash Lele. The argument is the same, namely a possibly major impact of the prior modelling on the resulting inference, in particular when some parameters are hardly identifiable, the more when the model is complex and when there are many parameters. And that “data cloning” being available since 2007, frequentist methods have “caught up” with Bayesian computational abilities.

Let me remind the reader that “data cloning” means constructing a sequence of Bayes estimators corresponding to the data being duplicated (or cloned) once, twice, &tc., until the point estimator stabilises. Since this corresponds to using increasing powers of the likelihood, the posteriors concentrate more and more around the maximum likelihood estimator. And even recover the Hessian matrix. This technique is actually older than 2007 since I proposed it in the early 1990’s under the name of prior feedback, with earlier occurrences in the literature like D’Epifanio (1989) and even the discussion of Aitkin (1991). A more efficient version of this approach is the SAME algorithm we developed in 2002 with Arnaud Doucet and Simon Godsill where the power of the likelihood is increased during iterations in a simulated annealing version (with a preliminary version found in Duflo, 1996).

I completely agree with the author that a hierarchical model does not have to be Bayesian: when the random parameters in the model are analysed as sources of additional variations, as for instance in animal breeding or ecology, and integrated out, the resulting model can be analysed by any statistical method. Even though one may wonder at the motivations for selecting this particular randomness structure in the model. And at an increasing blurring between what is prior modelling and what is sampling modelling as the number of levels in the hierarchy goes up. This rather amusing set of slides somewhat misses a few points, in particular the ability of data cloning to overcome identifiability and multimodality issues. Indeed, as with all simulated annealing techniques, there is a practical difficulty in avoiding the fatal attraction of a local mode using MCMC techniques. There are thus high chances data cloning ends up in the “wrong” mode. Moreover, when the likelihood is multimodal, it is a general issue to decide which of the modes is most relevant for inference. In which sense is the MLE more objective than a Bayes estimate, then? Further, the impact of a prior on some aspects of the posterior distribution can be tested by re-running a Bayesian analysis with different priors, including empirical Bayes versions or, why not?!, data cloning, in order to understand where and why huge discrepancies occur. This is part of model building, in the end.

maximum likelihood: an introduction

Posted in Books, Statistics with tags , , , , on December 20, 2014 by xi'an

“Basic Principle 0. Do not trust any principle.” L. Le Cam (1990)

Here is the abstract of a International Statistical Rewiew 1990 paper by Lucien Le Cam on maximum likelihood. ISR keeping a tradition of including an abstract in French for every paper, Le Cam (most presumably) wrote his own translation [or maybe wrote the French version first], which sounds much funnier to me and so I cannot resist posting both, pardon my/his French! [I just find “Ce fait” rather unusual, as I would have rather written “Ceci fait”…]:

Maximum likelihood estimates are reported to be best under all circumstances. Yet there are numerous simple examples where they plainly misbehave. One gives some examples for problems that had not been invented for the purpose of annoying maximum likelihood fans. Another example, imitated from Bahadur, has been specially created with just such a purpose in mind. Next, we present a list of principles leading to the construction of good estimates. The main principle says that one should not believe in principles but study each problem for its own sake.

L’auteur a ouï dire que la méthode du maximum de vraisemblance est la meilleure méthode d’estimation. C’est bien vrai, et pourtant la méthode se casse le nez sur des exemples bien simples qui n’avaient pas été inventés pour le plaisir de montrer que la méthode peut être très désagréable. On en donne quelques-uns, plus un autre, imité de Bahadur et fabriqué exprès pour ennuyer les admirateurs du maximum de vraisemblance. Ce fait, on donne une savante liste de principes de construction de bons estimateurs, le principe principal étant qu’il ne faut pas croire aux principes.

The entire paper is just as witty, as in describing the mixture model as “contaminated and not fit to drink”! Or in “Everybody knows that taking logarithms is unfair”. Or, again, in “biostatisticians, being complicated people, prefer to work out not with the dose y but with its logarithm”… And a last line: “One possibility is that there are too many horse hairs in e”.

Statistical modeling and computation [book review]

Posted in Books, R, Statistics, University life with tags , , , , , , , , , , , , , on January 22, 2014 by xi'an

Dirk Kroese (from UQ, Brisbane) and Joshua Chan (from ANU, Canberra) just published a book entitled Statistical Modeling and Computation, distributed by Springer-Verlag (I cannot tell which series it is part of from the cover or frontpages…) The book is intended mostly for an undergrad audience (or for graduate students with no probability or statistics background). Given that prerequisite, Statistical Modeling and Computation is fairly standard in that it recalls probability basics, the principles of statistical inference, and classical parametric models. In a third part, the authors cover “advanced models” like generalised linear models, time series and state-space models. The specificity of the book lies in the inclusion of simulation methods, in particular MCMC methods, and illustrations by Matlab code boxes. (Codes that are available on the companion website, along with R translations.) It thus has a lot in common with our Bayesian Essentials with R, meaning that I am not the most appropriate or least unbiased reviewer for this book. Continue reading

optimal estimation of parameters (book review)

Posted in Books, Statistics with tags , , , , , , , on September 12, 2013 by xi'an

As I had read some of Jorma Rissanen’s papers in the early 1990’s when writing The Bayesian Choice, I was quite excited to learn that Rissanen had written a book on the optimal estimation of parameters, where he presents and develops his own approach to statistical inference (estimation and testing). As explained in the Preface this was induced by having to deliver the 2009 Shannon Lecture at the Information Theory Society conference.

Very few statisticians have been studying information theory, the result of which, I think, is the disarray of the present discipline of statistics.” J. Rissanen (p.2)

Now that I have read the book (between Venezia in the peaceful and shaded Fundamenta Sacca San Girolamo and Hong Kong, so maybe in too a leisurely and off-handed manner), I am not so excited… It is not that the theory presented in optimal estimation of parameters is incomplete or ill-presented: the book is very well-written and well-designed, if in a highly personal (and borderline lone ranger) style. But the approach Rissanen advocates, namely maximum capacity as a generalisation of maximum likelihood, does not seem to relate to my statistical perspective and practice. Even though he takes great care to distance himself from Bayesian theory by repeating that the prior distribution is not necessary for his theory of optimal estimation (“priors are not needed in the general MDL principle”, p.4). my major source of incomprehension lies with the choice of incorporating the estimator within the data density to produce a new density, as in

\hat{f}(x) = f(x|\hat{\theta}(x)) / \int f(x|\hat{\theta}(x))\,\text{d}x\,.

Indeed, this leads to (a) replace a statistical model with a structure that mixes the model and the estimation procedure and (b) peak the new distribution by always choosing the most appropriate (local) value of the parameter. For a normal sample with unknown mean θ, this produces for instance to a joint normal distribution that is degenerate since

\hat{f}(x)\propto f(x|\bar{x}).

(For a single observation it is not even defined.) In a similar spirit, Rissanen defines this estimated model for dynamic data in a sequential manner, which means in the end that x1 is used n times, x2 n-1 times, and so on.., This asymmetry does not sound logical, especially when considering sufficiency.

…the misunderstanding that the more parameters there are in the model the better it is because it is closer to the `truth’ and the `truth’ obviously is not simple” J. Rissanen (p.38)

Another point of contention with the approach advocated in optimal estimation of parameters is the inherent discretisation of the parameter space, which seems to exclude large dimensional spaces and complex models. I somehow subscribe to the idea that a given sample (hence a given sample size) induces a maximum precision in the estimation that can be translated into using a finite number of parameter values, but the implementation suggested in the book is essentially unidimensional. I also find the notion of optimality inherent to the statistical part of optimal estimation of parameters quite tautological as it ends up being a target that leads to the maximum likelihood estimator (or its pseudo-Bayesian counterpart).

The BIC criterion has neither information nor a probability theoretic interpretation, and it does not matter which measure for consistency is selected.” J. Rissanen (p.64)

The first part of the book is about coding and information theory; it amounts in my understanding to a justification of the Kullback-Leibler divergence, with an early occurrence (p.27) of the above estimation distribution. (The channel capacity is the normalising constant of this weird density.)

“…in hypothesis testing [where] the assumptions that the hypotheses are  `true’ has misguided the entire field by generating problems which do not exist and distorting rational solutions to problems that do exist.” J. Rissanen (p.41)

I have issues with the definition of confidence intervals as they rely on an implicit choice of a measure and have a constant coverage that decreases with the parameter dimension. This notion also seem to clash with the subsequent discretisation of the parameter space. Hypothesis testing à la Rissanen reduces to an assessment of a goodness of fit, again with fixed coverage properties. Interestingly, the acceptance and rejection regions are based on two quantities, the likelihood ratio and the KL distance (p. 96), which leads to a delayed decision if they do not agree wrt fixed bounds.

“A drawback of the prediction formulas is that they require the knowledge of the ARMA parameters.” J. Rissanen (p.141)

A final chapter on sequential (or dynamic) models reminded me that Rissanen was at the core of inventing variable order Markov chains. The remainder of this chapter provides some properties of the sequential normalised maximum likelihood estimator advocated by the author in the same spirit as the earlier versions.  The whole chapter feels (to me) somewhat disconnected from

In conclusion, Rissanen’s book is a definitely  interesting  entry on a perplexing vision of statistics. While I do not think it will radically alter our understanding and practice of statistics, it is worth perusing, if only to appreciate there are still people (far?) out there attempting to bring a new vision of the field.

A misleading title…

Posted in Books, R, Statistics, University life with tags , , , , , , , , , , on September 5, 2011 by xi'an

When I received this book, Handbook of fitting statistical distributions with R, by Z. Karian and E.J. Dudewicz,  from/for the Short Book Reviews section of the International Statistical Review, I was obviously impressed by its size (around 1700 pages and 3 kilos…). From briefly glancing at the table of contents, and the list of standard distributions appearing as subsections of the first chapters, I thought that the authors were covering different estimation/fitting techniques for most of the standard distributions. After taking a closer look at the book, I think the cover is misleading in several aspects: this is not a handbook (a.k.a. a reference book), it does not cover standard statistical distributions, the R input is marginal, and the authors only wrote part of the book, since about half of the chapters are written by other authors…

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Feedback on data cloning

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , , , , , on September 22, 2010 by xi'an

Following some discussions I had last week at Banff about data cloning, I re-read the 2007 “Data cloning” paper published in Ecology Letters by Lele, Dennis, and Lutscher. Once again, I see a strong similarity with our 2002 Statistics and Computing SAME algorithm, as well as with the subsequent (and equally similar) “A multiple-imputation Metropolis version of the EM algorithm” published in Biometrika by Gaetan and Yao in 2003—Biometrika to which Arnaud and I had earlier and unsuccessfully submitted this unpublished technical report on the convergence of the SAME algorithm… (The SAME algorithm is also described in detail in the 2005 book Inference in Hidden Markov Models, Chapter 13.)

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