Archive for inference

from statistical evidence to evidence of causality

Posted in Books, Statistics with tags , , , , , , , , , on December 24, 2013 by xi'an

I took the opportunity of having to wait at a local administration a long while today (!) to read an arXived paper by Dawid, Musio and Fienberg on the−both philosophical and practical−difficulty to establish the probabilities of the causes of effects. The first interesting thing about the paper is that it relates to the Médiator drug scandal that took place in France in the past year and still is under trial: thanks to the investigations of a local doctor, Irène Frachon, the drug was exposed as an aggravating factor for heart disease. Or maybe the cause. The case-control study of Frachon summarises into a 2×2 table with a corrected odds ratio of 17.1. From there, the authors expose the difficulties of drawing inference about causes of effects, i.e. causality, an aspect of inference that has always puzzled me. (And the paper led me to search for the distinction between odds ratio and risk ratio.)

“And the conceptual and implementational difficulties that we discuss below, that beset even the simplest case of inference about causes of effects, will be hugely magnified when we wish to take additional account of such policy considerations.”

A third interesting notion in the paper is the inclusion of counterfactuals. My introduction to counterfactuals dates back to a run in the back-country roads around Ithaca, New York, when George told me about a discussion paper from Phil he was editing for JASA on that notion with his philosopher neighbour Steven Schwartz as a discussant. (It was a great run, presumably in the late Spring. And the best introduction I could dream of!) Now, the paper starts from the counterfactual perspective to conclude that inference is close to impossible in this setting. Within my limited understanding, I would see that as a drawback of using counterfactuals, rather than of drawing inference about causes. If the corresponding statistical model is nonindentifiable, because one of the two responses is always missing, the model seems inappropriate. I am also surprised at the notion of “sufficiency” used in the paper, since it sounds like the background information cancels the need to account for the treatment (e.g., aspirin) decision.  The fourth point is the derivation of bounds on the probabilities of causation, despite everything! Quite an interesting read thus!

machine learning [book review]

Posted in Books, R, Statistics, University life with tags , , , , , , , on October 21, 2013 by xi'an

I have to admit the rather embarrassing fact that Machine Learning, A probabilistic perspective by Kevin P. Murphy is the first machine learning book I really read in detail…! It is a massive book with close to 1,100 pages and I thus hesitated taking it with me around, until I grabbed it in my bag for Warwick. (And in the train to Argentan.) It is also massive in its contents as it covers most (all?) of what I call statistics (but visibly corresponds to machine learning as well!). With a Bayesian bent most of the time (which is the secret meaning of probabilistic in the title).

“…we define machine learning as a set of methods that can automatically detect patterns in data, and then use the uncovered patterns to predict future data, or to perform other kinds of decision making under uncertainty (such as planning how to collect more data!).” (p.1)

Apart from the Introduction—which I find rather confusing for not dwelling on the nature of errors and randomness and on the reason for using probabilistic models (since they are all wrong) and charming for including a picture of the author’s family as an illustration of face recognition algorithms—, I cannot say I found the book more lacking in foundations or in the breadth of methods and concepts it covers than a “standard” statistics book. In short, this is a perfectly acceptable statistics book! Furthermore, it has a very relevant and comprehensive selection of references (sometimes favouring “machine learning” references over “statistics” references!). Even the vocabulary seems pretty standard to me. All this makes me wonder why we at all distinguish between the two domains, following Larry Wasserman’s views (for once!) that the difference is mostly in the eye of the beholder, i.e. in which department one teaches… Which was already my perspective before I read the book but it comforted me even further. And the author agrees as well (“The probabilistic approach to machine learning is closely related to the field of statistics, but differs slightly in terms of its emphasis and terminology”, p.1). Let us all unite!

[..part 2 of the book review to appear tomorrow...]

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.

Defence of model-based inference

Posted in Statistics, University life with tags , , , , , on January 13, 2010 by xi'an

A tribune—to which I contributed—about the virtues of statistical inference in phylogeography  just appeared in Molecular Ecology. (The whole paper seems to be available on line as I can access it.) It has been written by 22 (!) contributors in response to Templeton’s recent criticism of ABC (and his defence of Nested Clade Analysis) in the same journal. My contribution to the paper is mostly based on the arguments posted here last March, namely that the paper was confusing ABC (which is a computational method) with Bayesian statistics. The paper as a whole goes beyond a “Bayesian defence” since not all authors are Bayesian. It supports a statistics based approach to phyleogeography, as reported in the abstract

Recent papers have promoted the view that model-based methods in general, and those based on Approximate Bayesian Computation (ABC) in particular, are flawed in a number of ways, and are therefore inappropriate for the analysis of phylogeographic data. These papers further argue that Nested Clade Phylogeographic Analysis (NCPA) offers the best approach in statistical phylogeography. In order to remove the confusion and misconceptions introduced by these papers, we justify and explain the reasoning behind model-based inference. We argue that ABC is a statistically valid approach, alongside other computational statistical techniques that have been successfully used to infer parameters and compare models in population genetics. We also examine the NCPA method and highlight numerous deficiencies, either when used with single or multiple loci. We further show that the ages of clades are carelessly used to infer ages of demographic events, that these ages are estimated under a simple model of panmixia and population stationarity but are then used under different and unspecified models to test hypotheses, a usage the invalidates these testing procedures. We conclude by encouraging researchers to study and use model-based inference in population genetics.

This will most presumably fail to end the debate between the proponents and the opponents of model-based inference in phylogenics and elsewhere, but the point was worth making…

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