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"Grunwald, Peter"
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The no-free-lunch theorems of supervised learning
The no-free-lunch theorems promote a skeptical conclusion that all possible machine learning algorithms equally lack justification. But how could this leave room for a learning theory, that shows that some algorithms are better than others? Drawing parallels to the philosophy of induction, we point out that the no-free-lunch results presuppose a conception of learning algorithms as purely data-driven. On this conception, every algorithm must have an inherent inductive bias, that wants justification. We argue that many standard learning algorithms should rather be understood as modeldependent: in each application they also require for input a model, representing a bias. Generic algorithms themselves, they can be given a model-relative justification.
Journal Article
Why optional stopping can be a problem for Bayesians
Recently, optional stopping has been a subject of debate in the Bayesian psychology community. Rouder (
Psychonomic Bulletin & Review
21
(2), 301–308,
2014
) argues that optional stopping is no problem for Bayesians, and even recommends the use of optional stopping in practice, as do (Wagenmakers, Wetzels, Borsboom, van der Maas & Kievit,
Perspectives on Psychological Science
7
, 627–633,
2012
). This article addresses the question of whether optional stopping is problematic for Bayesian methods, and specifies under which circumstances and in which sense it is and is not. By slightly varying and extending Rouder’s (
Psychonomic Bulletin & Review
21
(2), 301–308,
2014
) experiments, we illustrate that, as soon as the parameters of interest are equipped with default or pragmatic priors—which means, in most practical applications of Bayes factor hypothesis testing—resilience to optional stopping can break down. We distinguish between three types of default priors, each having their own specific issues with optional stopping, ranging from no-problem-at-all (type 0 priors) to quite severe (type II priors).
Journal Article
Immobilized Biocatalysts
[...]I owe particular thanks to all the authors who contributed their excellent papers to this Special Issue that is comprised of twenty-nine articles, among them six reviews, covering many important aspects of this topic together with a variety of new approaches. The enzymes, the immobilization of which is described in the different articles, belong to the enzyme classes EC 1 (glucose oxidase, cyclohexanone monooxygenase, horseradish peroxidase, ketoreductase, glucose dehydrogenase, and laccase), EC 2 (hypoxanthine–guanine–xanthine phosphoribosyltransferase and aminotransferase), EC 3 (cellulase, β-amylase, various lipases, invertase, endo-β-N-acetylglucosaminidase, β-fructofuranosidase, and the protease ficin), and to EC 4 (hydroxynitril lyase). [34] prepared immobilized β-amylase in the form of cross-linked enzyme aggregates using bovine serum albumin or soy protein isolate as feeder proteins to reduce diffusion problems and to successfully obtain maltose from converting the residual starch contained in cassava bagasse. [...]the Special Issue “Immobilized Biocatalysts” should be of great interest for all those involved in the various aspects of this topic, which are discussed in the contributions and review articles.
Journal Article
Contextuality of Misspecification and Data-Dependent Losses
We elaborate on Watson and Holmes' observation that misspecification is contextual: a model that is wrong can still be adequate in one prediction context, yet grossly inadequate in another. One can incorporate such phenomena by adopting a generalized posterior, in which the likelihood is multiplied by an exponentiated loss. We argue that Watson and Holmes' characterization of such generalized posteriors does not really explain their good practical performance, and we provide an alternative explanation which suggests a further extension of the method.
Journal Article
Game Theory, Maximum Entropy, Minimum Discrepancy and Robust Bayesian Decision Theory
We describe and develop a close relationship between two problems that have customarily been regarded as distinct: that of maximizing entropy, and that of minimizing worst-case expected loss. Using a formulation grounded in the equilibrium theory of zero-sum games between Decision Maker and Nature, these two problems are shown to be dual to each other, the solution to each providing that to the other. Although Topsøe described this connection for the Shannon entropy over 20 years ago, it does not appear to be widely known even in that important special case. We here generalize this theory to apply to arbitrary decision problems and loss functions. We indicate how an appropriate generalized definition of entropy can be associated with such a problem, and we show that, subject to certain regularity conditions, the above-mentioned duality continues to apply in this extended context. This simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust Bayes acts. We also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions. This leads to an extension, to arbitrary discrepancies, of a well-known minimax theorem for the case of Kullback-Leibler divergence (the \"redundancy-capacity theorem\" of information theory). For the important case of families of distributions having certain mean values specified, we develop simple sufficient conditions and methods for identifying the desired solutions. We use this theory to introduce a new concept of \"generalized exponential family\" linked to the specific decision problem under consideration, and we demonstrate that this shares many of the properties of standard exponential families. Finally, we show that the existence of an equilibrium in our game can be rephrased in terms of a \"Pythagorean property\" of the related divergence, thus generalizing previously announced results for Kullback-Leibler and Bregman divergences.
Journal Article
Making Decisions Using Sets of Probabilities: Updating, Time Consistency, and Calibration
We consider how an agent should update her beliefs when her beliefs are represented by a set P of probability distributions, given that the agent makes decisions using the minimax criterion, perhaps the best-studied and most commonly-used criterion in the literature. We adopt a game-theoretic framework, where the agent plays against a bookie, who chooses some distribution from P. We consider two reasonable games that differ in what the bookie knows when he makes his choice. Anomalies that have been observed before, like time inconsistency, can be understood as arising because different games are being played, against bookies with different information. We characterize the important special cases in which the optimal decision rules according to the minimax criterion amount to either conditioning or simply ignoring the information. Finally, we consider the relationship between updating and calibration when uncertainty is described by sets of probabilities. Our results emphasize the key role of the rectangularity condition of Epstein and Schneider.
Journal Article