20,160 research outputs found
Entropy Concentration and the Empirical Coding Game
We give a characterization of Maximum Entropy/Minimum Relative Entropy
inference by providing two `strong entropy concentration' theorems. These
theorems unify and generalize Jaynes' `concentration phenomenon' and Van
Campenhout and Cover's `conditional limit theorem'. The theorems characterize
exactly in what sense a prior distribution Q conditioned on a given constraint,
and the distribution P, minimizing the relative entropy D(P ||Q) over all
distributions satisfying the constraint, are `close' to each other. We then
apply our theorems to establish the relationship between entropy concentration
and a game-theoretic characterization of Maximum Entropy Inference due to
Topsoe and others.Comment: A somewhat modified version of this paper was published in Statistica
Neerlandica 62(3), pages 374-392, 200
A Tight Excess Risk Bound via a Unified PAC-Bayesian-Rademacher-Shtarkov-MDL Complexity
We present a novel notion of complexity that interpolates between and
generalizes some classic existing complexity notions in learning theory: for
estimators like empirical risk minimization (ERM) with arbitrary bounded
losses, it is upper bounded in terms of data-independent Rademacher complexity;
for generalized Bayesian estimators, it is upper bounded by the data-dependent
information complexity (also known as stochastic or PAC-Bayesian,
complexity. For
(penalized) ERM, the new complexity reduces to (generalized) normalized maximum
likelihood (NML) complexity, i.e. a minimax log-loss individual-sequence
regret. Our first main result bounds excess risk in terms of the new
complexity. Our second main result links the new complexity via Rademacher
complexity to entropy, thereby generalizing earlier results of Opper,
Haussler, Lugosi, and Cesa-Bianchi who did the log-loss case with .
Together, these results recover optimal bounds for VC- and large (polynomial
entropy) classes, replacing localized Rademacher complexity by a simpler
analysis which almost completely separates the two aspects that determine the
achievable rates: 'easiness' (Bernstein) conditions and model complexity.Comment: 38 page
Almost the Best of Three Worlds: Risk, Consistency and Optional Stopping for the Switch Criterion in Nested Model Selection
We study the switch distribution, introduced by Van Erven et al. (2012),
applied to model selection and subsequent estimation. While switching was known
to be strongly consistent, here we show that it achieves minimax optimal
parametric risk rates up to a factor when comparing two nested
exponential families, partially confirming a conjecture by Lauritzen (2012) and
Cavanaugh (2012) that switching behaves asymptotically like the Hannan-Quinn
criterion. Moreover, like Bayes factor model selection but unlike standard
significance testing, when one of the models represents a simple hypothesis,
the switch criterion defines a robust null hypothesis test, meaning that its
Type-I error probability can be bounded irrespective of the stopping rule.
Hence, switching is consistent, insensitive to optional stopping and almost
minimax risk optimal, showing that, Yang's (2005) impossibility result
notwithstanding, it is possible to `almost' combine the strengths of AIC and
Bayes factor model selection.Comment: To appear in Statistica Sinic
A survey on fractional variational calculus
Main results and techniques of the fractional calculus of variations are
surveyed. We consider variational problems containing Caputo derivatives and
study them using both indirect and direct methods. In particular, we provide
necessary optimality conditions of Euler-Lagrange type for the fundamental,
higher-order, and isoperimetric problems, and compute approximated solutions
based on truncated Gr\"{u}nwald--Letnikov approximations of Caputo derivatives.Comment: This is a preprint of a paper whose final and definite form is in
'Handbook of Fractional Calculus with Applications. Vol 1: Basic Theory', De
Gruyter. Submitted 29-March-2018; accepted, after a revision, 13-June-201
Taylor- and fugacity expansion for the effective center model of QCD at finite density
Using the effective center model of QCD we test series expansions for finite
chemical potential . In particular we study two variants of Taylor
expansion as well as the fugacity series. The effective center model has a dual
representation where the sign problem is absent and reliable Monte Carlo
simulations are possible at arbitrary . We use the results from the dual
simulation as reference data to assess the Taylor- and fugacity series
approaches. We find that for most of parameter space fugacity expansion is the
best (but also numerically most expensive) choice for reproducing the dual
simulation results, while conventional Taylor expansion is reliable only for
very small . We also discuss the results of a modified Taylor expansion in
which at the same numerical effort clearly outperforms the
conventional Taylor series.Comment: presented at the 31st International Symposium on Lattice Field Theory
(Lattice 2013), 29 July - 3 August 2013, Mainz, Germany. Reference adde
Optional Stopping with Bayes Factors: a categorization and extension of folklore results, with an application to invariant situations
It is often claimed that Bayesian methods, in particular Bayes factor methods
for hypothesis testing, can deal with optional stopping. We first give an
overview, using elementary probability theory, of three different mathematical
meanings that various authors give to this claim: (1) stopping rule
independence, (2) posterior calibration and (3) (semi-) frequentist robustness
to optional stopping. We then prove theorems to the effect that these claims do
indeed hold in a general measure-theoretic setting. For claims of type (2) and
(3), such results are new. By allowing for non-integrable measures based on
improper priors, we obtain particularly strong results for the practically
important case of models with nuisance parameters satisfying a group invariance
(such as location or scale). We also discuss the practical relevance of
(1)--(3), and conclude that whether Bayes factor methods actually perform well
under optional stopping crucially depends on details of models, priors and the
goal of the analysis.Comment: 29 page
On derivations with respect to finite sets of smooth functions
The purpose of this paper is to show that functions that derivate the
two-variable product function and one of the exponential, trigonometric or
hyperbolic functions are also standard derivations. The more general problem
considered is to describe finite sets of differentiable functions such that
derivations with respect to this set are automatically standard derivations
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