arXiv subjects

Peter Grünwald

Publications and source records attributed to Peter Grünwald.

45 records · Page 3Linked to original sources

arXiv2019

Effective second-order correlation function and single-photon detection

Quantum-optical research on semiconductor single-photon sources puts special emphasis on the measurement of the second-order correlation function $g^{(2)}(τ)$, arguing that $g^{(2)}(0)<1/2$ implies the source field represents a good single-photon light source. We analyze the gain of information from $g^{(2)}(0)$ with respect to single photons. Any quantum state, for which the second-order correlation function falls below $1/2$, has a nonzero projection on the single-photon Fock state. The amplitude $p$ of this projection is arbitrary, independent of $g^{(2)}(0)$. However, one can extract a lower bound on the single-to-multi-photon-projection ratio. A vacuum contribution in the quantum state of light artificially increases the value of $g^{(2)}(0)$, cloaking actual single-photon projection. Thus, we propose an effective second-order correlation function $\tilde g^{(2)}(0)$, which takes the influence of vacuum into account and also yields lower and upper bounds on $p$. We consider the single-photon purity as a standard figure-of merit in experiments, reinterpret it within our results and provide an effective version of that physical quantity. Besides comparing different experimental and theoretical results, we also provide a possible measurement scheme for determining $\tilde g^{(2)}(0)$.

quant-ph↗

arXiv2018

Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It

We empirically show that Bayesian inference can be inconsistent under misspecification in simple linear regression problems, both in a model averaging/selection and in a Bayesian ridge regression setting. We use the standard linear model, which assumes homoskedasticity, whereas the data are heteroskedastic, and observe that the posterior puts its mass on ever more high-dimensional models as the sample size increases. To remedy the problem, we equip the likelihood in Bayes' theorem with an exponent called the learning rate, and we propose the Safe Bayesian method to learn the learning rate from the data. SafeBayes tends to select small learning rates as soon the standard posterior is not `cumulatively concentrated', and its results on our data are quite encouraging.

math.ST↗

arXiv2017

Excitonic giant-dipole potentials in cuprous oxide

In this work we predict the existence of a novel species of Wannier excitons when exposed to crossed electric and magnetic fields. In particular, we present a theory of giant-dipole excitons in $\textrm{Cu}_2\rm O$ in crossed fields. Within our theoretical approach we perform a pseudoseparation of the center-of-mass motion for the field-dressed excitonic species, thereby obtaining an effective single-particle Hamiltonian for the relative motion. For arbitrary gauge fields we exactly separate the gauge-dependent kinetic energy terms from the effective single-particle interaction potential. Depending on the applied field strengths and the specific field orientation, the potential for the relative motion of electron and hole exhibits an outer well at spatial separations up to several micrometers and depths up to $380\, μ\rm eV$, leading to possible permanent excitonic electric dipole moments of around three million Debye.

cond-mat.mes-hall↗

arXiv2016

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 $\log\log n$ 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.

math.ST↗

arXiv2016

Combining Adversarial Guarantees and Stochastic Fast Rates in Online Learning

We consider online learning algorithms that guarantee worst-case regret rates in adversarial environments (so they can be deployed safely and will perform robustly), yet adapt optimally to favorable stochastic environments (so they will perform well in a variety of settings of practical importance). We quantify the friendliness of stochastic environments by means of the well-known Bernstein (a.k.a. generalized Tsybakov margin) condition. For two recent algorithms (Squint for the Hedge setting and MetaGrad for online convex optimization) we show that the particular form of their data-dependent individual-sequence regret guarantees implies that they adapt automatically to the Bernstein parameters of the stochastic environment. We prove that these algorithms attain fast rates in their respective settings both in expectation and with high probability.

cs.LG↗

arXiv2016

Safe Probability

We formalize the idea of probability distributions that lead to reliable predictions about some, but not all aspects of a domain. The resulting notion of `safety' provides a fresh perspective on foundational issues in statistics, providing a middle ground between imprecise probability and multiple-prior models on the one hand and strictly Bayesian approaches on the other. It also allows us to formalize fiducial distributions in terms of the set of random variables that they can safely predict, thus taking some of the sting out of the fiducial idea. By restricting probabilistic inference to safe uses, one also automatically avoids paradoxes such as the Monty Hall problem. Safety comes in a variety of degrees, such as "validity" (the strongest notion), "calibration", "confidence safety" and "unbiasedness" (almost the weakest notion).

stat.ME↗

arXiv2016

Deviations of the exciton level spectrum in cuprous oxide from the hydrogen series

Recent high-resolution absorption spectroscopy on excited excitons in cuprous oxide [Nature (London) 514, 343 (2014)] has revealed significant deviations of their spectrum from that of the ideal hydrogen-like series. Here we show that the complex band dispersion of the crystal, which determines the kinetic energy of electrons and holes, strongly affects the exciton binding energy. Specifically, we show that the nonparabolicity of the band dispersion is the main cause of the deviation from the hydrogen series. Experimental data collected from high-resolution absorption spectroscopy in electric fields validate the assignment of the deviation to the nonparabolicity of the band dispersion.

cond-mat.mes-hall↗

arXiv2011

Adaptive Hedge

Most methods for decision-theoretic online learning are based on the Hedge algorithm, which takes a parameter called the learning rate. In most previous analyses the learning rate was carefully tuned to obtain optimal worst-case performance, leading to suboptimal performance on easy instances, for example when there exists an action that is significantly better than all others. We propose a new way of setting the learning rate, which adapts to the difficulty of the learning problem: in the worst case our procedure still guarantees optimal performance, but on easy instances it achieves much smaller regret. In particular, our adaptive method achieves constant regret in a probabilistic setting, when there exists an action that on average obtains strictly smaller loss than all other actions. We also provide a simulation study comparing our approach to existing methods.

stat.ML↗

arXiv2010

Prequential Plug-In Codes that Achieve Optimal Redundancy Rates even if the Model is Wrong

We analyse the prequential plug-in codes relative to one-parameter exponential families M. We show that if data are sampled i.i.d. from some distribution outside M, then the redundancy of any plug-in prequential code grows at rate larger than 1/2 ln(n) in the worst case. This means that plug-in codes, such as the Rissanen-Dawid ML code, may behave inferior to other important universal codes such as the 2-part MDL, Shtarkov and Bayes codes, for which the redundancy is always 1/2 ln(n) + O(1). However, we also show that a slight modification of the ML plug-in code, "almost" in the model, does achieve the optimal redundancy even if the the true distribution is outside M.

cs.IT↗