arXiv subjects

Robert L. Wolpert

Publications and source records attributed to Robert L. Wolpert.

14 recordsLinked to original sources

arXiv2024

Bayesian inference: More than Bayes's theorem

Bayesian inference gets its name from *Bayes's theorem*, expressing posterior probabilities for hypotheses about a data generating process as the (normalized) product of prior probabilities and a likelihood function. But Bayesian inference uses all of probability theory, not just Bayes's theorem. Many hypotheses of scientific interest are *composite hypotheses*, with the strength of evidence for the hypothesis dependent on knowledge about auxiliary factors, such as the values of nuisance parameters (e.g., uncertain background rates or calibration factors). Many important capabilities of Bayesian methods arise from use of the law of total probability, which instructs analysts to compute probabilities for composite hypotheses by *marginalization* over auxiliary factors. This tutorial targets relative newcomers to Bayesian inference, aiming to complement tutorials that focus on Bayes's theorem and how priors modulate likelihoods. The emphasis here is on marginalization over parameter spaces -- both how it is the foundation for important capabilities, and how it may motivate caution when parameter spaces are large. Topics covered include the difference between likelihood and probability, understanding the impact of priors beyond merely shifting the maximum likelihood estimate, and the role of marginalization in accounting for uncertainty in nuisance parameters, systematic error, and model misspecification.

stat.ME↗

arXiv2020

Approximate Bayesian Computation for Finite Mixture Models

Finite mixture models are used in statistics and other disciplines, but inference for mixture models is challenging due, in part, to the multimodality of the likelihood function and the so-called label switching problem. We propose extensions of the Approximate Bayesian Computation-Population Monte Carlo (ABC-PMC) algorithm as an alternative framework for inference on finite mixture models. There are several decisions to make when implementing an ABC-PMC algorithm for finite mixture models, including the selection of the kernels used for moving the particles through the iterations, how to address the label switching problem, and the choice of informative summary statistics. Examples are presented to demonstrate the performance of the proposed ABC-PMC algorithm for mixture modeling. The performance of the proposed method is evaluated in a simulation study and for the popular recessional velocity galaxy data.

stat.ME↗

arXiv2020

Improving Exoplanet Detection Power: Multivariate Gaussian Process Models for Stellar Activity

The radial velocity method is one of the most successful techniques for detecting exoplanets. It works by detecting the velocity of a host star induced by the gravitational effect of an orbiting planet, specifically the velocity along our line of sight, which is called the radial velocity of the star. Low-mass planets typically cause their host star to move with radial velocities of 1 m/s or less. By analyzing a time series of stellar spectra from a host star, modern astronomical instruments can in theory detect such planets. However, in practice, intrinsic stellar variability (e.g., star spots, convective motion, pulsations) affects the spectra and often mimics a radial velocity signal. This signal contamination makes it difficult to reliably detect low-mass planets. A principled approach to recovering planet radial velocity signals in the presence of stellar activity was proposed by Rajpaul et al. (2015). It uses a multivariate Gaussian process model to jointly capture time series of the apparent radial velocity and multiple indicators of stellar activity. We build on this work in two ways: (i) we propose using dimension reduction techniques to construct new high-information stellar activity indicators; and (ii) we extend the Rajpaul et al. (2015) model to a larger class of models and use a power-based model comparison procedure to select the best model. Despite significant interest in exoplanets, previous efforts have not performed large-scale stellar activity model selection or attempted to evaluate models based on planet detection power. In the case of main sequence G2V stars, we find that our method substantially improves planet detection power compared to previous state-of-the-art approaches.

astro-ph.IM↗

arXiv2018

Statistical challenges in the search for dark matter

The search for the particle nature of dark matter has given rise to a number of experimental, theoretical and statistical challenges. Here, we report on a number of these statistical challenges and new techniques to address them, as discussed in the DMStat workshop held Feb 26 - Mar 3 2018 at the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, Alberta.

hep-ph↗

arXiv2018

Model-free inference on extreme dependence via waiting times

A variety of methods have been proposed for inference about extreme dependence for multivariate or spatially-indexed stochastic processes and time series. Most of these proceed by first transforming data to some specific extreme value marginal distribution, often the unit Fréchet, then fitting a family of max-stable processes to the transformed data and exploring dependence within the framework of that model. The marginal transformation, model selection, and model fitting are all possible sources of misspecification in this approach. We propose an alternative model-free approach, based on the idea that substantial information on the strength of tail dependence and its temporal structure are encoded in the distribution of the waiting times between exceedances of high thresholds at different locations. We propose quantifying the strength of extremal dependence and assessing uncertainty by using statistics based on these waiting times. The method does not rely on any specific underlying model for the process, nor on asymptotic distribution theory. The method is illustrated by applications to climatological, financial, and electrophysiology data. To put the proposed approach within the context of the existing literature, we construct a class of spacetime-indexed stochastic processes whose waiting time distributions are available in closed form by endowing the support points in de Haan's spectral representation of max-stable processes with random birth times, velocities, and lifetimes, and applying Smith's model to these processes. We show that waiting times in this model are stochatically decreasing in mean speed, and the sample mean of the waiting times obeys a central limit theorem with a uniform convergence rate under mild conditions. This indicates that our procedure can be implemented in this setting using standard $t$ statistics and associated hypothesis tests.

math.ST↗

arXiv2017

Multistage Monte-Carlo simulation of jet modification in a static medium

The modification of hard jets in an extended static medium held at a fixed temperature is studied using three different Monte-Carlo event generators (LBT, MATTER, MARTINI). Each event generator contains a different set of assumptions regarding the energy and virtuality of the partons within a jet versus the energy scale of the medium, and hence, applies to a different epoch in the space-time history of the jet evolution. For the first time, modeling is developed where a jet may sequentially transition from one generator to the next, on a parton-by-parton level, providing a detailed simulation of the space-time evolution of medium modified jets over a much broader dynamic range than has been attempted previously in a single calculation. Comparisons are carried out for different observables sensitive to jet quenching, including the parton fragmentation function and the azimuthal distribution of jet energy around the jet axis. The effect of varying the boundary between different generators is studied and a theoretically motivated criterion for the location of this boundary is proposed. The importance of such an approach with coupled generators to the modeling of jet quenching is discussed.

nucl-th↗

arXiv2016

The Longevity of Lava Dome Eruptions

Understanding the duration of past, on-going and future volcanic eruptions is an important scientific goal and a key societal need. We present a new methodology for forecasting the duration of on-going and future lava dome eruptions based on a database (DomeHaz) recently compiled by the authors. The database includes duration and composition for 177 such eruptions, with "eruption" defined as the period encompassing individual episodes of dome growth along with associated quiescent periods during which extrusion pauses but unrest continues. In a key finding we show that probability distributions for dome eruption durations are both heavy-tailed and composition-dependent. We construct Objective Bayes statistical models featuring heavy-tailed Generalized Pareto distributions with composition-specific parameters to make forecasts about the durations of new and on-going eruptions that depend on both eruption duration-to-date and composition. Our Bayesian predictive distributions reflect both uncertainty about model parameter values (epistemic uncertainty) and the natural variability of the geologic processes (aleatoric uncertainty). The results are illustrated by presenting likely trajectories for fourteen dome-building eruptions on-going in 2015. Full representation of the uncertainty is presented for two key eruptions, Soufri{é}re Hills Volcano in Montserrat (10--139 years, median 35yr) and Sinabung, Indonesia (1--17 years, median 4yr). Uncertainties are high, but, importantly, quantifiable. This work provides for the first time a quantitative and transferable method and rationale on which to base long-term planning decisions for lava dome forming volcanoes, with wide potential use and transferability to forecasts of other types of eruptions and other adverse events across the geohazard spectrum.

stat.AP↗

arXiv2015

ACME: A Partially Periodic Estimator of Avian & Chiropteran Mortality at Wind Turbines

Estimating the mortality of birds and bats at wind turbines based on periodic carcass counts is challenging because carcasses may be removed by scavengers or may be missed in investigators' searches, leading to undercounting. Existing mortality estimators intended to correct for this offer wildly different estimates when search intervals are short. We introduce a new estimator that includes many existing ones as special cases but extends and improves them in three ways to reflect phenomena discovered in the field: * Decreasing removal rate by scavengers as carcasses age; * Diminishing proficiency of Field Technicians in discovering carcasses as they age; * Possibility that some (but not all) carcasses arriving in earlier search periods may be discovered in the current period. It is this feature that makes the new estimator "partially periodic". Both point estimates and 50% and 90% Objective Bayes interval estimates are provided for mortality. The new ACME mortality estimator will significantly improve the accuracy and credibility of mortality estimates for birds and bats near wind turbines.

stat.AP↗

arXiv2015

Quantifying properties of hot and dense QCD matter through systematic model-to-data comparison

We systematically compare an event-by-event heavy-ion collision model to data from the Large Hadron Collider. Using a general Bayesian method, we probe multiple model parameters including fundamental quark-gluon plasma properties such as the specific shear viscosity $η/s$, calibrate the model to optimally reproduce experimental data, and extract quantitative constraints for all parameters simultaneously. The method is universal and easily extensible to other data and collision models.

nucl-th↗

arXiv2015

Geometric Representations of Random Hypergraphs

A parametrization of hypergraphs based on the geometry of points in $\mathbf{R}^d$ is developed. Informative prior distributions on hypergraphs are induced through this parametrization by priors on point configurations via spatial processes. This prior specification is used to infer conditional independence models or Markov structure of multivariate distributions. Specifically, we can recover both the junction tree factorization as well as the hyper Markov law. This approach offers greater control on the distribution of graph features than Erdös-Rényi random graphs, supports inference of factorizations that cannot be retrieved by a graph alone, and leads to new Metropolis\slash Hastings Markov chain Monte Carlo algorithms with both local and global moves in graph space. We illustrate the utility of this parametrization and prior specification using simulations.

math.ST↗

arXiv2013

A template for describing intrinsic GRB pulse shapes

A preliminary study of a set of well-isolated pulses in GRB light curves indicates that simple pulse models, with smooth and monotonic pulse rise and decay regions, are inadequate. Examining the residuals of fits of pulses to such models suggests the following patterns of departure from the smooth pulse model of Norris et al. (2005): A Precursor Shelf occurs prior to or concurrent with the exponential Rapid Rise. The pulse reaches maximum intensity at the Peak Plateau, then undergoes a Rapid Decay. The decay changes into an Extended Tail. Pulses are almost universally characterized by hard-to-soft evolution, arguing that the new pulse features reflect a single physical phenomenon, rather than artifacts of pulse overlap.

astro-ph.HE↗

arXiv2011

Stochastic expansions using continuous dictionaries: Lévy adaptive regression kernels

This article describes a new class of prior distributions for nonparametric function estimation. The unknown function is modeled as a limit of weighted sums of kernels or generator functions indexed by continuous parameters that control local and global features such as their translation, dilation, modulation and shape. Lévy random fields and their stochastic integrals are employed to induce prior distributions for the unknown functions or, equivalently, for the number of kernels and for the parameters governing their features. Scaling, shape, and other features of the generating functions are location-specific to allow quite different function properties in different parts of the space, as with wavelet bases and other methods employing overcomplete dictionaries. We provide conditions under which the stochastic expansions converge in specified Besov or Sobolev norms. Under a Gaussian error model, this may be viewed as a sparse regression problem, with regularization induced via the Lévy random field prior distribution. Posterior inference for the unknown functions is based on a reversible jump Markov chain Monte Carlo algorithm. We compare the Lévy Adaptive Regression Kernel (LARK) method to wavelet-based methods using some of the standard test functions, and illustrate its flexibility and adaptability in nonstationary applications.

math.ST↗

arXiv2011

Bayesian nonparametric models for peak identification in MALDI-TOF mass spectroscopy

We present a novel nonparametric Bayesian approach based on Lévy Adaptive Regression Kernels (LARK) to model spectral data arising from MALDI-TOF (Matrix Assisted Laser Desorption Ionization Time-of-Flight) mass spectrometry. This model-based approach provides identification and quantification of proteins through model parameters that are directly interpretable as the number of proteins, mass and abundance of proteins and peak resolution, while having the ability to adapt to unknown smoothness as in wavelet based methods. Informative prior distributions on resolution are key to distinguishing true peaks from background noise and resolving broad peaks into individual peaks for multiple protein species. Posterior distributions are obtained using a reversible jump Markov chain Monte Carlo algorithm and provide inference about the number of peaks (proteins), their masses and abundance. We show through simulation studies that the procedure has desirable true-positive and false-discovery rates. Finally, we illustrate the method on five example spectra: a blank spectrum, a spectrum with only the matrix of a low-molecular-weight substance used to embed target proteins, a spectrum with known proteins, and a single spectrum and average of ten spectra from an individual lung cancer patient.

stat.AP↗

arXiv2009

On the Supremum of Certain Families of Stochastic Processes

We consider a family of stochastic processes $\{X_t^ε, t \in T\}$ on a metric space $T$, with a parameter $ε\downarrow 0$. We study the conditions under which \lim_{\e \to 0} ¶\Big(\sup_{t \in T} |X_t^\e| < δ\Big) =1 when one has the \textit{a priori} estimate on the modulus of continuity and the value at one point. We compare our problem to the celebrated Kolmogorov continuity criteria for stochastic processes, and finally give an application of our main result for stochastic intergrals with respect to compound Poisson random measures with infinite intensity measures.

math.PR↗