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A non-commutative Bayes' theorem

Using a diagrammatic reformulation of Bayes' theorem, we provide a necessary and sufficient condition for the existence of Bayesian inference in the setting of finite-dimensional C* -algebras. In other words, we prove an analogue of Bayes' theorem in the joint classical and quantum context. Our analogue is justified by recent advances in categorical probability theory, which have provided an abstract formulation of the classical Bayes' theorem. In the process, we further develop non-commutative almost everywhere equivalence and illustrate its important role in non-commutative Bayesian inversion. The construction of such Bayesian inverses, when they exist, involves solving a positive semidefinite matrix completion problem for the Choi matrix. This gives a solution to the open problem of constructing Bayesian inversion for completely positive unital maps acting on density matrices that do not have full support. In conclusion, we illustrate how the procedure works for several examples relevant to quantum information theory.

97 MATHEMATICS AND COMPUTING↗

Constraining transfer cross sections using Bayes' theorem

Background: Being able to rigorously quantify the uncertainties in reaction models is crucial to moving this field forward. Even though Bayesian methods are becoming increasingly popular in nuclear theory, they have yet to be implemented and applied in reaction theory problems. Purpose: The purpose of this work is to investigate, using Bayesian methods, the uncertainties in the optical potentials generated from fits to elastic-scattering data and the subsequent uncertainties in the transfer predictions. We also study the differences in two reaction models where the parameters are constrained in a similar manner, as well as the impact of reducing the experimental error bars on the data used to constrain the parameters. Method: We use Bayes' theorem combined with a Markov chain Monte Carlo to determine posterior distributions for the parameters of the optical model, constrained by neutron-, proton-, and/or deuteron-target elastic scattering. These potentials are then used to predict transfer cross sections within the adiabatic wave approximation or the distorted-wave Born approximation. Results: We study a number of reactions involving deuteron projectiles with energies in the range 10–25 MeV/nucleon on targets with mass A = 48–208. The case of 48 Ca(d,p) 49 Ca transfer to the ground state is described in detail. A comparative study of the effect of the size of experimental errors is also performed. Five transfer reactions are studied, and their results are then compiled in order to systematically identify trends.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Application of Bayes' theorem for pulse shape discrimination

A Bayesian approach is proposed for pulse shape discrimination of photons and neutrons in liquid organic scinitillators. Instead of drawing a decision boundary, each pulse is assigned a photon or neutron confidence probability. In addition, this allows for photon and neutron classification on an event-by-event basis. The sum of those confidence probabilities is used to estimate the number of photon and neutron instances in the data. An iterative scheme, similar to an expectation-maximization algorithm for Gaussian mixtures, is used to infer the ratio of photons-to-neutrons in each measurement. Therefore, the probability space adapts to data with varying photon-to-neutron ratios. A time-correlated measurement of Am–Be and separate measurements of 137 Cs, 60 Co and 232 Th photon sources were used to construct libraries of neutrons and photons. These libraries were then used to produce synthetic data sets with varying ratios of photons-to-neutrons. Probability weighted method that we implemented was found to maintain neutron acceptance rate of up to 90% up to photon-to-neutron ratio of 2000, and performed 9% better than the decision boundary approach. Furthermore, the iterative approach appropriately changed the probability space with an increasing number of photons which kept the neutron population estimate from unrealistically increasing.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Constraining the dark energy equation of state using Bayes theorem and the Kullback–Leibler divergence

Data-driven model-independent reconstructions of the dark energy equation of state w(z) are presented using Planck 2015 era CMB, BAO, SNIa and Lyman-α data. These reconstructions identify the w(z) behaviour supported by the data and show a bifurcation of the equation of state posterior in the range 1.5 < z < 3. Although the concordance ΛCDM model is consistent with the data at all redshifts in one of the bifurcated spaces, in the other a supernegative equation of state (also known as ‘phantom dark energy’) is identified within the 1.5σ confidence intervals of the posterior distribution. In order to identify the power of different datasets in constraining the dark energy equation of state, we use a novel formulation of the Kullback–Leibler divergence. Moreover, this formalism quantifies the information the data add when moving from priors to posteriors for each possible dataset combination. The SNIa and BAO datasets are shown to provide much more constraining power in comparison to the Lyman-α datasets. Furthermore, SNIa and BAO constrain most strongly around redshift range 0.1 - 0.5, whilst the Lyman-α data constrains weakly over a broader range. We do not attribute the supernegative favouring to any particular dataset, and note that the ΛCDM model was favoured at more than 2 log-units in Bayes factors over all the models tested despite the weakly preferred w(z) structure in the data.

79 ASTRONOMY AND ASTROPHYSICS↗

BAYES’ THEOREM AND EARLY SOLAR SHORT-LIVED RADIONUCLIDES: THE CASE FOR AN UNEXCEPTIONAL ORIGIN FOR THE SOLAR SYSTEM

The presence of excesses of short-lived radionuclides in the early solar system evidenced in meteorites has been taken as testament to close encounters with exotic nucleosynthetic sources, including supernovae or AGB stars. An analysis of the likelihoods associated with different sources of these extinct nuclides in the early solar system indicates that, rather than being exotic, their abundances were typical of star-forming regions like those observed today in the Galaxy. The radiochemistry of the early solar system is therefore unexceptional, being the consequence of extensive averaging of solids from molecular clouds.

79 ASTRONOMY AND ASTROPHYSICS↗

Accurate Holdup Calculations with Predictive Modeling & Data Integration

In facilities that process special nuclear material (SNM) it is important to account accurately for the fissile material that enters and leaves the plant. Although there are many stages and processes through which materials must be traced and measured, the focus of this project is material that is “held-up” in equipment, pipes, and ducts during normal operation and that can accumulate over time into significant quantities. Accurately estimating the holdup is essential for proper SNM accounting (vis-à-vis nuclear non-proliferation), criticality and radiation safety, waste management, and efficient plant operation. Usually it is not possible to directly measure the holdup quantity and location, so these must be inferred from measured radiation fields, primarily gamma and less frequently neutrons. Current methods to quantify holdup, i.e. Generalized Geometry Holdup (GGH), primarily rely on simple source configurations and crude radiation transport models aided by ad hoc correction factors. This project seeks an alternate method of performing measurement-based holdup calculations using a predictive model that employs state-of-the-art radiation transport codes capable of accurately simulating such situations. Inverse and data assimilation methods use the forward transport model to search for a source configuration that best matches the measured data and simultaneously provide an estimate of the level of confidence in the correctness of such configuration. In this work the holdup problem is re-interpreted as an inverse problem that is under-determined, hence may permit multiple solutions. A probabilistic approach is applied to solving the resulting inverse problem. This approach rates possible solutions according to their plausibility given the measurements and initial information. This is accomplished through the use of Bayes’ Theorem that resolves the issue of multiple solutions by giving an estimate of the probability of observing each possible solution. To use Bayes’ Theorem, one must have a model y(x) that maps the state variables x (the solution in this case) to the measurements y. In this case, the unknown state variables are the configuration and composition of the heldup SNM. The measurements are the detector readings. Thus, the natural model is neutral-particle radiation transport where a wealth of computational tools exists for performing these simulations accurately and efficiently. The combination of predictive model and Bayesian inference forms the Data Integration with Modeled Predictions (DIMP) method that serves as foundation for this project. The cost functional describing the model-to-data misfit is computed via a norm created by the inverse of the covariance matrix of the model parameters and responses. Since the model y(x) for the holdup problem is nonlinear, a nonlinear optimization on Q is conducted via Newton-type iterative methods to find the optimal values of the model parameters x. This project comprised a collaboration between NC State University (NCSU), the University of South Carolina (USC), and Oak Ridge National Laboratory (ORNL). The project was originally proposed in seven main tasks with an eighth contingency task to be performed if time and funding permitted; in fact time did not permit commencement of the contingency task and it was not performed. The remaining tasks involved holdup analysis with gamma detection strategies and separately with neutrons based on coincidence counting. Early in the project, and upon consultation with experts in coincidence counting it became evident that this approach is not viable for holdup applications and this task was replaced with an alternative, but valuable investigation that was carried out by the USC partner. Nevertheless, the experimental 4 measurements at ORNL of both gamma and neutron sources for the purpose of constructing Detector Response Functions (DRFs) with the associated uncertainties were indeed completed.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Advancing the Theory of Nuclear Data Evaluations [Abstract]

We present recent advances in the R-matrix formalism as well as the Bayesian evaluation framework for improved nuclear data evaluations. The advances in the R matrix formalism include: 1) direct processes, 2) doorway, as well as multistep, processes, and 3) various forms of the Reich-Moore approximation for eliminated capture channels. Furthermore, to address unreasonably small posterior uncertainties often encountered in nuclear data evaluations of large data sets using the conventional form of the Bayes’ theorem, we introduce imperfections (of the data or the model) as a formal evaluation tool for taming the evaluated uncertainties in harmony with Bayes’ theorem. These theoretical advances were motivated by the nuclear data evaluations of differential resolved resonance cross section data using the code SAMMY, as well as the integral benchmark experiments using the SCALE code system, being performed at Oak Ridge National Laboratory for the Nuclear Criticality Safety Program. Some pedagogical applications of the new formalism, as well as a snapshot of the SAMMY modernization efforts, will be presented.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Bayesian Monte Carlo Evaluation of Imperfect (n, 233 U) Data and Model

Conventional nuclear data evaluation methods using generalized linear least squares make the following assumptions: prior and posterior probability distribution functions (PDFs) of all model parameters and data are normal (Gaussian); the linear approximation is sufficiently accurate to minimize the cost function (even for nonlinear models); the model (e.g., of neutron cross section) and experimental data (including covariance data) are without defect and prior PDFs of parameters and measured data are known perfectly. Neglect of covariance between model parameters and measured data in conventional evaluations contributes to imperfections. These assumptions are inherent to the generalized linear least squares minimization method commonly used for resolved resonance region neutron cross section evaluations but are often not justified due to the presence of non-normal PDFs, nonlinear models (e.g., R-matrix formalism), and inherent imperfections in data and models (e.g., imperfect covariance data). Here, these assumptions are removed in a mathematical framework of Bayes’ theorem, which is implemented using the Metropolis-Hastings Monte Carlo method. Most importantly, new parameters are introduced to parameterize discrepancies between the theoretical model and measured data to quantify judgement about discrepancies or imperfections in a reproducible manner. An evaluation of 233U in the eV region using the ENDF-B/VIII.0 library and transmission data (Guber et al.) is presented, and posterior parameters are compared to those obtained by conventional evaluation methods. This example illustrates the effects of removing the most harmful assumption: that of model-data perfection.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Generalized Bayesian Framework for Evaluation of Integral Benchmark Experiments [Slides]

This presentation is on generalized Bayesian framework for evaluation of integral benchmark experiments. This presentation starts off with assumptions and approximations used with Bayes Theorem. then an overview of approximations used by ORNL codes, and Generalized Bayesian Monto Carlo (GBMC). The presentation then details out a precise framework. This presentation then concludes with considerations.

97 MATHEMATICS AND COMPUTING↗

Advances in Nuclear Data Evaluation Theory for NCSP [Slides]

The generalized form of the Bayes’ Theorem improves UQ for NCSP of differential and/or integral data evaluations and of applications. It is being implemented in an API for use by SCALE, SAMMY, etc. Extended R-matrix formalism improves evaluations of RRR including direct reactions, direct capture, and the Reich-Moore approx., doorway state reactions with applications to RPI data on lead isotopes. Extended R-Matrix formalism will be implemented into SAMMY post modernization.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Bayesian Monte-Carlo Evaluation Framework of Differential and Integral Data [Slides]

This presentation discusses the Bayesian Monte Carlo evaluation framework of differential and integral data, specifically in relation to two methods that are based on Bayes theorem, which assumes that all PDFs are normal (Gaussian) and that they use a linear approximation for IBEs. The presentation states that basic components of the MC evaluation framework include the computation of random MC ensemble from ENDF File 32, the simulation of IBEs and R-matrix cross section compared to experimental data, and the computation of weighted averages. It also states that the application to U-233 indicates deviation from the conventional linear approximation. In conclusion, evaluation framework will require MCMC method e.g., Metropolis-Hastings (M.-H.).

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Optimization of an exchange-correlation density functional for water

We describe a method, that we call data projection onto parameter space (DPPS), to optimize an energy functional of the electron density, so that it reproduces a dataset of experimental magnitudes. Our scheme, based on Bayes theorem, constrains the optimized functional not to depart unphysically from existing ab initio functionals. The resulting functional maximizes the probability of being the “correct” parameterization of a given functional form, in the sense of Bayes theory. The application of DPPS to water sheds new light on why density functional theory has performed rather poorly for liquid water, on what improvements are needed, and on the intrinsic limitations of the generalized gradient approximation to electron exchange and correlation. Finally, we present tests of our water-optimized functional, that we call vdW-DF-w, showing that it performs very well for a variety of condensed water systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Bayesian evidence and predictivity of the inflationary paradigm

In this paper we consider the issue of paradigm evaluation by applying Bayes' theorem along the following nested hierarchy of progressively more complex structures: i) parameter estimation (within a model), ii) model selection and comparison (within a paradigm), iii) paradigm evaluation. In such a hierarchy the Bayesian evidence works both as the posterior's normalization at a given level and as the likelihood function at the next level up. Whilst raising no objections to the standard application of the procedure at the two lowest levels, we argue that it should receive a considerable modification when evaluating paradigms, when testability and fitting data are equally important. By considering toy models we illustrate how models and paradigms that are difficult to falsify are always favoured by the Bayes factor. We argue that the evidence for a paradigm should not only be high for a given dataset, but exceptional with respect to what it would have been, had the data been different. With this motivation we propose a measure which we term predictivity , as well as a prior to be incorporated into the Bayesian framework, penalising unpredictivity as much as not fitting data. We apply this measure to inflation seen as a whole, and to a scenario where a specific inflationary model is hypothetically deemed as the only one viable as a result of information alien to cosmology (e.g. Solar System gravity experiments, or particle physics input). We conclude that cosmic inflation is currently hard to falsify, but that this could change were external/additional information to cosmology to select one of its many models. We also compare this state of affairs to bimetric varying speed of light cosmology.

79 ASTRONOMY AND ASTROPHYSICS↗

Determination of proton PDF uncertainties with Markov chain Monte Carlo

We present an analysis of parton distribution functions (PDFs) of the proton using Markov chain Monte Carlo (MCMC) methods. The MCMC approach naturally implements Bayes’ theorem and, thus, provides a means to directly sample the underlying probability distribution—in this case, the probability distribution of the PDF parameters. This allows for a straightforward propagation of the resulting uncertainties into any PDF-dependent observable, preserving their simple probabilistic interpretation. In our analysis we include a broad set of deep inelastic scattering data from HERA, BCDMS and NMC experiments along with the Drell-Yan, 𝑊 and 𝑍 boson data from LHC and Tevatron experiments, which combined with theoretical calculations at next-to-next-to-leading order in QCD allow for realistic determination of PDFs. The main focus of this analysis is to explore alternative methods for PDF uncertainty estimation that are more firmly grounded in statistical principles. We show that the flexibility of the Bayes framework, allowing one, e.g., to account for non-Gaussianity or inconsistencies of datasets, is crucial to extract realistic uncertainties when such assumptions are not fulfilled. We also demonstrate that MCMC allows one to determine the Δ⁢𝜒 2 value corresponding to a given confidence level in the sample, which can, in turn, be used as a statistically well-founded tolerance criterion used in the Hessian method, thus addressing one of its main long-standing drawbacks.

Risse, Peter Clemens [Universität Münster (Germany↗

Markov Chain Monte Carlo Used in Parameter Inference of Magnetic Resonance Spectra

In this paper, we use Boltzmann statistics and the maximum likelihood distribution derived from Bayes’ Theorem to infer parameter values for a Pake Doublet Spectrum, a lineshape of historical significance and contemporary relevance for determining distances between interacting magnetic dipoles. A Metropolis Hastings Markov Chain Monte Carlo algorithm is implemented and designed to find the optimum parameter set and to estimate parameter uncertainties. In conclusion, the posterior distribution allows us to define a metric on parameter space that induces a geometry with negative curvature that affects the parameter uncertainty estimates, particularly for spectra with low signal to noise.

43 PARTICLE ACCELERATORS↗

A dataset of eco-evidence tools to inform early-stage environmental impact assessments of hydropower development

The datasets described herein provide the foundation for a decision support prototype (DSP) toolkit aimed at assisting stakeholders in determining evidence of which aspects of river ecosystems have been impacted by hydropower. The DSP toolkit and its application are presented and described in the article “Evidence-based indicator approach to guide preliminary environmental impact assessments of hydropower development” [1]. Development of the DSP and the output for decision support centralize around 42 river function indicators describing the dimensionality of river ecosystems through six main categories: biota and biodiversity, water quality, hydrology, geomorphology, land cover, and river connectivity. Three main tools are represented in the DSP: A science-based questionnaire (SBQ), an environmental envelope model (EEM), and a river function linkage assessment tool (RFLAT). The SBQ is a structured survey-style questionnaire whose objective is to provide evidence of which indicators have been impacted by hydropower. Based on a global literature review, 140 questions were developed from general hypotheses regarding the impacts of dams on rivers. The EEM is a model to predict the likelihood of hydropower impacting indicators based on a several variables. The intended use of the EEM is for situations of new hydropower development where results of the SBQ are incomplete or highly uncertain. The EEM was developed through the compilation of a dataset containing attributes of dams, reservoirs, and geospatial information on environmental concerns, which was combined with data on ecological indicators documented at those sites through literature review. The model operates through 247 “envelopes” and weighting factors, representing the individual effect of each variable on each indicator, all available through spreadsheets. Finally, the RFLAT is a tool to examine causal relationships amongst indicators. Inter-indicator relationships were hypothesized based on literature review and summarized into node and edge datasets to represent the structure of a graphical network. Bayes theorem was used estimate conditional probabilities of inter-indicator relationships based on the output of the SBQ. Nodes and edges were imported into R programming environment to visualize ecological indicator networks. The datasets can be expanded upon and enriched with more detailed questions for the SBQ, building upon the EEM with to develop more sophisticated models, and identifying new relationships for the RFALT. Additionally, once the tools are applied to numerous hydropower developments, the output of the tools (e.g. evidence of impacted indicators) becomes a very useful dataset for meta-analyses of hydropower impacts.

13 HYDRO ENERGY↗