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2018 | Book

Quantum Phases in Entropic Dynamics Read chapter Read first chapter

Bayesian Inference and Maximum Entropy Methods in Science and Engineering

MaxEnt 37, Jarinu, Brazil, July 09–14, 2017

Editors: Prof. Dr. Adriano Polpo, Prof. Dr. Julio Stern, Prof. Dr. Francisco Louzada, Rafael Izbicki, Hellinton Takada

Publisher: Springer International Publishing

Book Series : Springer Proceedings in Mathematics & Statistics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

These proceedings from the 37th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering (MaxEnt 2017), held in São Carlos, Brazil, aim to expand the available research on Bayesian methods and promote their application in the scientific community. They gather research from scholars in many different fields who use inductive statistics methods and focus on the foundations of the Bayesian paradigm, their comparison to objectivistic or frequentist statistics counterparts, and their appropriate applications.

Interest in the foundations of inductive statistics has been growing with the increasing availability of Bayesian methodological alternatives, and scientists now face much more difficult choices in finding the optimal methods to apply to their problems. By carefully examining and discussing the relevant foundations, the scientific community can avoid applying Bayesian methods on a merely ad hoc basis.

For over 35 years, the MaxEnt workshops have explored the use of Bayesian and Maximum Entropy methods in scientific and engineering application contexts. The workshops welcome contributions on all aspects of probabilistic inference, including novel techniques and applications, and work that sheds new light on the foundations of inference. Areas of application in these workshops include astronomy and astrophysics, chemistry, communications theory, cosmology, climate studies, earth science, fluid mechanics, genetics, geophysics, machine learning, materials science, medical imaging, nanoscience, source separation, thermodynamics (equilibrium and non-equilibrium), particle physics, plasma physics, quantum mechanics, robotics, and the social sciences. Bayesian computational techniques such as Markov chain Monte Carlo sampling are also regular topics, as are approximate inferential methods. Foundational issues involving probability theory and information theory, as well as novel applications of inference to illuminate the foundations of physical theories, are also of keen interest.

Table of Contents

Frontmatter
Quantum Phases in Entropic Dynamics
Abstract
In the Entropic Dynamics framework, the dynamics is driven by maximizing entropy subject to appropriate constraints. In this work, we bring Entropic Dynamics one-step closer to full equivalence with quantum theory by identifying constraints that lead to wave functions that remain single-valued even for multi-valued phases by recognizing the intimate relation between quantum phases, gauge symmetry, and charge quantization.
Nicholas Carrara, Ariel Caticha
Bayesian Approach to Variable Splitting Forward Models
Abstract
Classical single additive noise forward model can be extended to account for different uncertainties by variable splitting models. For example, one can distinguish between observation noise and forward model uncertainty or even to account for other forward model uncertainties. In this paper, we consider different cases and propose to use the Bayesian approach to handle them. As a by-product, we see that when MAP estimator is used we can find the same kind of optimization algorithms as Alternating Direction Method of Multipliers (ADMM) or Iterative Shrinkage Thresholding Algorithm (ISTA) optimization ones. However, the Bayesian approach gives us the tools to go further by estimating the hyperparameters of the inversion problems which are often crucial in real applications.
Ali Mohammad-Djafari, Mircea Dumitru, Camille Chapdelaine, Li Wang
Prior Shift Using the Ratio Estimator
Abstract
Several machine learning applications use classifiers as a way of quantifying the prevalence of positive class labels in a target dataset, a task named quantification. For instance, a naive a way of determining what proportion of people like a given product with no labeled reviews is to (i) train a classifier based on the Google Shopping reviews to predict whether a user likes a product given its review, and then (ii) apply this classifier to Facebook/Google+ posts about that product. It is well known that such a two-step approach, named Classify and Count, fails because of dataset shift, and thus, several improvements have been recently proposed under an assumption named prior shift. Unfortunately, these methods only explore the relationship between the covariates and the response via classifiers. Moreover, the literature lacks in the theoretical foundation to improve these techniques. We propose a new family of estimators named Ratio Estimator which is able to explore the relationship between the cov ariates and the response using any function \( g: \mathscr {X} \rightarrow \mathbb {R}\) and not only classifiers. We show that for some choices of g, our estimator matches standard estimators used in the literature. We also explore alternative ways of constructing functions g that lead to estimators with good performance, and compare them using real datasets. Finally, we provide a theoretical analysis of the method.
Afonso Vaz, Rafael Izbicki, Rafael Bassi Stern
Bayesian Meta-Analytic Measure
Abstract
Meta-analysis is a procedure that combines results from studies (or experiments) with a common interest: inferences about an unknown parameter. We present a meta-analytic measure based on a combination of the posterior density functions obtained in each of the studies. Clearly, the point of view is from a Bayesian perspective. The measure preserves both the heterogeneity between and within the studies, and it is assumed that the all of the data from each study are available.
Camila B. Martins, Carlos A. de B. Pereira, Adriano Polpo
Feature Selection from Local Lift Dependence-Based Partitions
Abstract
The classical approach to feature selection consists in minimizing a cost function of the estimated joint distribution of the variable of interest and the feature vectors. However, in order to estimate the joint distribution, and therefore, the cost function, it is necessary to discretize the variables, so that feature selection algorithms are partition dependent, as they depend on the partitions in which the variables are discretized. In this framework, this paper aims to propose a systematic approach to the discretization of random vectors, which is based on the Local Lift Dependence. Our approach allows an interpretation of the local dependence between the variable of interest and the selected features, so that it is possible to outline the kind of dependence that exists between them. The proposed approach is applied to study the dependence between the performances on entrance exam subjects and on first semester courses of University of São Paulo Statistics and Computer Science undergraduate programs.
Diego Marcondes, Adilson Simonis, Junior Barrera
Probabilistic Inference of Surface Heat Flux Densities from Infrared Thermography
Abstract
In nuclear fusion research, based on the magnetic confinement, the determination of the heat flux density distribution onto the plasma facing components is important. The heat load poses the threat of damaging the components. The heat flux distribution is a footprint of the transport mechanisms in the plasma, which are still to be understood. Obtaining the heat flux density is an ill-posed problem. Most common is a measurement of the surface temperature by means of infrared thermography. Solving the heat diffusion equation in the target material with measured temperature information as boundary condition allows to determine the surface heat load distribution. A Bayesian analysis tool is developed as an alternative to deterministic tools, which aim for fast evaluation. The probabilistic evaluation uses adaptive kernels to model the heat flux distribution. They allow for self-consistent determination of the effective Degree of Freedom, depending on the quality of the measurement. This is beneficial, as the signal-to-noise ratio depends on the surface temperatures, ranging from room temperatures up to the melting point of tungsten.
D. Nille, U. von Toussaint, B. Sieglin, M. Faitsch
Schrödinger’s Zebra: Applying Mutual Information Maximization to Graphical Halftoning
Abstract
The graphical process of halftoning is, fundamentally, a communication process: an image made from a continuous set of possible grays, for example, is to be represented recognizably by elements that are only black or white. With this in mind, we ask what a halftoning algorithm would look like that maximizes the mutual information between images and their halftoned renditions. Here, we find such an algorithm and explore its properties. The algorithm is inherently probabilistic and bears an information theoretic similarity to features of quantum mechanical measurements, so we dub the method quantum halftoning. The algorithm provides greater discrimination of medium gray shades, and less so very dark or very light shades, as we show via both the algorithm’s mathematical structure and examples of its application. We note, in passing, some generalized applications of this algorithm. Finally, we conclude by showing that our methodology offers a tool to investigate Bayesian priors of the human visual system, and spell out a scheme to use the results of this paper to do so.
Antal Spector-Zabusky, Donald Spector
Regression of Fluctuating System Properties: Baryonic Tully–Fisher Scaling in Disk Galaxies
Abstract
In various interesting physical systems, important properties or dynamics display a strongly fluctuating behavior that can best be described using probability distributions. Examples are fluid turbulence, plasma instabilities, textured images, porous media and cosmological structure. In order to quantitatively compare such phenomena, a similarity measure between distributions is needed, such as the Rao geodesic distance on the corresponding probabilistic manifold. This can form the basis for validation of theoretical models against experimental data and classification of regimes, but also for regression between fluctuating properties. This is the primary motivation for geodesic least squares (GLS) as a robust regression technique, with general applicability. In this contribution, we further clarify this motivation and we apply GLS to Tully–Fisher scaling of baryonic mass vs. rotation velocity in disk galaxies. We show that GLS is well suited to estimate the coefficients and tightness of the scaling. This is relevant for constraining galaxy formation models and for testing alternatives to the Lambda cold dark matter cosmological model.
Geert Verdoolaege
Bayesian Portfolio Optimization for Electricity Generation Planning
Abstract
Nowadays, there are several electricity generation technologies based on the different sources, such as wind, biomass, gas, coal, and so on. Considering the uncertainties associated with the future costs of such technologies is crucial for planning purposes. In the literature, the allocation of resources in the available technologies have been solved as a mean-variance optimization problem using the expected costs and the correspondent covariance matrix. However, in practice, the expected values and the covariance matrix of interest are not exactly known parameters. Consequently, the optimal allocations obtained from the mean-variance optimization are not robust to possible errors in the estimation of such parameters. Additionally, there are specialists in the electricity generation technologies participating in the planning process and, obviously, the consideration of useful prior information based on their previous experience is of utmost importance. The Bayesian models consider not only the uncertainty in the parameters, but also the prior information from the specialists. In this paper, we introduce the Bayesian mean-variance optimization to solve the electricity generation planning problem using both improper and proper prior distributions for the parameters. In order to illustrate our approach, we present an application comparing the Bayesian with the naive mean-variance optimal portfolios.
Hellinton H. Takada, Julio M. Stern, Oswaldo L. V. Costa, Celma de O. Ribeiro
Bayesian Variable Selection Methods for Log-Gaussian Cox Processes
Abstract
Point patterns are very common in present days of many researchers. The desire to understand the spatial distribution and investigate connections between point patterns and p covariates, that is possibly associated with the event of interest, arises naturally. Generally, not all of the p covariates are useful. Therefore it would be handy to identify the covariate which is, and just use those. Variable selection is an important step when setting a parsimonious model and still occupies the minds of many statisticians. In this work, we investigated Bayesian variable selection methods in the context of point pattern. This work concentrated on the following methods: Kuo and Mallick, Gibbs Variable Selection, and Stochastics Search Variable Selection for log-Gaussian Cox processes. The methods were evaluated in several scenarios: with a different number of covariates that should be included in the model, absence, and presence of multicollinearity and fixed and random effect model. Our results suggest that the three methods, specially Stochastics Search Variable Selection, can work very well with the absence of multicollinearity. We implemented the methods in BUGS.
Jony Arrais Pinto Junior, Patrícia Viana da Silva
Effect of Hindered Diffusion on the Parameter Sensitivity of Magnetic Resonance Spectra
Abstract
Magnetic Resonance spectroscopy is a powerful tool for elucidating the details of molecular dynamics. In many important applications, a model of hindered diffusion is useful for summarizing the complex dynamics of ordered media, such as a liquid crystalline environment, as well as the dynamics of proteins in solution or confined to a membrane. In previous work, we have shown how the sensitivity of a magnetic resonance spectrum to the details of molecular dynamics depends on the symmetries of the magnetic tensors for the relevant interactions, e.g., Zeeman, hyperfine, or quadrupolar interactions. If the hindered diffusion is modeled as arising from an orienting potential, then the parameter sensitivity of the magnetic resonance spectrum may be studied by generalizations of methods we have introduced in previous work. In particular, we will show how lineshape calculations using eigenfunction expansions of solutions of the diffusion equation, can be used as inputs to an information-geometric approach to parameter sensitivity estimation. We illustrate our methods using model systems drawn from Nuclear Magnetic Resonance, Electron Spin Resonance, and Nuclear Quadrupole Resonance.
Keith A. Earle, Troy Broderick, Oleks Kazakov
The Random Bernstein Polynomial Smoothing Via ABC Method
Abstract
In recent years, many statistical inference problems have been solved by using Markov Chain Monte Carlo (MCMC) techniques. However, it is necessary to derivate the analytical form for the likelihood function. Although the level of computing has increased steadily, there is a limitation caused by the difficulty or the misunderstanding of how computing the likelihood function. The Approximate Bayesian Computation (ABC) method dispenses the use of the likelihood function by simulating candidates of posterior distributions and using an algorithm to accept or reject the proposed candidates. This work presents an alternative nonparametric estimation method of smoothing empirical distributions with random Bernstein polynomials via ABC method. The Bernstein prior is obtained by rewriting the Bernstein polynomial in terms of k mixtures / m mixtures of beta densities and mixing weights. A study of simulation and a real example are presented to illustrate the method proposed.
Leandro A. Ferreira, Victor Fossaluza
Mean Field Studies of a Society of Interacting Agents
Abstract
We model a society of agents that interact in pairs by exchanging for/against opinions about issues using an algorithm obtained with methods of Bayesian inference and maximum entropy. The agents gauge the incoming information with respect to the mistrust attributed to the other agents. There is no underlying lattice and all agents interact among themselves. The interaction pair can be described as a dynamics along the gradient of the logarithm of the evidence. By using a symmetric version of the two-body interactions we introduce a Hamiltonian for the whole society. Knowledge of the expected value of the Hamiltonian is relevant information for the state of the society. In the case of uniform mistrust, independent of the pair of agents, the phase diagram of the society in a mean field approximation shows a phase transition that separates an ordered phase where opinions are to a large extent shared by the agents and a disordered phase of dissension of opinions.
Lucas Silva Simões, Nestor Caticha
The Beginnings of Axiomatic Subjective Probability
Abstract
We study the origins of the axiomatization of subjective probabilities. Starting with the problem of how to measure subjective probabilities, our main goal was to search for the first explicit uses of the definition of subjective probability using betting odds or ratios, i.e., using the Dutch book argument, as it is presently known. We have found two authors prior to Ramsey (The foundations of mathematics and other logical essays. Routledge & Kegan Paul, 1931, [43]) and de Finetti (Fund Math 17:298–329, 1931, [20]) that used the mentioned definition: Émile Borel, in an article of 1924, and Jean-Baptiste Estienne, a French army officer, in a series of four articles published in 1903 and 1904. We tried to identify, in the references given by Borel and Estienne, inspirations common to Ramsey and de Finetti in order to determine, in the literature on the probability of the beginning of the last century, at least some elements that point to specific events that lead to the referred axiomatization. To the best of our knowledge, the genesis of the axiomatic approach in the subjective school was not traced yet, and this untold history can give us a better understanding of recent developments and help us, as applied scientists, in future works.
Marcio A. Diniz, Sandro Gallo
Model Selection in the Sparsity Context for Inverse Problems in Bayesian Framework
Abstract
The Bayesian approach is considered for inverse problems with a typical forward model accounting for errors and a priori sparse solutions. Solutions with sparse structure are enforced using heavy-tailed prior distributions. The particular case of such prior expressed via normal variance mixtures with conjugate laws for the mixing distribution is the main interest of this paper. Such a prior is considered in this paper, namely, the Student-t distribution. Iterative algorithms are derived via posterior mean estimation. The mixing distribution parameters appear in updating equations and are also used for the initialization. For the choice of mixing distribution parameters, three model selection strategies are considered: (i) parameters approximating the mixing distribution with Jeffrey law, i.e., keeping the mixing distribution well defined but as close as possible to the Jeffreys priors, (ii) based on the prior distribution form, fixing the parameters corresponding to the form inducing the most sparse solution and (iii) based on the sparsity mechanism, fixing the hyperparameters using the statistical measures of the mixing and prior distribution. For each strategy of model selection, the theoretical advantages and drawbacks are discussed and the corresponding simulations are reported for a 1D direct sparsity application in a biomedical context. We show that the third strategy seems to provide the best parameter selection strategy for this context.
Mircea Dumitru, Li Wang, Ali Mohammad-Djafari, Nicolas Gac
Sample Size Calculation Using Decision Theory
Abstract
Decision Theory and Bayesian Inference have an important role to solve some common problems in research and practice in the medical field. These decisions may be from different natures and can consider several factors, such as the cost to carry out the study and each sample unit and, especially, the risks for the patients involved. Here, the estimation of sample size calculation considers the cost of sampling units and clinically relevant size of the credible interval for difference between groups. By fixing a probability to the HPD region, the Bayes’ Risk is calculated for each sample size possible and it is chosen the optimal sample size, that minimizes the risk. In addition, a second solution is presented by setting the amplitude of the credible interval, leaving its probability free. It is considered a Normal distribution for data with unknown mean and fixed variance (Normal prior) and the case where both mean and variance are unknown (Normal-Inverse Gamma prior). It is presented as a solution considering the statistical distribution of sufficient statistics. In scenarios with no analytical solutions, the optimal sample sizes are presented using Monte Carlo methods.
Milene Vaiano Farhat, Nicholas Wagner Eugenio, Victor Fossaluza
Utility for Significance Tests
Abstract
The range of possible readings among and within the statistical inference, in addition to the relevance of these in the applied context, justify the extensive literature analyzing and comparing the main methodologies. However, the fact that each approach is built upon their own structures, varying even the spaces in which they are evaluated, limit the conclusions to the specified scenarios. As a solution for that, in the context of hypotheses tests, we work with the decision theory, which provides a unique language to incorporate the logic of each existent philosophy. For such, after discussing the main points of the frequentist and Bayesian inference, the main approaches are presented, particularly regarding to precise hypotheses, and then unify by the decision-theoretic viewpoint. Additionally, by through this perspective we analyze, interpret and compare the loss functions of some precise approaches, in the context of significance tests.
Nathália Demetrio Vasconcelos Moura, Sergio Wechsler
Probabilistic Equilibrium: A Review on the Application of MAXENT to Macroeconomic Models
Abstract
The concept of equilibrium is central to many macroeconomic models. However, after the 2008 crisis, many of the most used macroeconomic models have been subject to criticism, after their failure in predicting and explaining the crisis. Over the last years, a response to this situation has been the proposal of new approaches to the study of macroeconomical systems, in particular, with the introduction of thermodynamics and statistical physics methods. In this paper, we offer a brief review of the application of the maximum entropy framework in macroeconomics, centered around the different interpretations of the equlibrium concept.
Paulo Hubert, Julio M. Stern
Full Bayesian Approach for Signal Detection with An Application to Boat Detection on Underwater Soundscape Data
Abstract
The problem of detecting a signal of known form in a noisy message is a long-studied problem. In this paper, we formulate it as the test of a sharp hypothesis, and propose the Full Bayesian significance test of Pereira and Stern as the tool for the job. We study the FBST in the signal detection problem using simulated data, and also using data from OceanPod, a hydrophone designed and operated by the Dynamics and Instrumentation Laboratory at EP-USP.
Paulo Hubert, Julio M. Stern, Linilson Padovese
Bayesian Support for Evolution: Detecting Phylogenetic Signal in a Subset of the Primate Family
Abstract
The theory of evolution states that the diversity of species can be explained by descent with modification. Therefore, all living beings are related through a common ancestor. This evolutionary process must have left traces in our molecular composition. In this work, we present a randomization procedure in order to determine if a group of five species of the primate family, namely, macaque, guereza, orangutan, chimpanzee, and human, has retained these traces in its molecules. First, we present the randomization methodology through two toy examples, which allow to understand its logic. We then carry out a DNA data analysis to assess if the group of primates contains phylogenetic information which links them in a joint evolutionary history. This is carried out by monitoring a Bayesian measure, called marginal likelihood, which we estimate by using nested sampling. We found that it would be unusual to get the relationship observed in the data among these primate species if they had not shared a common ancestor. The results are in total agreement with the theory of evolution.
Patricio Maturana Russel
A Comparison of Two Methods for Obtaining a Collective Posterior Distribution
Abstract
Bayesian inference is a powerful method that allows individuals to update their knowledge about any phenomenon when more information about it becomes available. In this paradigm, before data is observed, an individual expresses his uncertainty about the phenomenon of interest through a prior probability distribution. Then, after data is observed, this distribution is updated using Bayes theorem. In many situations, however, one desires to evaluate the knowledge of a group rather than of a single individual. In this case, a way to combine information from different sources is by mixing their uncertainty. The mixture can be done in two ways: before or after the data is observed. Although in both cases, we achieve a collective posterior distribution, they can be substantially different. In this work, we present several comparisons between these two approaches with noninformative priors and use the Kullback–Leibler’s divergence to quantify the amount of information that is gained by each collective distribution.
Rafael Catoia Pulgrossi, Natalia Lombardi Oliveira, Adriano Polpo, Rafael Izbicki
A Nonparametric Bayesian Approach for the Two-Sample Problem
Abstract
In this work, we propose a novel nonparametric Bayesian approach to the so-called two-sample problem. Let \(X_1, \ldots , X_n\) and \(Y_1, \ldots , Y_m\) be two independent i.i.d samples generated from \(P_1\) and \(P_2\), respectively. Using a nonparametric prior distribution for \((P_1,P_2)\), we propose a new evidence index for the null hypothesis \(H_0: P_1 = P_2\) based on the posterior distribution of the distance \(d(P_1, P_2)\) between \(P_1\) and \(P_2\). This evidence index is easy to compute, has an intuitive interpretation, and can also be justified from a Bayesian decision-theoretic framework. We provide a simulation study to show that our method achieves greater power than the Kolmogorov–Smirnov and the Wilcoxon tests in several settings. Finally, we apply the method to a dataset on Alzheimer’s disease.
Rafael de C. Ceregatti, Rafael Izbicki, Luis Ernesto B. Salasar
Covariance Modeling for Multivariate Spatial Processes Based on Separable Approximations
Abstract
The computational treatment of high dimensionality problems is a challenge. In the context of geostatistics, analyzing multivariate data requires the specification of the cross-covariance function, which defines the dependence between the components of a response vector for all locations in the spatial domain. However, the computational cost to make inference and predictions can be prohibitive. As a result, the use of complex models might be unfeasible. In this paper, we consider a flexible nonseparable covariance model for multivariate spatiotemporal data and present a way to approximate the full covariance matrix from two separable matrices of minor dimensions. The method is applied only in the likelihood computation, keeping the interpretation of the original model. We present a simulation study comparing the inferential and predictive performance of our proposal and we see that the approximation provides important gains in computational efficiency without presenting substantial losses in predictive terms.
Rafael S. Erbisti, Thais C. O. Fonseca, Mariane B. Alves
Uncertainty Quantification and Cumulative Distribution Function: How are they Related?
Abstract
Uncertainty is described by the cumulative distribution function (CDF). Using, the CDF one describes all the main cases: the discrete case, the case when a absolutely continuous probability density exists, and the singular case, when it does not, or combinations of the three preceding cases. The reason one does not see any mention of uncertainty quantification in classical books, as Feller’s and Chung’s, is that they found no reason to call a CDF by another name. However, one has to acknowledge that to use a CDF to describe uncertainty is clumsy. The comparison of CDF to see which is more uncertain is not evident. One feels that there must be a simpler way. Why not to use some small set of statistics to reduce a CDF to a simpler measure, easier to grasp? This seems a great idea and, indeed, one finds it in the literature. Several books deal with the problem. We focus the discussion on three main cases: (1) to use mean and standard deviation to construct an envelope with them; (2) to use coefficient of variation; (3) to use Shannon entropy, a number, that could allow an ordering for the uncertainties of all CDF that have entropy, a most desirable thing. The reductions (to replace the CDF for a small set of statistics) may indeed work in some cases. But they do not always work and, moreover, the different measures they define may not be compatible. That is, the ordering of uncertainty may vary depending what set one chooses. So the great idea does not work so far, but they are happily used in the literature. One of the objectives of this paper is to show, with examples, that the three reductions used to “measure” uncertainties are not compatible. The reason it took so long to find out the mistake is that these reductions methods are applied to very complex problem that hide well the unsuitability of the reductions. Once one tests them with simpler examples one clearly sees their inadequacy. So, let us safely continue to use the CDF while a good reduction is not found!
Roberta Lima, Rubens Sampaio
Maximum Entropy Analysis of Flow Networks with Structural Uncertainty (Graph Ensembles)
Abstract
This study examines MaxEnt methods for probabilistic inference of the state of flow networks, including pipe flow, electrical and transport networks, subject to physical laws and observed moments. While these typically assume networks of invariant graph structure, we here consider higher-level MaxEnt schemes, in which the network structure constitutes part of the uncertainty in the problem specification. In physics, most studies on the statistical mechanics of graphs invoke the Shannon entropy \(H_G^{Sh} = - \sum \nolimits _{\Omega _G} P(G) \ln P(G)\), where G is the graph and \(\Omega _G\) is the graph ensemble. We argue that these should adopt the relative entropy \(H_G = - \sum \nolimits _{\Omega _G} P(G) \ln {P(G)}/{Q(G)}\), where Q(G) is the graph prior associated with the graph macrostate G. By this method, the user is able to employ a simplified accounting over graph macrostates rather than need to count individual graphs. Using combinatorial methods, we here derive a variety of graph priors for different graph ensembles, using different macrostate partitioning schemes based on the node or edge counts. A variety of such priors are listed herein, for ensembles of undirected or directed graphs.
Robert K. Niven, Michael Schlegel, Markus Abel, Steven H. Waldrip, Roger Guimera
Optimization Employing Gaussian Process-Based Surrogates
Abstract
The optimization of complex plasma-wall interaction and material science models is tantamount with long-running and expensive computer simulations. This indicates the use of surrogate-based methods in the optimization process. A Gaussian process (GP)-based Bayesian adaptive exploration method has been developed and validated on mock examples. The self-consistent adjustment of hyperparameters according to the information present in the data turns out to be the main benefit from the Bayesian approach. While the overall properties and performance is favorable (in terms of expensive function evaluations), the optimal balance between local and global exploitation still mandates further research for strongly multimodal optimization problems.
R. Preuss, U. von Toussaint
Bayesian and Maximum Entropy Analyses of Flow Networks with Non-Gaussian Priors and Soft Constraints
Abstract
We have recently developed new maximum entropy (MaxEnt) and Bayesian methods for the analysis of flow networks, including pipe flow, electrical and transportation networks. Both methods of inference update a prior probability density function (pdf) with new information, in the form of data or constraints, to obtain a posterior pdf for the system. We here examine the effects of non-Gaussian prior pdfs, including truncated normal and beta distributions, both analytically and by the use of numerical examples, to explore the differences and similarities between the MaxEnt and Bayesian formulations. We also examine ‘soft constraints’ imposed within the prior.
Steven H. Waldrip, Robert K. Niven
Using the Z-Order Curve for Bayesian Model Comparison
Abstract
BayeSys is an MCMC-based program that can be used to perform Bayesian model comparison for problems with atomic models. To sample distributions with more than one parameter, BayeSys uses the Hilbert curve to index the multidimensional parameter space using one very large integer. While the Hilbert curve maintains locality well, computations to translate back and forth between parameter coordinates and Hilbert curve indexes are time-consuming. The Z-order curve is an alternative SFC with faster transformation algorithms. This work presents an efficient bitmask-based algorithm for performing the Z-order curve transformations for an arbitrary number of parameter space dimensions and integer bit-lengths. We compare results for an exponential decay separation problem evaluated using BayeSys with both the Hilbert and Z-order curves. We demonstrate that no appreciable precision penalty is incurred by using the Z-order curve, and there is a significant increase in time efficiency.
R. Wesley Henderson, Paul M. Goggans
Metadata
Title
Bayesian Inference and Maximum Entropy Methods in Science and Engineering
Editors
Prof. Dr. Adriano Polpo
Prof. Dr. Julio Stern
Prof. Dr. Francisco Louzada
Rafael Izbicki
Hellinton Takada
Copyright Year
2018
Electronic ISBN
978-3-319-91143-4
Print ISBN
978-3-319-91142-7
DOI
https://doi.org/10.1007/978-3-319-91143-4

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