Advanced Algorithmic Implementations of Bayesian Inference in Cosmology
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Jewell, Jeffrey↗
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Genetic Algorithm for Optimization of Neural Networks for Bayesian Inference of Model Uncertainty
The objective of this work was to develop a genetic optimization algorithm that can design a neural network capable of producing uncertainty estimates along with predictions. This algorithm is necessary because the inclusion of uncertainty modeling in a neural network greatly complicates the network’s design space, making the development of a converging model extremely difficult and time consuming. The genetic algorithm presented in this work uses a number of value ranges for various configurable neural network parameters to create a randomly generated population of network architectures. The initially generated population is then evolved over the course of several generations, with the best performing models breeding to produce novel network configurations. Mutations are randomly applied to the network designs to facilitate the development of adaptations beneficial to the task being performed. An experiment was conducted to validate the proposed algorithm, in which the genetic optimizer was tasked with producing a neural network capable of predicting the sound pressure level (SPL) resulting from jet-surface interaction (JSI) noise. The data used for this task was generated at the NASA Glenn Research Center in the Aero-Acoustic Propulsion Laboratory. Starting with an initial population size of 35 randomly generated networks, and evolved over the course of 10 generations, the genetic algorithm produced a design able to predict SPL as a result of JSI noise within 0.272 dB, on average.
Genetic algorithm↗
Bayesian inference of texture and elastic constants for additively manufactured cobalt-nickel components using Resonance Ultrasound Spectroscopy
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Jeffrey Rossin↗
Bayesian Inference of Asteroid Physical Properties: Application to Impact Scenarios
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risk assessment↗
On Bayesian Inductive Inference & Predictive Estimation
We investigate Bayesian inference and the Principle of Maximum Entropy (PME) as methods for doing inference under uncertainty. This investigation is primarily through concrete examples that have been previously investigated in the literature. We find that it is possible to do Bayesian inference and PME inference using the same information, despite claims to the contrary, but that the results are not directly comparable. This is because Bayesian inference yields a probability density function (pdf) over the unknown model parameters, whereas PME yields point estimates. If mean estimates are extracted from the Bayesian pdfs, the resulting parameter estimates can differ radically from the PME values and also from the Maximum Likelihood values. We conclude that these differences are due to the Bayesian inference not assuming anything beyond the given prior probabilities and the data, whereas PME implicitly assumes that the given constraints are the only constraints that are operating. Since this assumption can be wrong, PME values may have to be revised when subsequent data shows evidence for more constraints. The entropy concentration previously "proved" by E. T. Jaynes is shown to be in error. Further, we show that PME is a generalized form of independence assumption, and so can be a very powerful method of inference when the variables being investigated are largely independent of each other.
Cheeseman, Peter↗
Ockham's razor and Bayesian analysis
'Ockham's razor', the ad hoc principle enjoining the greatest possible simplicity in theoretical explanations, is presently shown to be justifiable as a consequence of Bayesian inference; Bayesian analysis can, moreover, clarify the nature of the 'simplest' hypothesis consistent with the given data. By choosing the prior probabilities of hypotheses, it becomes possible to quantify the scientific judgment that simpler hypotheses are more likely to be correct. Bayesian analysis also shows that a hypothesis with fewer adjustable parameters intrinsically possesses an enhanced posterior probability, due to the clarity of its predictions.
Jefferys, William H.↗
On the theory and methods of statistical inference
Bayesian estimation, decision theory, least squares method, maximum likelihood, and other mathematical techniques of statistical inference theory
MATHEMATICAL MODEL↗
Understanding the Scalability of Bayesian Network Inference using Clique Tree Growth Curves
Bayesian networks (BNs) are used to represent and efficiently compute with multi-variate probability distributions in a wide range of disciplines. One of the main approaches to perform computation in BNs is clique tree clustering and propagation. In this approach, BN computation consists of propagation in a clique tree compiled from a Bayesian network. There is a lack of understanding of how clique tree computation time, and BN computation time in more general, depends on variations in BN size and structure. On the one hand, complexity results tell us that many interesting BN queries are NP-hard or worse to answer, and it is not hard to find application BNs where the clique tree approach in practice cannot be used. On the other hand, it is well-known that tree-structured BNs can be used to answer probabilistic queries in polynomial time. In this article, we develop an approach to characterizing clique tree growth as a function of parameters that can be computed in polynomial time from BNs, specifically: (i) the ratio of the number of a BN's non-root nodes to the number of root nodes, or (ii) the expected number of moral edges in their moral graphs. Our approach is based on combining analytical and experimental results. Analytically, we partition the set of cliques in a clique tree into different sets, and introduce a growth curve for each set. For the special case of bipartite BNs, we consequently have two growth curves, a mixed clique growth curve and a root clique growth curve. In experiments, we systematically increase the degree of the root nodes in bipartite Bayesian networks, and find that root clique growth is well-approximated by Gompertz growth curves. It is believed that this research improves the understanding of the scaling behavior of clique tree clustering, provides a foundation for benchmarking and developing improved BN inference and machine learning algorithms, and presents an aid for analytical trade-off studies of clique tree clustering using growth curves.
Mengshoel, Ole Jakob↗
Understanding the Scalability of Bayesian Network Inference Using Clique Tree Growth Curves
One of the main approaches to performing computation in Bayesian networks (BNs) is clique tree clustering and propagation. The clique tree approach consists of propagation in a clique tree compiled from a Bayesian network, and while it was introduced in the 1980s, there is still a lack of understanding of how clique tree computation time depends on variations in BN size and structure. In this article, we improve this understanding by developing an approach to characterizing clique tree growth as a function of parameters that can be computed in polynomial time from BNs, specifically: (i) the ratio of the number of a BN s non-root nodes to the number of root nodes, and (ii) the expected number of moral edges in their moral graphs. Analytically, we partition the set of cliques in a clique tree into different sets, and introduce a growth curve for the total size of each set. For the special case of bipartite BNs, there are two sets and two growth curves, a mixed clique growth curve and a root clique growth curve. In experiments, where random bipartite BNs generated using the BPART algorithm are studied, we systematically increase the out-degree of the root nodes in bipartite Bayesian networks, by increasing the number of leaf nodes. Surprisingly, root clique growth is well-approximated by Gompertz growth curves, an S-shaped family of curves that has previously been used to describe growth processes in biology, medicine, and neuroscience. We believe that this research improves the understanding of the scaling behavior of clique tree clustering for a certain class of Bayesian networks; presents an aid for trade-off studies of clique tree clustering using growth curves; and ultimately provides a foundation for benchmarking and developing improved BN inference and machine learning algorithms.
Mengshoel, Ole J.↗
Bayesian Super-Resolved Surface Reconstruction From Multiple Images
Bayesian inference has been wed successfully for many problems where the aim is to infer the parameters of a model of interest. In this paper we formulate the three dimensional reconstruction problem as the problem of inferring the parameters of a surface model from image data, and show how Bayesian methods can be used to estimate the parameters of this model given the image data. Thus we recover the three dimensional description of the scene. This approach also gives great flexibility. We can specify the geometrical properties of the model to suit our purpose, and can also use different models for how the surface reflects the light incident upon it. In common with other Bayesian inference problems, the estimation methodology requires that we can simulate the data that would have been recoded for any values of the model parameters. In this application this means that if we have image data we must be able to render the surface model. However it also means that we can infer the parameters of a model whose resolution can be chosen irrespective of the resolution of the images, and may be super-resolved. We present results of the inference of surface models from simulated aerial photographs for the case of super-resolution, where many surface elements project into a single pixel in the low-resolution images.
Smelyanskiy, V. N.↗
Computational Inference of Vibratory System with Incomplete Modal Information Using Parallel, Interactive and Adaptive Markov Chains
Inverse analysis of vibratory system is an important subject in fault identification, model updating, and robust design and control. It is challenging subject because 1) the problem is oftentimes underdetermined while the measurements are limited and/or incomplete; 2) many combinations of parameters may yield results that are similar with respect to actual response measurements; and 3) uncertainties inevitably exist. The aim of this research is to leverage upon computational intelligence through statistical inference to facilitate an enhanced, probabilistic framework using incomplete modal response measurement. This new framework is built upon efficient inverse identification through optimization, whereas Bayesian inference is employed to account for the effect of uncertainties. To overcome the computational cost barrier, we adopt Markov chain Monte Carlo (MCMC) to characterize the target function/distribution. Instead of using single Markov chain in conventional Bayesian approach, we develop a new sampling theory with multiple parallel, interactive and adaptive Markov chains and incorporate it into Bayesian inference. This can harness the collective power of these Markov chains to realize the concurrent search of multiple local optima. The number of required Markov chains and their respective initial model parameters are automatically determined via Monte Carlo simulation-based sample pre-screening followed by K-means clustering analysis. These enhancements can effectively address the aforementioned challenges in finite element inverse analysis. The validity of this framework is systematically demonstrated through case studies.
K Zhou↗
Optimal Estimation Framework for Ocean Color Atmospheric Correction and Pixel-level Uncertainty Quantification
Ocean color remote sensing requires compensation for atmospheric scattering and absorption (aerosol, Rayleigh, and trace gases), referred to as atmospheric correction (AC). AC allows inference of parameters such as spectrally resolved remote sensing reflectance ( R rs )(λ) ; sr 1 ) at the ocean surface from the top-of-atmosphere reflectance. Often, the uncertainty of this process is not fully explored. Bayesian inference techniques provide a simultaneous AC and uncertainty assessment via a full posterior distribution of the relevant variables, given the prior distribution of those variables and the radiative transfer (RT) likelihood function. Given uncertainties in the algorithm inputs, the Bayesian framework enables better constraints on the AC process by using the complete spectral information compared to traditional approaches that use only a subset of bands for AC. This paper investigates a Bayesian inference research method (Optimal Estimation, OE) for ocean color AC by simultaneously retrieving atmospheric and ocean properties using all visible and near-infrared spectral bands. The OE algorithm analytically approximates the posterior distribution of parameters based on normality assumptions and provides a potentially viable operational algorithm with a reduced computational expense. We developed a Neural Network (NN) RT forward model look-up-table-based emulator to increase algorithm efficiency further and thus speed up the likelihood computations. We then applied the OE algorithm to synthetic data and observations from the MODerate resolution Imaging Spectroradiometer (MODIS) on NASA’s Aqua spacecraft. We compared the R rs )(λ) retrieval and its uncertainty estimates from the OE method with in-situ validation data from the SeaWiFS Bio-optical Archive and Storage System (SeaBASS) and Aerosol Robotic Network Ocean Color (AERONET-OC) datasets. The OE algorithm improved R rs )(λ) estimates relative to the NASA standard operational algorithm by improving all statistical metrics at 443, 555, and 667 nm. Unphysical negative R rs )(λ) , which often appear in complex water conditions, was reduced by a factor of 3. The OE-derived pixel-level R rs )(λ) uncertainty estimates were also assessed relative to in-situ data and were shown to have skill.
Atmospheric correction↗
Probabilistic Damage Characterization Using the Computationally-Efficient Bayesian Approach
This work presents a computationally-ecient approach for damage determination that quanti es uncertainty in the provided diagnosis. Given strain sensor data that are polluted with measurement errors, Bayesian inference is used to estimate the location, size, and orientation of damage. This approach uses Bayes' Theorem to combine any prior knowledge an analyst may have about the nature of the damage with information provided implicitly by the strain sensor data to form a posterior probability distribution over possible damage states. The unknown damage parameters are then estimated based on samples drawn numerically from this distribution using a Markov Chain Monte Carlo (MCMC) sampling algorithm. Several modi cations are made to the traditional Bayesian inference approach to provide signi cant computational speedup. First, an ecient surrogate model is constructed using sparse grid interpolation to replace a costly nite element model that must otherwise be evaluated for each sample drawn with MCMC. Next, the standard Bayesian posterior distribution is modi ed using a weighted likelihood formulation, which is shown to improve the convergence of the sampling process. Finally, a robust MCMC algorithm, Delayed Rejection Adaptive Metropolis (DRAM), is adopted to sample the probability distribution more eciently. Numerical examples demonstrate that the proposed framework e ectively provides damage estimates with uncertainty quanti cation and can yield orders of magnitude speedup over standard Bayesian approaches.
Warner, James E.↗
Near Real-Time Probabilistic Damage Diagnosis Using Surrogate Modeling and High Performance Computing
This work investigates novel approaches to probabilistic damage diagnosis that utilize surrogate modeling and high performance computing (HPC) to achieve substantial computational speedup. Motivated by Digital Twin, a structural health management (SHM) paradigm that integrates vehicle-specific characteristics with continual in-situ damage diagnosis and prognosis, the methods studied herein yield near real-time damage assessments that could enable monitoring of a vehicle's health while it is operating (i.e. online SHM). High-fidelity modeling and uncertainty quantification (UQ), both critical to Digital Twin, are incorporated using finite element method simulations and Bayesian inference, respectively. The crux of the proposed Bayesian diagnosis methods, however, is the reformulation of the numerical sampling algorithms (e.g. Markov chain Monte Carlo) used to generate the resulting probabilistic damage estimates. To this end, three distinct methods are demonstrated for rapid sampling that utilize surrogate modeling and exploit various degrees of parallelism for leveraging HPC. The accuracy and computational efficiency of the methods are compared on the problem of strain-based crack identification in thin plates. While each approach has inherent problem-specific strengths and weaknesses, all approaches are shown to provide accurate probabilistic damage diagnoses and several orders of magnitude computational speedup relative to a baseline Bayesian diagnosis implementation.
Warner, James E.↗
Probabilistic Calibration of Expensive Models using Efficiently Trained Surrogates
Calibration of computational models in the presence of uncertainty is often cast as a Bayesian inference problem and solved via sampling methods, e.g., Markov chain Monte Carlo. When the computational model is expensive, this task becomes intractable due to the large number of samples required to accurately estimate the posterior distribution of the calibration parameters. A popular solution to this problem is to use machine learning to develop a faster-to-evaluate, lower-fidelity substitute for the original model to serve as a surrogate while solving the inference problem. Although considered an offline cost, generating training data to construct this surrogate model can still be an expensive task in practice. An active learning algorithm is presented that focuses training on improving surrogate accuracy specifically in and around the bulk of the posterior distribution, as this is where the model is exercised during calibration. Candidate samples are drawn from families of distributions related to an approximation of the posterior. The sample maximizing predictive variance is then selected for evaluation by the original computational model, yielding a label for the training point. Iterating this approach increases efficiency relative to space filling designs (e.g., Latin hypercube sampling) by avoiding low probability points. Practical considerations are discussed, including the benefits of using a sequential Monte Carlo sampling approach, convergence heuristics, and the importance of both exploration and exploitation given that the true posterior is unknown a priori.
uncertainty quantification↗
Shock-Stationary Application of Pseudoshock Models During High-Amplitude Combustion-Driven Unsteadiness
The isolator pseudo-shock provides necessary compression within a dual-mode scramjet engine and buffers the engine system against unstart. Quasi-1D flux-conserved models are the state-of-the-art reduced-order model for optimization and online control of dual-mode scramjet engines. The stability and efficacy of this modeling approach is evaluated against data from a combustor-driven direct-connect experiment. The experiment exhibited strong combustor-driven unsteadiness that produced upstream propagating weak shocks into the isolator, interacting with the pseudo-shock. While this configuration resulted in unsteadiness that is atypical of standard operation, the experiment provided an opportunity to evaluate the modeling techniques in highly transient states. Such transients could occur during maneuvering or result from unexpected combustor events. A flexible quasi-1D formulation, the Fievet flux-conserved model is fit using Bayesian inference in the laboratory and shock-stationary reference frames. Model performance is analyzed using the Bayesian posteriors and model evaluations over the measured shock train speed range. It is concluded that to produce consistent isolator pressure profile estimates in this unsteady environment, the model must be implemented in a shock-stationary reference frame. Implementing this conclusion in model-based engine controllers may reduce needed unstart safety margins and increase maximum performance.
Bayesian↗
The inference of mantle viscosity from an inversion of the Fennoscandian relaxation spectrum
The theory of Peltier (1976) for mantle viscosity is adopted, within the framework of nonlinear Bayesian Inference, to invert the Fennoscandian relaxation spectra derived by McConnell (1968). A set of rigorous constraints which all models for the viscosity variation beneath Fennoscandian must satisfy is derived. These constraints are used to test the plausibility of a wide class of viscosity models. It is shown that a model with a weak asthenosphere overlying an isoviscous 10 exp 21 Pa s deep mantle provides to good fit to the relaxation spectrum. This is also true of models with a thin sublithospheric low-viscosity zone overlying a two-layer deep mantle with a moderate jump in viscosity across 670 km depth, and models with a viscosity jump of between four and six across isoviscous upper and lower mantle regions.
Mitrovica, J. X.↗