scholarly journals Dynamical Sampling with Langevin Normalization Flows

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1096 ◽  
Author(s):  
Minghao Gu ◽  
Shiliang Sun ◽  
Yan Liu
Keyword(s):  
Monte Carlo ◽  
Sampling Method ◽  
Sampling Methods ◽  
Probability Distributions ◽  
Unbiased Estimation ◽  
Rapid Convergence ◽  
Target Distribution ◽  
Leibler Divergence ◽  
Langevin Diffusions ◽  
Bayesian Machine Learning

In Bayesian machine learning, sampling methods provide the asymptotically unbiased estimation for the inference of the complex probability distributions, where Markov chain Monte Carlo (MCMC) is one of the most popular sampling methods. However, MCMC can lead to high autocorrelation of samples or poor performances in some complex distributions. In this paper, we introduce Langevin diffusions to normalization flows to construct a brand-new dynamical sampling method. We propose the modified Kullback-Leibler divergence as the loss function to train the sampler, which ensures that the samples generated from the proposed method can converge to the target distribution. Since the gradient function of the target distribution is used during the process of calculating the modified Kullback-Leibler, which makes the integral of the modified Kullback-Leibler intractable. We utilize the Monte Carlo estimator to approximate this integral. We also discuss the situation when the target distribution is unnormalized. We illustrate the properties and performances of the proposed method on varieties of complex distributions and real datasets. The experiments indicate that the proposed method not only takes the advantage of the flexibility of neural networks but also utilizes the property of rapid convergence to the target distribution of the dynamics system and demonstrate superior performances competing with dynamics based MCMC samplers.

scholarly journals Seismic Tomography Using Variational Inference Methods

2020 ◽  
Author(s):  
Xin Zhang ◽  
Andrew Curtis
Keyword(s):  
Monte Carlo ◽  
Probability Distribution ◽  
Seismic Tomography ◽  
Posterior Probability ◽  
Sampling Methods ◽  
Probability Distributions ◽  
Monte Carlo Sampling ◽  
Variational Inference ◽  
Posterior Probability Distribution ◽  
Inference Methods

<p><span>In a variety of geoscientific applications we require maps of subsurface properties together with the corresponding maps of uncertainties to assess their reliability. Seismic tomography is a method that is widely used to generate those maps. Since tomography is significantly nonlinear, Monte Carlo sampling methods are often used for this purpose, but they are generally computationally intractable for large data sets and high-dimensionality parameter spaces. To extend uncertainty analysis to larger systems, we introduce variational inference methods to conduct seismic tomography. In contrast to Monte Carlo sampling, variational methods solve the Bayesian inference problem as an optimization problem yet still provide fully nonlinear, probabilistic results. This is achieved by minimizing the Kullback-Leibler (KL) divergence between approximate and target probability distributions within a predefined family of probability distributions.</span></p><p><span>We introduce two variational inference methods: automatic differential variational inference (ADVI) and Stein variational gradient descent (SVGD). In ADVI a Gaussian probability distribution is assumed and optimized to approximate the posterior probability distribution. In SVGD a smooth transform is iteratively applied to an initial probability distribution to obtain an approximation to the posterior probability distribution. At each iteration the transform is determined by seeking the steepest descent direction that minimizes the KL-divergence. </span></p><p><span>We apply the two variational inference methods to 2D travel time tomography using both synthetic and real data, and compare the results to those obtained from two different Monte Carlo sampling methods: Metropolis-Hastings Markov chain Monte Carlo (MH-McMC) and reversible jump Markov chain Monte Carlo (rj-McMC). The results show that ADVI provides a biased approximation because of its Gaussian approximation, whereas SVGD produces more accurate approximations to the results of MH-McMC. In comparison rj-McMC produces smoother mean velocity models and lower standard deviations because the parameterization used in rj-McMC (Voronoi cells) imposes prior restrictions on the pixelated form of models: all pixels within each Voronoi cell have identical velocities. This suggests that the results of rj-McMC need to be interpreted in the light of the specific prior information imposed by the parameterization. Both variational methods estimate the posterior distribution at significantly lower computational cost, provided that gradients of parameters with respect to data can be calculated efficiently. We therefore expect that the methods can be applied fruitfully to many other types of geophysical inverse problems.</span></p>

scholarly journals Comparative Study of Time-Domain Fatigue Assessments for an Offshore Wind Turbine Jacket Substructure by Using Conventional Grid-Based and Monte Carlo Sampling Methods

Energies ◽  
2018 ◽  
Vol 11 (11) ◽  
pp. 3112 ◽  
Author(s):  
Chi-Yu Chian ◽  
Yi-Qing Zhao ◽  
Tsung-Yueh Lin ◽  
Bryan Nelson ◽  
Hsin-Haou Huang
Keyword(s):  
Monte Carlo ◽  
Wind Turbine ◽  
Fatigue Damage ◽  
Sampling Method ◽  
Sampling Methods ◽  
Offshore Wind ◽  
Offshore Wind Turbine ◽  
Statistical Uncertainty ◽  
High Convergence Rate ◽  
Grid Based

Currently, in the design standards for environmental sampling to assess long-term fatigue damage, the grid-based sampling method is used to scan a rectangular grid of meteorological inputs. However, the required simulation cost increases exponentially with the number of environmental parameters, and considerable time and effort are required to characterise the statistical uncertainty of offshore wind turbine (OWT) systems. In this study, a K-type jacket substructure of an OWT was modelled numerically. Time rather than frequency-domain analyses were conducted because of the high nonlinearity of the OWT system. The Monte Carlo (MC) sampling method is well known for its theoretical convergence, which is independent of dimensionality. Conventional grid-based and MC sampling methods were applied for sampling simulation conditions from the probability distributions of four environmental variables. Approximately 10,000 simulations were conducted to compare the computational efficiencies of the two sampling methods, and the statistical uncertainty of the distribution of fatigue damage was assessed. The uncertainty due to the stochastic processes of the wave and wind loads presented considerable influence on the hot-spot stress of welded tubular joints of the jacket-type substructure. This implies that more simulations for each representative short-term environmental condition are required to derive the characteristic fatigue damage. The characteristic fatigue damage results revealed that the MC sampling method yielded the same error level for Grids 1 and 2 (2443 iterations required for both) after 1437 and 516 iterations for K- and KK-joint cases, respectively. This result indicated that the MC method has the potential for a high convergence rate.

A Novel Quota Sampling Algorithm for Generating Representative Random Samples given Small Sample Size

2013 ◽  
Vol 2 (1) ◽  
pp. 97-113 ◽  
Author(s):  
Ahmed M. Fouad ◽  
Mohamed Saleh ◽  
Amir F. Atiya
Keyword(s):  
Monte Carlo ◽  
Sample Size ◽  
Sampling Method ◽  
Optimization Problems ◽  
Small Sample Size ◽  
Probability Distributions ◽  
Small Sample ◽  
Sampling Algorithm ◽  
Quasi Monte Carlo ◽  
Quota Sampling

In this paper, a novel algorithm is proposed for sampling from discrete probability distributions using the probability proportional to size sampling method, which is a special case of Quota sampling method. The motivation for this study is to devise an efficient sampling algorithm that can be used in stochastic optimization problems -- when there is a need to minimize the sample size. Several experiments have been conducted to compare the proposed algorithm with two widely used sample generation methods, the Monte Carlo using inverse transform, and quasi-Monte Carlo algorithms. The proposed algorithm gave better accuracy than these methods, and in terms of time complexity it is nearly of the same order.

scholarly journals A Neural Network MCMC Sampler That Maximizes Proposal Entropy

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 269
Author(s):  
Zengyi Li ◽  
Yubei Chen ◽  
Friedrich T. Sommer
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Monte Carlo ◽  
Markov Chain ◽  
Network Architecture ◽  
Probability Distributions ◽  
Natural Images ◽  
Mcmc Methods ◽  
Proposal Distribution ◽  
Target Distribution

Markov Chain Monte Carlo (MCMC) methods sample from unnormalized probability distributions and offer guarantees of exact sampling. However, in the continuous case, unfavorable geometry of the target distribution can greatly limit the efficiency of MCMC methods. Augmenting samplers with neural networks can potentially improve their efficiency. Previous neural network-based samplers were trained with objectives that either did not explicitly encourage exploration, or contained a term that encouraged exploration but only for well structured distributions. Here we propose to maximize proposal entropy for adapting the proposal to distributions of any shape. To optimize proposal entropy directly, we devised a neural network MCMC sampler that has a flexible and tractable proposal distribution. Specifically, our network architecture utilizes the gradient of the target distribution for generating proposals. Our model achieved significantly higher efficiency than previous neural network MCMC techniques in a variety of sampling tasks, sometimes by more than an order magnitude. Further, the sampler was demonstrated through the training of a convergent energy-based model of natural images. The adaptive sampler achieved unbiased sampling with significantly higher proposal entropy than a Langevin dynamics sample. The trained sampler also achieved better sample quality.

scholarly journals Implicitly adaptive importance sampling

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Topi Paananen ◽  
Juho Piironen ◽  
Paul-Christian Bürkner ◽  
Aki Vehtari
Keyword(s):  
Importance Sampling ◽  
Sampling Method ◽  
Sampling Methods ◽  
Probability Distributions ◽  
Adaptive Importance Sampling ◽  
Importance Sampling Method ◽  
Leave One Out ◽  
Standard Probability ◽  
Better Than ◽  
Current Proposal

AbstractAdaptive importance sampling is a class of techniques for finding good proposal distributions for importance sampling. Often the proposal distributions are standard probability distributions whose parameters are adapted based on the mismatch between the current proposal and a target distribution. In this work, we present an implicit adaptive importance sampling method that applies to complicated distributions which are not available in closed form. The method iteratively matches the moments of a set of Monte Carlo draws to weighted moments based on importance weights. We apply the method to Bayesian leave-one-out cross-validation and show that it performs better than many existing parametric adaptive importance sampling methods while being computationally inexpensive.

Estimation of Regional Parameters in a Macro Scale Hydrological Model

2001 ◽  
Vol 32 (3) ◽  
pp. 161-180 ◽  
Author(s):  
Kolbjørn Engeland ◽  
Lars Gottschalk ◽  
Lena Tallaksen
Keyword(s):  
Additional Data ◽  
Hydrological Model ◽  
Sampling Methods ◽  
Probability Distributions ◽  
Repeated Application ◽  
Landscape Characteristics ◽  
Time Period ◽  
Macro Scale ◽  
Model Element ◽  
Regular Sampling

Macro-scale hydrological modelling implies a repeated application of a model within an area using regional parameters. These parameters are based on climate and landscape characteristics, and they are used to calculate the water balance in ungauged areas. The regional parameters ought to be robust and not too dependent of the catchment and time period used for calibration. The ECOMAG model is applied for the NOPEX-region as a macro-scale hydrological model distributed on a 2×2 km2 grid. Each model element is assigned parameters according to soil and vegetation classes. A Bayesian methodology is followed. An objective function describing the fit between observed and simulated values is used to describe the likelihood of the parameters. Using Baye's theorem these likelihoods are used to update the probability distributions of the parameters using additional data, being it either an additional year of streamflow or an additional streamflow station. Two sampling methods are used, regular sampling and Metropolis-Hastings sampling. The results show that regional parameters exist according to some predefined criteria. The probability distribution of the parameters shows a decreasing variance as data from new catchments are used for updating. A few parameters do, however, not exhibit this property, and they are therefore not suitable in a regional context.

scholarly journals Stochastic Order and Generalized Weighted Mean Invariance

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 662
Author(s):  
Mateu Sbert ◽  
Jordi Poch ◽  
Shuning Chen ◽  
Víctor Elvira
Keyword(s):  
Monte Carlo ◽  
Probability Distributions ◽  
Stochastic Order ◽  
Arithmetic Mean ◽  
Stochastic Orders ◽  
Arithmetic Means ◽  
Increasing Convex Order ◽  
Monte Carlo Techniques ◽  
Multiple Importance Sampling ◽  
Invariance Properties

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.

Research on Monte Carlo Perturbation Calculation Methods Applied in Reactor Physics

2009 ◽  
Author(s):  
Ze-guang Li ◽  
Kan Wang ◽  
Gang-lin Yu
Keyword(s):  
Monte Carlo ◽  
Differential Operator ◽  
Sampling Method ◽  
Perturbation Methods ◽  
Calculation Model ◽  
Perturbation Calculation ◽  
Reactor Physics ◽  
Tracking Method ◽  
Monte Carlo Techniques ◽  
Sensitivity Studies

In the reactor design and analysis, there is often a need to calculate the effects caused by perturbations of temperature, components and even structure of reactors on reactivity. And in sensitivity studies, uncertainty analysis of target quantities and unclear data adjustment, perturbation calculations are also widely used. To meet the need of different types of reactors (complex, multidimensional systems), Monte Carlo perturbation methods have been developed. In this paper, several kinds of perturbation methods are investigated. Specially, differential operator sampling method and correlated tracking method are discussed in details. MCNP’s perturbation calculation capability is discussed by calculating certain problems, from which some conclusions are obtained on the capabilities of the differential operator sampling method used in the perturbation calculation model of MCNP. Also, a code using correlated tracking method has been developed to solve certain problems with cross-section changes, and the results generated by this code agree with the results generated by straightforward Monte Carlo techniques.

Reliability Assessment of Power System Using Importance Sampling Technique Based on Layer Optimization Simulation

2011 ◽  
Vol 88-89 ◽  
pp. 554-558 ◽  
Author(s):  
Bin Wang
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Power System ◽  
Importance Sampling ◽  
System Reliability ◽  
Sampling Method ◽  
Sampling Technique ◽  
Test System ◽  
Importance Sampling Method ◽  
Importance Sampling Technique

An improved importance sampling method with layer simulation optimization is presented in this paper. Through the solution sequence of the components’ optimum biased factors according to their importance degree to system reliability, the presented technique can further accelerate the convergence speed of the Monte-Carlo simulation. The idea is that the multivariate distribution’ optimization of components in power system is transferred to many steps’ optimization based on importance sampling method with different optimum biased factors. The practice is that the components are layered according to their importance degree to the system reliability before the Monte-Carlo simulation, the more forward, the more important, and the optimum biased factors of components in the latest layer is searched while the importance sampling is carried out until the demanded accuracy is reached. The validity of the presented is verified using the IEEE-RTS79 test system.

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