Bayesian neural network. The application example here is a real case study of several ore types from a polymetallic sulfide Oct 24, 2023 · Neural networks (NNs) are primarily developed within the frequentist statistical framework. Nodes that interact are connected by edges in the direction of influence; the edge A→B implies that A Bayesian training of neural network models avoids overfitting, provides proper quantification of uncertainty, and allows learning of high-level properties of the data, such as degree of smoothness, relevance of features, and additivity. Oct 16, 2021 · Bayesian neural network (BNN) combines neural network with Bayesian inference. Bayesian Neural Network(BNN) can generate probability interpretability for deep learning, and quantify uncertainty. Simply speaking, in BNN, we treat the weights and outputs as the variables and we are finding their marginal distributions that best fit the data. Sep 16, 2021 · As exact Bayesian inference for neural networks is computationally intractable, all of the published BNN approaches are in fact clever approximations. 2. 1. Oct 1, 2023 · A Bayesian neural network, unlike a traditional network, treats its weights as random variables as described previously. It covers the theoretical foundations, the practical applications, and the open problems of Bayesian neural networks. BNs have been widely applied for machine learning in many fields, ranging from forensic science [95] to bioinformatics [96] to fault diagnosis [97] and neuroscience [98], [43]. Each data server is assumed to provide local neural network weights, which are modeled through our framework. To address these challenges, Bayesian neural networks (BNNs) have emerged as a compelling extension of conventional neural networks Sep 25, 2019 · A standard vanilla neural network has matrices of parameters that are fixed or constant. The optimal solution is the one with the highest value of evidence. 2 Standard and Bayesian Neural Networks. BNNs are comprised of a Probabilistic Model and a Neural Network. as random variables; w eights are assumed to have a true v alue that is just. Federated learning's poor performance in the presence of heterogeneous data remains one of the most pressing issues in the field. Bayesian inference allows us to learn a probability distribution over possible neural networks. Similarly Bayesian Neural Networks with. g. C. [ 30 ] Unlike the traditional neural networks, a Bayesian neural network is equivalent to an integrated model that considers an infinite number of adapted training sets as well as the uncertainty of the samples, which makes the final prediction results more robust. Feedforward neural networks. 2021. Nevertheless, frequentist NNs lack the capability to provide uncertainties in the predictions, and hence their robustness can not be adequately assessed. This article reviews the recent advances and challenges of Bayesian neural networks, a class of deep learning methods that incorporate uncertainty and prior knowledge. Another innovation of our proposed study consists in enhancing the accuracy of the Bayesian classifier via intelligent sampling algorithms. Among others, one can cite the automatic tuning of regularization coefficients, the selection of the most important input variables, the derivation of an uncertainty interval on the model output and the possibility to perform a Oct 4, 2020 · This article proposes a novel end-to-end FDI framework, which adopts a recently developed Bayesian recurrent neural network (BRNN) architecture (Gal and Ghahramani, 2016b). However, it is unclear whether these priors accurately reflect our true beliefs about the weight distributions or give optimal performance. Implementing the BNN, we first put the prior distribution \ (p (\omega )\) on all possible parameters before seeing the data. Denker and Yann leCun. Bayesian Neural Network A Bayesian neural network is a neural network with a prior distribution on its weights (Neal, 2012) . jl with Lux. Oct 7, 2021 · 3. (2) to obtain a dropout RNN unit, as given by Eq. (13). jl to implement implementing a classification algorithm. To find better priors, we study summary statistics of neural Bayesian Networks2. drop-in replacements of Convolutional, Linear and LSTM layers to corresponding Bayesian layers. Personalized federated learning departs from the conventional paradigm in which all clients employ While these models are not deep per se, there are many ways in which they connect to Bayesian deep learning, which merits their appearance in this thesis. Jan 1, 2024 · Neural network prediction of sound quality via domain knowledge-based data augmentation and Bayesian approach with small data sets Mech. ymssp. 1 Neural Networks Before discussing a Bayesian perspective of NNs, it is important to briefly survey the fundamentals of neural computation and to define the notation to be used throughout the chapter. These parameters are estimated using backpropagation, and gradient descent. For instance, in developing the Bayesian RNN (BRNN) model, Eq. With Bayesian methods, we can generalize learning to include learning the appropriate model size and even model type. In a bayesian neural network the weights take on probability distributions. Bayesian Deep Learning. We have adopted the Bayesian neural network framework to obtain posterior densities from Laplace approximation. Jun 22, 2020 · 2. A Bayesian neural network (BNN) places a prior distribution on its parameters. I will also discuss how bridging Probabilistic Programming and Deep Learning can open up very interesting avenues to explore in future Mar 14, 2019 · This article demonstrates how to implement and train a Bayesian neural network with Keras following the approach described in Weight Uncertainty in Neural Networks (Bayes by Backprop). Nov 5, 2023 · 2. 12. However, their computational Key features: dnn_to_bnn(): Seamless conversion of model to be Uncertainty-aware with single line of code. Two approaches to fit Bayesian neural networks (BNN) · The variational inference (VI) approximation for BNNs · The Monte Carlo dropout approximation for BNNs · TensorFlow Probability (TFP) variational layers to build VI-based BNNs · Using Keras to implement Monte Carlo dropout in BNNs Bayesian networks are a type of Probabilistic Graphical Model that can be used to build models from data and/or expert opinion. Bayesian neural networks are a form of deep learning algorithm trained using Bayesian statistics, which give a stochastic character to the network components. Both feedforward neural network and LSTM neural network can be considered as regression models in general, and these models are trained to relate a series of inputs X = X 1, X 2, ⋯, X N with their corresponding outputs Y = Y 1, Y 2, ⋯, Y N. Compared to a conventional DNN, which gives a definite point prediction for each given input, a BNN returns a distribution of predictions, which qualitatively corresponds to the aggregate prediction of Aug 28, 2015 · A Bayesian network is a graph in which nodes represent entities such as molecules or genes. Our new class of architectures is designed to improve the limitations of existing architectures around computational efficiency at training and inference time. Jan 23, 2018 · This paper describes and discusses Bayesian Neural Network (BNN). First, by minimizing the deviations between the predictions and data, the This choice was found to be advantageous for Bayesian techniques—a result that may be related to the known asymptotic behavior of Bayesian neural networks as non-parametric models . We now present a number of illustrative applications in neuroscience and the industry. A bayesian neural network actually estimates a distribution over each parameter or weight in the neural network matrix. A Bayesian neural network (BayNN), whose weights are represented by probability distributions, is experimentally Dec 21, 2022 · Bayesian Neural Networks are a specific type of neural networks trained in the light of the Bayesian paradigm, being capable to quantify uncertainty associated with the underlying processes. 2 Variational Inference. In practice, one needs to employ sampling methods to approximate the posterior distribution/integrals encountered in the Bayesian setting. , 08883270 , 157 ( 2021 ) , Article 107713 , 10. Jun 30, 2019 · Bayesian Neural Networks (LSTM): implementation. By using a distribution of weights Bayesian Neural Network We borrow this tutorial from the official Turing Docs. A Bayesian Neural Network (BNN) is simply posterior inference applied to a neural network architecture. ipynb : An additional example showing how the same linear model can be implemented using NumPyro to take advantage of its state-of-the-art MCMC algorithms (in this case Dec 1, 2021 · In this section, we explain the structure of feedforward neural networks (NNs) and Bayesian modeling prior to discussing uncertainty in detail. Apr 1, 2022 · Meanwhile, a Bayesian Neural Network (BNN) (Hasz, 2019) can take advantage of Bayesian statistics to provide model uncertainty in its predictions. 3 Prediction of thermal error May 1, 2008 · The Bayesian approach to modelling offers significant advantages over classical neural network (NN) learning methods. 107713 Jun 8, 2018 · A Bayesian network is a directed acyclic graph in which each edge corresponds to a conditional dependency, and each node corresponds to a unique random variable. –a directed, acyclic graph (link ≈“directly influences”) –a conditional distribution for each node given its parents: P(X. 1016/j. Here we take a whistle-sto The neural network represents this relationship as a function f ( x, θ), where θ are the weights and biases of the network. The resulting algorithm mitigates overfitting, enables learning from small datasets, and tells us how uncertain our predictions are. Consider a set of candidate models Hi that could include neural networks with different numbers of hidden units, RBF networks and Aug 9, 2023 · What is the Bayesian Neural Network? List of Bayesian Neural Network components: Dataset D with predictors X (for example, images) and labels Y (for example, classes). Sep 28, 2023 · Neural networks have achieved remarkable performance across various problem domains, but their widespread applicability is hindered by inherent limitations such as overconfidence in predictions, lack of interpretability, and vulnerability to adversarial attacks. We present Credal Bayesian Deep Learning (CBDL). Syst. The process of finding these distributions is called marginalization. With the designed analog switching characteristics, the RRAM device can realize the function of sampling from a tunable probabilistic distribution. 68098. , P (C;A;H;I )) by specifying local conditional distributions (e. , stochastic artificial neural networks trained using Bayesian methods. e it is condition independent. This example uses Bayes by backpropagation (also known as Bayes by backprop) to estimate the distribution of the weights of a neural network. For problems of small to moderate size, Bayesian training can be practically implemented using Markov chain Monte Carlo (MCMC) methods based on Hamiltonian For the first time, this paper develops a novel stochastic computing method by utilizing the inherent random noises of analog RRAM. , p(i j a )). May 28, 2019 · In federated learning problems, data is scattered across different servers and exchanging or pooling it is often impractical or prohibited. We have May 1, 2022 · Bayesian neural networks are able to better represent the uncertainty regarding the unknown. In this study, the aforementioned dropout is applied in the basic units of each of the three deep neural networks to develop the Bayesian deep neural network. 1. We will show how the explicit parameterization of Lux enables first-class composability with packages which expect flattened out parameter vectors. A very efficient way of seeing the Bayes theorem is the following: “The Bayes theorem is the mathematical theorem that explains why if all the cars in the world are blue then my car has to be blue, but just because my car is blue it doesn’t Mar 15, 2023 · 1. Likelihood P(D|θ) or P(Y |X, θ) represented with a categorical softmax distribution on logits calculated by a neural network (NN) parameterized with θ, for example, softmax A neural network diagram with one input layer, one hidden layer, and an output layer. , during training). Neal “ A Practical Bayesian Framework for Backprop Networks,” by David J. nl Jul 14, 2020 · A tutorial for deep learning users to design, implement, train, use and evaluate Bayesian Neural Networks, i. Apr 2, 2021 · Neural networks are the backbone of deep learning. With standard neural networks, the weights between the different layers of the network take single values. This tutorial provides deep learning practitioners with an overview of the relevant literature and a complete toolset to design, implement, train, use and evaluate Bayesian neural networks, i. See full list on jorisbaan. Personalized federated learning departs from the conventional paradigm in which all clients employ Bayesian Neural Networks As we know, xed basis functions are limited. 2 Bayesian Neural Networks. Neural networks have achieved remarkable performance across various problem domains, but their widespread applicability is hindered by Bayesian Neural Networks As we know, xed basis functions are limited. Magris M, Shabani M, Iosifidis A (2022a) Bayesian bilinear neural network for predicting the mid-price dynamics in limit-order book markets. We scrutinize four of the most popular algorithms in the area: Bayes by Backprop, Probabilistic Backpropagation, Monte Carlo Dropout, Variational Adam. Mar 4, 2021 · 5. 4 and Tensorflow 1. For two use cases discussed above, it can be achieved like below: Neural Network is trained a number of times on different hyper-parameter combinations and the accuracies are captured & stored. Bayesian neural networks merge these fields. Lets start by importing the Oct 2, 2021 · This is a neural network whose parameters are distributions rather than point values; the network is trained with Bayesian inference to estimate a posterior over the model’s parameters. 2. 0. We develop a Bayesian nonparametric framework for federated learning with neural networks. Feb 19, 2023 · Uncertainty quantification and robustness to distribution shifts are important goals in machine learning and artificial intelligence. Can we combine the advantages of neural nets and Bayesian models? Bayesian neural networks (BNNs) Place a prior on the weights of the network, e. The MLP serves as the . Although Bayesian Neural Networks (BNNs) allow for uncertainty in the predictions to be assessed, different sources of uncertainty are indistinguishable. The BNN was improved in terms of three aspects: numerical parameters, input layer, and network structure. To be precise, a prior distribution is specified for each weight and bias. The paper showcases a few different applications of them for classification and regression problems. Deep Neural Networks are non-linear function approximators and the state of the art in pattern recognition for unstructured data (audio, images, text, video) But they do have limitations. From the Publisher: Artificial "neural networks" are now widely used as flexible models for regression classification applications, but questions remain regarding what these models Oct 27, 2021 · How we leverage the concept of Bayesian inference to update the probability distribution of model weights and outputs. Sep 28, 2023 · This comprehensive primer presents a systematic introduction to the fundamental concepts of neural networks and Bayesian inference, elucidating their synergistic integration for the development of BNNs and provides an overview of commonly employed priors. We then develop an inference approach that allows us to Apr 3, 2023 · A Probabilistic Neural Network (PNN) is a type of feed-forward ANN in which the computation-intensive backpropagation is not used It’s a classifier that can estimate the pdf of a given set of data. They are a type of neural network whose parameters and predictions are both probabilistic. They can be used for a wide range of tasks including diagnostics, reasoning, causal modeling, decision making under uncertainty, anomaly detection, automated insight and prediction. Jun 16, 2020 · The value of Bayesian neural network is maximum 0. The paper provides an overview of the relevant literature and a complete toolset for Bayesian neural networks. 3. A Bayesian Neural Network (BNN) is an Artificial Neural Network (ANN) trained with Bayesian Inference (Jospin et al. Consider a data set \(\{(\mathbf{x}_n, y_n)\}\) , where each data point comprises of features \(\mathbf{x}_n\in\mathbb{R}^D\) and output \(y_n\in\mathbb{R}\) . –a set of nodes, one per variable. , when the output can be evaluated at different levels of accuracy and 3. In recent years, the Bayesian neural networks are gathering a lot of attention. “ Bayesian Learning for Neural Networks,” by Radford M. This distribution is a basic building block in a Bayesian neural network. It represents a single hidden layer, i. Jun 15, 2021 · In Bayesian Optimization, an initial set of input/output combination is generally given as said above or may be generated from the function. This survey will focus on the primary network structure of interest, the Multi-Layer Perceptron (MLP) network. 2022 ). The target audience comprises statisticians with a potential background in Bayesian methods but lacking deep learning expertise, as well as machine learners proficient in deep neural networks but with limited exposure to Bayesian statistics. Nov 8, 2022 · This work is an attempt to improve the Bayesian neural network (BNN) for studying photoneutron yield cross sections as a function of the charge number Z, mass number A, and incident energy $$\\varepsilon$$ ε . Additionally, there is a growing bibliography available on research materials relating to Bayesian Learning for Neural Networks shows that Bayesian methods allow complex neural network models to be used without fear of the "overfitting" that can occur with traditional neural network learning methods. MacKay. deep-learning reproducible-research regression pytorch uncertainty classification uncertainty-neural-networks bayesian-inference mcmc variational-inference hmc bayesian-neural-networks langevin-dynamics approximate-inference local-reparametrization-trick kronecker-factored-approximation mc-dropout bayes-by-backprop out-of-distribution-detection Jan 9, 2017 · The Bayesian network is different from the Neural Network in that it is explicit reasoning, even though probabilistic and hence could have multiple stable states based on each step being revisited and modified within legal values, just like an algorithm. Because of their huge parameter space, however, inferring the posterior is even more difficult than usual. May 12, 2021 · Another method to obtain a more robust and reliable model with a small dataset is the Bayesian neural network (BNN) (Gal and Ghahramani, 2015). First, the structure of a single-hidden-layer neural network (NN) [17] is presented, and then extended to the case Apr 4, 2020 · Bayesian Neural Networks are often optimized by sampling the loss many times on the same batch before optimizing and proceeding, which occurs to compensate the randomness over the weights and avoid optimizing them over a loss influenced by outliers. Apr 1, 2001 · For neural networks, the Bayesian approach was pioneered in Buntine and Weigend, 1991, MacKay, 1992, Neal, 1992, and reviewed in Bishop, 1995, MacKay, 1995, Neal, 1996. The prediction accuracy η of the three models is not much different, both above 80%. The implementation is kept simple for illustration purposes and uses Keras 2. The intent of such a design is to combine the strengths of Neural Networks and Stochastic modeling. 928. e. A simple, graphical notation for conditional independence assertions and hence for compact specification of full joint distributions Syntax. In the Bayesian framework, the network parameters are random variables following certain distributions rather than having a single fixed value, enabling modeling uncertainty about data and model. Stochastic Artificial Neural Networks trained using Bayesian methods. BLiTZ’s variational_estimator decorator also powers the neural network with the sample_elbo Aug 25, 2023 · We present the application of the physics-informed neural network (PINN) approach in Bayesian formulation. The uncertainty in the weights is encoded in a Normal variational distribution specified by the parameters A_scale and A_mean. In the following, we are going to present how GP priors can be parameterised by deep neural networks (DNNs) (Section 2), how GPs can be stacked to build deeper models (Section 3) and how DNNs can themselves turn into GPs or be approximated by Feb 12, 2021 · Bayesian Neural Network Priors Revisited. In the following, we provide a quick overview of ANNs and their typical estimation based on Backpropagation (Sect. Different techniques and methods in real-life scenarios to tackle the unknown distribution problem. In a Bayesian neural network, we represent the weights and biases as probability distributions, so f ( x, θ) becomes a probability distribution over possible outputs: p ( y | x, D) = ∫ p ( y | x, θ) p ( θ | D) d θ. Due to its feature of joint probability, the probability in Bayesian Belief Network is derived, based on a condition — P Feb 5, 2024 · In this paper, we present Partially Stochastic Infinitely Deep Bayesian Neural Networks, a novel family of architectures that integrates partial stochasticity into the framework of infinitely deep neural networks. Review: probabilistic inference. Heuristically, CBDL allows to train an (uncountably Jun 22, 2020 · 2. In particular, we first combine attribute data with spatial information as auxiliary variables to forecast reservoir thickness using a neural network. Neural Networks exhibit continuous function approximator Bayesian neural networks, catering to both statisticians and machine learning practitioners. Sep 5, 2020 · Bayesian Belief Network is a graphical representation of different probabilistic relationships among random variables in a particular set. " 04a-Bayesian-Neural-Network-Classification. In the frequentist setting presented abov e, the model weights are not treated. iSParents(X. For each model or fit, the evidence is computed, which is a measure that classifies the hypothesis. In Table 1 , we show the average accuracy over all tasks at the end of training on the last task, as well as the average ECE at that point for real-valued HiddenLayer. p(t jx; ) = N(t;f (x);˙2) Inside of PP, a lot of innovation is focused on making things scale using Variational Inference. The proposed FDI framework is fundamentally different from the two types of frameworks that have been previously used in the NN-based fault detection literature. Feb 25, 2024 · Bayesian Neural Network For Personalized Federated Learning Parameter Selection. p( ) = N( ;0; I) In practice, typically separate variance for each layer De ne an observation model, e. An API to convert deterministic deep neural network (dnn) model of any architecture to Bayesian deep neural network (bnn) model, simplifying the model definition i. The ultimate goal of BNN is to quantify the uncertainty introduced by the models in terms of outputs and weights so as Feb 17, 2021 · In this chapter, we introduce the concept of Bayesian Neural Network and motivate the reader, presenting its gains over the classical neural networks. The proposed methodology is relevant in emerging applicative settings Dec 5, 2023 · A Bayesian neural network approach to Multi-fidelity surrogate modelling. A Bayesian neural network (BNN) is a type of deep learning network that uses Bayesian methods to quantify the uncertainty in the predictions of a deep learning network. Two lectures ago, we talked about modeling: how can we use Bayesian networks to represent real-world problems. Baptiste Kerleguer (DAM/DIF, CMAP), Claire Cannamela (DAM/DIF), Josselin Garnier (CMAP) This paper deals with surrogate modelling of a computer code output in a hierarchical multi-fidelity context, i. Despite their success, they are often implemented in a frequentist scheme, meaning they are unable to reason about uncertainty in their predictions A Bayesian network (also known as a Bayes network, Bayes net, belief network, or decision network) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). The comprehensive analysis shows that the prediction ability of the three models is good, and the best performance is the Bayesian neural network model. and quantify the uncertainty associated with deep neural network predictions. p(t jx; ) = N(t;f (x);˙2) Apr 1, 2020 · Bayesian neural network. Each algorithm has its peculiarities and Aug 26, 2022 · Bayesian Neural Network and DNN were able to classify 90% of the samples correctly, followed by Logistic Regression which was able to classify 76% of the samples correctly. To do this, we Bayesian belief network is key computer technology for dealing with probabilistic events and to solve a problem which has uncertainty. We will use Turing. Jun 22, 2020 · Neural Networks (NNs) have provided state-of-the-art results for many challenging machine learning tasks such as detection, regression and classification across the domains of computer vision, speech recognition and natural language processing. 5. Bayesian networks in machine learning. Jul 21, 2020 · “ Transforming Neural-Net Output Levels to Probability Distributions,” by John S. numpyro. We then develop an inference approach that allows us to Two approaches to fit Bayesian neural networks (BNN) · The variational inference (VI) approximation for BNNs · The Monte Carlo dropout approximation for BNNs · TensorFlow Probability (TFP) variational layers to build VI-based BNNs · Using Keras to implement Monte Carlo dropout in BNNs Feb 25, 2024 · Bayesian Neural Network For Personalized Federated Learning Parameter Selection. Signal Process. Bayesian inferences permit quantification of uncertainty and thus, enable the development of robust machine learning models. A Bayesian network allows us to de ne a joint probability distribution over many variables (e. The proposed framework in this study will accomplish this by learning the history of equipment failure to predict the RUL of a component within uncertainty bounds. 3 Bayesian Neural Network. With neural networks, the main difficulty in model building is controlling the complexity of the model. [1] While it is one of several forms of causal notation, causal networks are special cases of Bayesian Dec 21, 2022 · Before understanding a Bayesian neural network, we should probably review a bit of the Bayes theorem. an affine transformation applied to a set of inputs X followed by a non-linearity. Formally, if an edge (A, B) exists in the graph connecting random variables A and B, it means that P(B|A) is a factor in the joint probability distribution, so we must know P(B|A) for all values of B and A in order to conduct inference. While standard neural networks often assign high confidence even to incorrect predictions, Bayesian neural networks can more accurately evaluate how likely their predictions are to be correct. Mar 15, 2023 · Mackay DJCProbable networks and plausible predictions—a review of practical Bayesian methods for supervised neural networksNetw Comput Neural Syst19956469 505 0834. We can define a Bayesian network as: "A Bayesian network is a probabilistic graphical model which represents a set of variables and their conditional dependencies using a directed acyclic graph. We would like to show you a description here but the site won’t allow us. We can approximately solve inference with a simple modification to standard neural network tools. ipynb: Implementing an MCMC algorithm to fit a Bayesian neural network for classification Further examples: 05-Linear-Model_NumPyro. Sep 15, 2023 · 3. [ 16 , 27 - 30 ] In one notable approach, Gal and coworkers showed that neural networks with dropout were an effective approach as a Bayesian approximation to model uncertainty. In this example, I will show how to use Variational Inference in PyMC to fit a simple Bayesian Neural Network. Bayesian methods to a neural network with a fixed number of units and a fixed architecture. As a result, the network functions as a probabilistic model where the weight distributions have been determined through variational inference (i. Sep 1, 2023 · Given the prominent advantages of Bayesian methods to model uncertainty, we propose in this paper a Bayesian neural network (BNN) model to predict reservoir thickness and quantify uncertainty. Apr 1, 2022 · Bayesian deep neural networks. Mengen Luo, Ercan Engin Kuruoglu. Dec 26, 2023 · In this paper, we propose and experimentally assess an innovative framework for scaling posterior distributions over different-curation datasets, based on Bayesian-Neural-Networks (BNN). PNNs are a scalable alternative to traditional backpropagation neural networks in classification and pattern recognition applications. What specific loss function we will use for Bayesian Neural Network to optimize the model. Conversely, the Bayesian neural networks (BNNs) naturally offer predictive uncertainty by applying Bayes' theorem. (1) is integrated into Eq. Isotropic Gaussian priors are the de facto standard for modern Bayesian neural network inference. Aug 26, 2022 · Bayesian Neural Network and DNN were able to classify 90% of the samples correctly, followed by Logistic Regression which was able to classify 76% of the samples correctly. 1 ). It is a classifier with no dependency on attributes i. Jan 9, 2017 · The Bayesian network is different from the Neural Network in that it is explicit reasoning, even though probabilistic and hence could have multiple stable states based on each step being revisited and modified within legal values, just like an algorithm. To compute the prediction of such a network, one marginalizes over this parameter posterior, which naturally folds in extrapolation uncertainty. bvtygywhbtusgixpsxok
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