Bayesian neural networks (BNNs) combine the expressive power of deep learning with the advantages of Bayesian formalism. In recent years, the analysis of wide, deep BNNs has provided theoretical insight into their priors and posteriors. However, we have no analogous insight into their posteriors under approximate inference. In this work, we show that mean-field variational inference entirely fails to model the data when the network width is large and the activation function is odd. Specifically, for fully-connected BNNs with odd activation functions and a homoscedastic Gaussian likelihood, we show that the optimal mean-field variational posterior predictive (i.e., function space) distribution converges to the prior predictive distribution as the width tends to infinity. We generalize aspects of this result to other likelihoods. Our theoretical results are suggestive of underfitting behavior previously observered in BNNs. While our convergence bounds are non-asymptotic and constants in our analysis can be computed, they are currently too loose to be applicable in standard training regimes. Finally, we show that the optimal approximate posterior need not tend to the prior if the activation function is not odd, showing that our statements cannot be generalized arbitrarily.

Since model bias and associated initialization shock are serious shortcomings that reduce prediction skills in state-of-the-art decadal climate prediction efforts, we pursue a complementary machine-learning-based approach to climate prediction. The example problem setting we consider consists of predicting natural variability of the North Atlantic sea surface temperature on the interannual timescale in the pre-industrial control simulation of the Community Earth System Model (CESM2). While previous works have considered the use of recurrent networks such as convolutional LSTMs and reservoir computing networks in this and other similar problem settings, we currently focus on the use of feedforward convolutional networks. In particular, we find that a feedforward convolutional network with a Densenet architecture is able to outperform a convolutional LSTM in terms of predictive skill. Next, we go on to consider a probabilistic formulation of the same network based on Stein variational gradient descent and find that in addition to providing useful measures of predictive uncertainty, the probabilistic (Bayesian) version improves on its deterministic counterpart in terms of predictive skill. Finally, we characterize the reliability of the ensemble of ML models obtained in the probabilistic setting by using analysis tools developed in the context of ensemble numerical weather prediction.

In the univariate setting, popular methods include Autoregressive Conditional Heteroskedastic models (ARCH) (Engle 1982) and Generalised GARCH (GARCH) models (Bollerslev 1986). ARCH and GARCH models are regression-based models estimated using maximum likelihood, and are capable of capturing stylised facts about financial time series such as volatility clustering (Bauwens et al. 2006). The ARCH(p) model describes the conditional volatility as a function of p lagged squared residuals, and similarly the GARCH(p,q) model include contributions due to the last q conditional variances. Many variants of the GARCH model have been proposed to better capture properties of financial time series, for example the EGARCH (Nelson 1991) and GJR-GARCH (Glosten et al. 1993) models were designed to capture the so-called leverage effect, which describes the negative relationship between asset price and volatility. In a multivariate setting, instead of modelling only time-varying conditional variances, for an n-dimensional system, we estimate the n n time-varying variance-covariance matrix. This allows us to investigate interactions between the volatility of different time series and whether there is a transmission of volatility (spillover effect) between markets (Bauwens et al. 2006, Erten et al. 2012). Popular multivariate GARCH models include the VEC model (Bollerslev et al. 1988), the

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Inhomogeneities in the aortic wall can lead to localized stress accumulations, possibly initiating dissection. In many cases, a dissection results from pathological changes such as fragmentation or loss of elastic fibers. But it has been shown that even the healthy aortic wall has an inherent heterogeneous microstructure. Some parts of the aorta are particularly susceptible to the development of inhomogeneities due to pathological changes, however, the distribution in the aortic wall and the spatial extent, such as size, shape, and type, are difficult to predict. Motivated by this observation, we describe the heterogeneous distribution of elastic fiber degradation in the dissected aortic wall using a stochastic constitutive model. For this purpose, random field realizations, which model the stochastic distribution of degraded elastic fibers, are generated over a non-equidistant grid. The random field then serves as input for a uni-axial extension test of the pathological aortic wall, solved with the finite-element (FE) method. To include the microstructure of the dissected aortic wall, a constitutive model developed in a previous study is applied, which also includes an approach to model the degradation of inter-lamellar elastic fibers. Then to assess the uncertainty in the output stress distribution due to this stochastic constitutive model, a convolutional neural network, specifically a Bayesian encoder-decoder, was used as a surrogate model that maps the random input fields to the output stress distribution obtained from the FE analysis. The results show that the neural network is able to predict the stress distribution of the FE analysis while significantly reducing the computational time. In addition, it provides the probability for exceeding critical stresses within the aortic wall, which could allow for the prediction of delamination or fatal rupture.

Learn Machine Learning from scratch, this course for beginner who want to learn the fundamental of machine learning and artificial intelligence.

Bayesian additive regression trees have seen increased interest in recent years due to their ability to combine machine learning techniques with principled uncertainty quantification. The Bayesian backfitting algorithm used to fit BART models, however, limits their application to a small class of models for which conditional conjugacy exists. In this article, we greatly expand the domain of applicability of BART to arbitrary \emph{generalized BART} models by introducing a very simple, tuning-parameter-free, reversible jump Markov chain Monte Carlo algorithm. Our algorithm requires only that the user be able to compute the likelihood and (optionally) its gradient and Fisher information. The potential applications are very broad; we consider examples in survival analysis, structured heteroskedastic regression, and gamma shape regression.

This is a course on Machine Learning, Deep Learning (Tensorflow PyTorch) and Bayesian Learning (yes all 3 topics in one place!!!). This is a course on Machine Learning, Deep Learning (Tensorflow PyTorch) and Bayesian Learning (yes all 3 topics in one place!!!). We start off by analysing data using pandas, and implementing some algorithms from scratch using Numpy. These algorithms include linear regression, Classification and Regression Trees (CART), Random Forest and Gradient Boosted Trees. We start off using TensorFlow for our Deep Learning lessons.

Evaluation metrics to analyze the performance of models Industry relevance of linear and logistic regression Mathematics behind KNN, SVM and Naive Bayes algorithms Implementation of KNN, SVM and Naive Bayes using sklearn Attribute selection methods- Gini Index and Entropy Mathematics behind Decision trees and random forest Boosting algorithms:- Adaboost, Gradient Boosting and XgBoost Different Algorithms for Clustering Different methods to deal with imbalanced data Correlation Filtering Content and Collaborative based filtering Singular Value Decomposition Different algorithms used for Time Series forecasting Hands on Real-World examples. To make sense out of this course, you should be well aware of linear algebra, calculus, statistics, probability and python programming language. To make sense out of this course, you should be well aware of linear algebra, calculus, statistics, probability and python programming language. This course is a perfect fit for you. This course will take you step by step into the world of Machine Learning.

We propose a network architecture capable of reliably estimating uncertainty of regression based predictions without sacrificing accuracy. The current state-of-the-art uncertainty algorithms either fall short of achieving prediction accuracy comparable to the mean square error optimization or underestimate the variance of network predictions. We propose a decoupled network architecture that is capable of accomplishing both at the same time. We achieve this by breaking down the learning of prediction and prediction interval (PI) estimations into a two-stage training process. We use a custom loss function for learning a PI range around optimized mean estimation with a desired coverage of a proportion of the target labels within the PI range. We compare the proposed method with current state-of-the-art uncertainty quantification algorithms on synthetic datasets and UCI benchmarks, reducing the error in the predictions by 23 to 34% while maintaining 95% Prediction Interval Coverage Probability (PICP) for 7 out of 9 UCI benchmark datasets. We also examine the quality of our predictive uncertainty by evaluating on Active Learning and demonstrating 17 to 36% error reduction on UCI benchmarks.