While theoretical results have been obtained for their heavy-tailedness, the full characterization of the prior predictive distribution (i.e. its density, CDF

Thank you for your feedback. The loss function should be selected based on the application, e.g., if We will add a discussion on loss functions and regularization in the final manuscript. Thank you for your helpful comments and suggestions. As mentioned in lines 75 and 89, different functional forms are deemed interpretable in different applications. Reviewer 3.) The theoretical justification of our framework was provided in Section 3.1, where we have shown that Our algorithm explores the Pareto front of simplicity vs. predictivity systematically We will add all the suggested references in the final the manuscript.

Recent works on Bayesian neural networks (BNNs) have highlighted the need to better understand the implications of using Gaussian priors in combination with the compositional structure of the network architecture. Similar in spirit to the kind of analysis that has been developed to devise better initialization schemes for neural networks (cf. He-or Xavier initialization), we derive a precise characterization of the prior predictive distribution of finite-width ReLU networks with Gaussian weights. While theoretical results have been obtained for their heavy-tailedness, the full characterization of the prior predictive distribution (i.e. its density, CDF and moments), remained unknown prior to this work. Our analysis, based on the Meijer-G function, allows us to quantify the influence of architectural choices such as the width or depth of the network on the resulting shape of the prior predictive distribution. We also formally connect our results to previous work in the infinite width setting, demonstrating that the moments of the distribution converge to those of a normal log-normal mixture in the infinite depth limit. Finally, our results provide valuable guidance on prior design: for instance, controlling the predictive variance with depth-and width-informed priors on the weights of the network.

Characterizing how neural network depth, width, and dataset size jointly impact model quality is a central problem in deep learning theory. We give here a complete solution in the special case of linear networks with output dimension one trained using zero noise Bayesian inference with Gaussian weight priors and mean squared error as a negative log-likelihood. For any training dataset, network depth, and hidden layer widths, we find non-asymptotic expressions for the predictive posterior and Bayesian model evidence in terms of Meijer-G functions, a class of meromorphic special functions of a single complex variable. Through novel asymptotic expansions of these Meijer-G functions, a rich new picture of the joint role of depth, width, and dataset size emerges. We show that linear networks make provably optimal predictions at infinite depth: the posterior of infinitely deep linear networks with data-agnostic priors is the same as that of shallow networks with evidence-maximizing data-dependent priors. This yields a principled reason to prefer deeper networks when priors are forced to be data-agnostic. Moreover, we show that with data-agnostic priors, Bayesian model evidence in wide linear networks is maximized at infinite depth, elucidating the salutary role of increased depth for model selection. Underpinning our results is a novel emergent notion of effective depth, given by the number of hidden layers times the number of data points divided by the network width; this determines the structure of the posterior in the large-data limit.

The COVID-19 corona virus has claimed 4.1 million lives, as of July 24, 2021. A variety of machine learning models have been applied to related data to predict important factors such as the severity of the disease, infection rate and discover important prognostic factors. Often the usefulness of the findings from the use of these techniques is reduced due to lack of method interpretability. Some recent progress made on the interpretability of machine learning models has the potential to unravel more insights while using conventional machine learning models. In this work, we analyze COVID-19 blood work data with some of the popular machine learning models; then we employ state-of-the-art post-hoc local interpretability techniques(e.g.- SHAP, LIME), and global interpretability techniques(e.g. - symbolic metamodeling) to the trained black-box models to draw interpretable conclusions. In the gamut of machine learning algorithms, regressions remain one of the simplest and most explainable models with clear mathematical formulation. We explore one of the most recent techniques called symbolic metamodeling to find the mathematical expression of the machine learning models for COVID-19. We identify Acute Kidney Injury (AKI), initial Albumin level (ALBI), Aspartate aminotransferase (ASTI), Total Bilirubin initial(TBILI) and D-Dimer initial (DIMER) as major prognostic factors of the disease severity. Our contributions are- (i) uncover the underlying mathematical expression for the black-box models on COVID-19 severity prediction task (ii) we are the first to apply symbolic metamodeling to this task, and (iii) discover important features and feature interactions.