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We provide a general framework for constructing probability distributions on Riemannian manifolds, taking advantage of area-preserving maps and isometries. Control over distributions' properties, such as parameters, symmetry and modality yield a family of flexible distributions that are straightforward to sample from, suitable for use within Monte Carlo algorithms and latent variable models, such as autoencoders. As an illustration, we empirically validate our approach by utilizing our proposed distributions within a variational autoencoder and a latent space network model. Finally, we take advantage of the generalized description of this framework to posit questions for future work., 34 pages, 9 figures. arXiv admin note: text overlap with arXiv:1804.00891 by other authors

Probabilistic machine learning models are often insufficient to help with decisions on interventions because those models find correlations - not causal relationships. If observational data is only available and experimentation are infeasible, the correct approach to study the impact of an intervention is to invoke Pearl's causality framework. Even that framework assumes that the underlying causal graph is known, which is seldom the case in practice. When the causal structure is not known, one may use out-of-the-box algorithms to find causal dependencies from observational data. However, there exists no method that also accounts for the decision-maker's prior knowledge when developing the causal structure either. The objective of this paper is to develop rational approaches for making decisions from observational data in the presence of causal graph uncertainty and prior knowledge from the decision-maker. We use ensemble methods like Bayesian Model Averaging (BMA) to infer set of causal graphs that can represent the data generation process. We provide decisions by computing the expected value and risk of potential interventions explicitly. We demonstrate our approach by applying them in different example contexts.

A rich class of network models associate each node with a low-dimensional latent coordinate that controls the propensity for connections to form. Models of this type are well established in the network analysis literature, where it is typical to assume that the underlying geometry is Euclidean. Recent work has explored the consequences of this choice and has motivated the study of models which rely on non-Euclidean latent geometries, with a primary focus on spherical and hyperbolic geometry. In this paper, we examine to what extent latent features can be inferred from the observable links in the network, considering network models which rely on spherical and hyperbolic geometries. For each geometry, we describe a latent space network model, detail constraints on the latent coordinates which remove the well-known identifiability issues, and present Bayesian estimation schemes. Thus, we develop computational procedures to perform inference for network models in which the properties of the underlying geometry play a vital role. Finally, we assess the validity of these models on real data., Comment: 46 pages, 14 figures

We adopt the statistical framework on robustness proposed by Watson and Holmes in 2016 and then tackle the practical challenges that hinder its applicability to network models. The goal is to evaluate how the quality of an inference for a network feature degrades when the assumed model is misspecified. Decision theory methods aimed to identify model missespecification are applied in the context of network data with the goal of investigating the stability of optimal actions to perturbations to the assumed model. Here the modified versions of the model are contained within a well defined neighborhood of model space. Our main challenge is to combine stochastic optimization and graph limits tools to explore the model space. As a result, a method for robustness on exchangeable random networks is developed. Our approach is inspired by recent developments in the context of robustness and recent works in the robust control, macroeconomics and financial mathematics literature and more specifically and is based on the concept of graphon approximation through its empirical graphon., 34 pages, 5 figures

Some of the most used sampling mechanisms that implicitly leverage a social network depend on tuning parameters; for instance, Respondent-Driven Sampling (RDS) is specified by the number of seeds and maximum number of referrals. We are interested in the problem of optimizing these sampling mechanisms with respect to their tuning parameters in order to optimize the inference on a population quantity, where such quantity is a function of the network and measurements taken at the nodes. This is done by formulating the problem in terms of decision theory and information theory, in turn. We discuss how the approaches discussed in this paper relate, via theoretical results, to other formalisms aimed to compare sampling designs, namely sufficiency and the Goel-DeGroot Criterion. The optimization procedure for different network sampling mechanisms is illustrated via simulations in the fashion of the ones used for Bayesian clinical trials., 21 pages, 1 figure