In inverse problems, we often have access to data consisting of paired samples $(x,y)\sim p_{X,Y}(x,y)$ where $y$ are partial observations of a physical system, and $x$ represents the unknowns of the problem. Under these circumstances, we can employ supervised training to learn a solution $x$ and its uncertainty from the observations $y$. We refer to this problem as the "supervised" case. However, the data $y\sim p_{Y}(y)$ collected at one point could be distributed differently than observations $y'\sim p_{Y}'(y')$, relevant for a current set of problems. In the context of Bayesian inference, we propose a two-step scheme, which makes use of normalizing flows and joint data to train a conditional generator $q_{\theta}(x|y)$ to approximate the target posterior density $p_{X|Y}(x|y)$. Additionally, this preliminary phase provides a density function $q_{\theta}(x|y)$, which can be recast as a prior for the "unsupervised" problem, e.g.~when only the observations $y'\sim p_{Y}'(y')$, a likelihood model $y'|x$, and a prior on $x'$ are known. We then train another invertible generator with output density $q'_{\phi}(x|y')$ specifically for $y'$, allowing us to sample from the posterior $p_{X|Y}'(x|y')$. We present some synthetic results that demonstrate considerable training speedup when reusing the pretrained network $q_{\theta}(x|y')$ as a warm start or preconditioning for approximating $p_{X|Y}'(x|y')$, instead of learning from scratch. This training modality can be interpreted as an instance of transfer learning. This result is particularly relevant for large-scale inverse problems that employ expensive numerical simulations.

In this paper, we provide a theoretical framework to analyze an agent who misinterprets or misperceives the true decision problem she faces. Within this framework, we show that a wide range of behavior observed in experimental settings manifest as failures to perceive implications, in other words, to properly account for the logical relationships between various payoff relevant contingencies. We present behavioral characterizations corresponding to several benchmarks of logical sophistication and show how it is possible to identify which implications the agent fails to perceive. Thus, our framework delivers both a methodology for assessing an agent's level of contingent thinking and a strategy for identifying her beliefs in the absence full rationality.

Information technology has spread widely, and extraordinarily large amounts of data have been made accessible to users, which has made it challenging to select data that are in accordance with user needs. For the resolution of the above issue, recommender systems have emerged, which much help users go through the process of decision-making and selecting relevant data. A recommender system predicts users behavior to be capable of detecting their interests and needs, and it often uses the classification technique for this purpose. It may not be sufficiently accurate to employ individual classification, where not all cases can be examined, which makes the method inappropriate to specific problems. In this research, group classification and the ensemble learning technique were used for increasing prediction accuracy in recommender systems. Another issue that is raised here concerns user analysis. Given the large size of the data and a large number of users, the process of user needs analysis and prediction (using a graph in most cases, representing the relations between users and their selected items) is complicated and cumbersome in recommender systems. Graph embedding was also proposed for resolution of this issue, where all or part of user behavior can be simulated through the generation of several vectors, resolving the problem of user behavior analysis to a large extent while maintaining high efficiency. In this research, individuals most similar to the target user were classified using ensemble learning, fuzzy rules, and the decision tree, and relevant recommendations were then made to each user with a heterogeneous knowledge graph and embedding vectors. This study was performed on the MovieLens datasets, and the obtained results indicated the high efficiency of the presented method.

Data Science and Machine Learning have become fundamental assets for companies and research institutions alike. As one of its fields, supervised classification allows for class prediction of new samples, learning from given training data. However, some properties can cause datasets to be problematic to classify. In order to evaluate a dataset a priori, data complexity metrics have been used extensively. They provide information regarding different intrinsic characteristics of the data, which serve to evaluate classifier compatibility and a course of action that improves performance. However, most complexity metrics focus on just one characteristic of the data, which can be insufficient to properly evaluate the dataset towards the classifiers' performance. In fact, class overlap, a very detrimental feature for the classification process (especially when imbalance among class labels is also present) is hard to assess. This research work focuses on revisiting complexity metrics based on data morphology. In accordance to their nature, the premise is that they provide both good estimates for class overlap, and great correlations with the classification performance. For that purpose, a novel family of metrics have been developed. Being based on ball coverage by classes, they are named after Overlap Number of Balls. Finally, some prospects for the adaptation of the former family of metrics to singular (more complex) problems are discussed.

Anomaly estimation, or the problem of finding a subset of a dataset that differs from the rest of the dataset, is a classic problem in machine learning and data mining. In both theoretical work and in applications, the anomaly is assumed to have a specific structure defined by membership in an $\textit{anomaly family}$. For example, in temporal data the anomaly family may be time intervals, while in network data the anomaly family may be connected subgraphs. The most prominent approach for anomaly estimation is to compute the Maximum Likelihood Estimator (MLE) of the anomaly. However, it was recently observed that for some anomaly families, the MLE is an asymptotically $\textit{biased}$ estimator of the anomaly. Here, we demonstrate that the bias of the MLE depends on the size of the anomaly family. We prove that if the number of sets in the anomaly family that contain the anomaly is sub-exponential, then the MLE is asymptotically unbiased. At the same time, we provide empirical evidence that the converse is also true: if the number of such sets is exponential, then the MLE is asymptotically biased. Our analysis unifies a number of earlier results on the bias of the MLE for specific anomaly families, including intervals, submatrices, and connected subgraphs. Next, we derive a new anomaly estimator using a mixture model, and we empirically demonstrate that our estimator is asymptotically unbiased regardless of the size of the anomaly family. We illustrate the benefits of our estimator on both simulated disease outbreak data and a real-world highway traffic dataset.

We introduce in this work a new method for online approximate Bayesian learning, whose main idea is to approximate the sequence $(\pi_t)_{t\geq 1}$ of posterior distributions by a sequence $(\tilde{\pi}_t)_{t\geq 1}$ which (i) can be estimated in an online fashion using sequential Monte Carlo methods and (ii) is shown to converge to the same distribution as the sequence $(\pi_t)_{t\geq 1}$, under weak assumptions on the statistical model at hand. In its simplest version, the proposed approach amounts to take for $(\tilde{\pi}_t)_{t\geq 1}$ the sequence of filtering distributions associated to a particular state-space model, and to approximate this sequence using a standard particle filter algorithm. We illustrate on several challenging examples the benefits of this procedure for online approximate Bayesian parameter inference, and with one real data example we show that its online predictive performance can significantly outperform that of stochastic gradient descent and of streaming variational Bayes.

The discovery of causal relations is a fundamental objective in many scientific endeavours. The process of the scientific method usually involves a conjecture, such as a causal graph or a set of equations, that explains observed phenomena. In practice, such a graph structure can be learned automatically from conditional independences in observational data via the PC/FCI algorithms (Spirtes, Glymour, and Scheines, 2000; Zhang, 2008). The crucial assumption in causal discovery is that directed edges in this learned graph express causal relations between variables. However, an immediate concern is whether directed graphs actually can simultaneously encode the causal semantics and the conditional independence constraints of a system.

Despite the success of classical traffic flow (e.g., second-order macroscopic) models and data-driven (e.g., Machine Learning - ML) approaches in traffic state estimation, those approaches either require great efforts for parameter calibrations or lack theoretical interpretation. To fill this research gap, this study presents a new modeling framework, named physics regularized Gaussian process (PRGP). This novel approach can encode physics models, i.e., classical traffic flow models, into the Gaussian process architecture and so as to regularize the ML training process. Particularly, this study aims to discuss how to develop a PRGP model when the original physics model is with discrete formulations. Then based on the posterior regularization inference framework, an efficient stochastic optimization algorithm is developed to maximize the evidence lowerbound of the system likelihood. To prove the effectiveness of the proposed model, this paper conducts empirical studies on a real-world dataset that is collected from a stretch of I-15 freeway, Utah. Results show the new PRGP model can outperform the previous compatible methods, such as calibrated physics models and pure machine learning methods, in estimation precision and input robustness.

Latent confounders---unobserved variables that influence both treatment and outcome---can bias estimates of causal effects. In some cases, these confounders are shared across observations, e.g. all students taking a course are influenced by the course's difficulty in addition to any educational interventions they receive individually. This paper shows how to semiparametrically model latent confounders that have this structure and thereby improve estimates of causal effects. The key innovations are a hierarchical Bayesian model, Gaussian processes with structured latent confounders (GP-SLC), and a Monte Carlo inference algorithm for this model based on elliptical slice sampling. GP-SLC provides principled Bayesian uncertainty estimates of individual treatment effect with minimal assumptions about the functional forms relating confounders, covariates, treatment, and outcome. Finally, this paper shows GP-SLC is competitive with or more accurate than widely used causal inference techniques on three benchmark datasets, including the Infant Health and Development Program and a dataset showing the effect of changing temperatures on state-wide energy consumption across New England.

Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation. The recent surge of transport maps offers a mathematical foundation and new insights for tackling this challenge by coupling intractable random variables with tractable reference random variables. This paper generalises a recently developed functional tensor-train (FTT) approximation of the inverse Rosenblatt transport [14] to a wide class of high-dimensional nonnegative functions, such as unnormalised probability density functions. First, we extend the inverse Rosenblatt transform to enable the transport to general reference measures other than the uniform measure. We develop an efficient procedure to compute this transport from a squared FTT decomposition which preserves the monotonicity. More crucially, we integrate the proposed monotonicity-preserving FTT transport into a nested variable transformation framework inspired by deep neural networks. The resulting deep inverse Rosenblatt transport significantly expands the capability of tensor approximations and transport maps to random variables with complicated nonlinear interactions and concentrated density functions. We demonstrate the efficacy of the proposed approach on a range of applications in statistical learning and uncertainty quantification, including parameter estimation for dynamical systems and inverse problems constrained by partial differential equations.