This empirical study proposes a novel methodology to measure users' perceived trust in an Explainable Artificial Intelligence (XAI) model. To do so, users' mental models are elicited using Fuzzy Cognitive Maps (FCMs). First, we exploit an interpretable Machine Learning (ML) model to classify suspected COVID-19 patients into positive or negative cases. Then, Medical Experts' (MEs) conduct a diagnostic decision-making task based on their knowledge and then prediction and interpretations provided by the XAI model. In order to evaluate the impact of interpretations on perceived trust, explanation satisfaction attributes are rated by MEs through a survey. Then, they are considered as FCM's concepts to determine their influences on each other and, ultimately, on the perceived trust. Moreover, to consider MEs' mental subjectivity, fuzzy linguistic variables are used to determine the strength of influences. After reaching the steady state of FCMs, a quantified value is obtained to measure the perceived trust of each ME. The results show that the quantified values can determine whether MEs trust or distrust the XAI model. We analyze this behavior by comparing the quantified values with MEs' performance in completing diagnostic tasks.

In the realm of data classification, broad learning system (BLS) has proven to be a potent tool that utilizes a layer-by-layer feed-forward neural network. It consists of feature learning and enhancement segments, working together to extract intricate features from input data. The traditional BLS treats all samples as equally significant, which makes it less robust and less effective for real-world datasets with noises and outliers. To address this issue, we propose the fuzzy BLS (F-BLS) model, which assigns a fuzzy membership value to each training point to reduce the influence of noises and outliers. In assigning the membership value, the F-BLS model solely considers the distance from samples to the class center in the original feature space without incorporating the extent of non-belongingness to a class. We further propose a novel BLS based on intuitionistic fuzzy theory (IF-BLS). The proposed IF-BLS utilizes intuitionistic fuzzy numbers based on fuzzy membership and non-membership values to assign scores to training points in the high-dimensional feature space by using a kernel function. We evaluate the performance of proposed F-BLS and IF-BLS models on 44 UCI benchmark datasets across diverse domains. Furthermore, Gaussian noise is added to some UCI datasets to assess the robustness of the proposed F-BLS and IF-BLS models. Experimental results demonstrate superior generalization performance of the proposed F-BLS and IF-BLS models compared to baseline models, both with and without Gaussian noise. Additionally, we implement the proposed F-BLS and IF-BLS models on the Alzheimers Disease Neuroimaging Initiative (ADNI) dataset, and promising results showcase the models effectiveness in real-world applications. The proposed methods offer a promising solution to enhance the BLS frameworks ability to handle noise and outliers.

Ensemble Kalman inversion (EKI) is a sequential Monte Carlo method used to solve inverse problems within a Bayesian framework. Unlike backpropagation, EKI is a gradient-free optimization method that only necessitates the evaluation of artificial neural networks in forward passes. In this study, we examine the effectiveness of EKI in training neural ordinary differential equations (neural ODEs) for system identification and control tasks. To apply EKI to optimal control problems, we formulate inverse problems that incorporate a Tikhonov-type regularization term. Our numerical results demonstrate that EKI is an efficient method for training neural ODEs in system identification and optimal control problems, with runtime and quality of solutions that are competitive with commonly used gradient-based optimizers.

The domain of machine learning is confronted with a crucial research area known as class imbalance learning, which presents considerable hurdles in the precise classification of minority classes. This issue can result in biased models where the majority class takes precedence in the training process, leading to the underrepresentation of the minority class. The random vector functional link (RVFL) network is a widely-used and effective learning model for classification due to its speed and efficiency. However, it suffers from low accuracy when dealing with imbalanced datasets. To overcome this limitation, we propose a novel graph embedded intuitionistic fuzzy RVFL for class imbalance learning (GE-IFRVFL-CIL) model incorporating a weighting mechanism to handle imbalanced datasets. The proposed GE-IFRVFL-CIL model has a plethora of benefits, such as $(i)$ it leverages graph embedding to extract semantically rich information from the dataset, $(ii)$ it uses intuitionistic fuzzy sets to handle uncertainty and imprecision in the data, $(iii)$ and the most important, it tackles class imbalance learning. The amalgamation of a weighting scheme, graph embedding, and intuitionistic fuzzy sets leads to the superior performance of the proposed model on various benchmark imbalanced datasets, including UCI and KEEL. Furthermore, we implement the proposed GE-IFRVFL-CIL on the ADNI dataset and achieved promising results, demonstrating the model's effectiveness in real-world applications. The proposed method provides a promising solution for handling class imbalance in machine learning and has the potential to be applied to other classification problems.

Variational inference (VI) is a method to approximate the computationally intractable posterior distributions that arise in Bayesian statistics. Typically, VI fits a simple parametric distribution to the target posterior by minimizing an appropriate objective such as the evidence lower bound (ELBO). In this work, we present a new approach to VI based on the principle of score matching, that if two distributions are equal then their score functions (i.e., gradients of the log density) are equal at every point on their support. With this, we develop score matching VI, an iterative algorithm that seeks to match the scores between the variational approximation and the exact posterior. At each iteration, score matching VI solves an inner optimization, one that minimally adjusts the current variational estimate to match the scores at a newly sampled value of the latent variables. We show that when the variational family is a Gaussian, this inner optimization enjoys a closed form solution, which we call Gaussian score matching VI (GSM-VI). GSM-VI is also a ``black box'' variational algorithm in that it only requires a differentiable joint distribution, and as such it can be applied to a wide class of models. We compare GSM-VI to black box variational inference (BBVI), which has similar requirements but instead optimizes the ELBO. We study how GSM-VI behaves as a function of the problem dimensionality, the condition number of the target covariance matrix (when the target is Gaussian), and the degree of mismatch between the approximating and exact posterior distribution. We also study GSM-VI on a collection of real-world Bayesian inference problems from the posteriorDB database of datasets and models. In all of our studies we find that GSM-VI is faster than BBVI, but without sacrificing accuracy. It requires 10-100x fewer gradient evaluations to obtain a comparable quality of approximation.

The problem of model selection is considered for the setting of interpolating estimators, where the number of model parameters exceeds the size of the dataset. Classical information criteria typically consider the large-data limit, penalizing model size. However, these criteria are not appropriate in modern settings where overparameterized models tend to perform well. For any overparameterized model, we show that there exists a dual underparameterized model that possesses the same marginal likelihood, thus establishing a form of Bayesian duality. This enables more classical methods to be used in the overparameterized setting, revealing the Interpolating Information Criterion, a measure of model quality that naturally incorporates the choice of prior into the model selection. Our new information criterion accounts for prior misspecification, geometric and spectral properties of the model, and is numerically consistent with known empirical and theoretical behavior in this regime.

In this work, we propose a novel prior learning method for advancing generalization and uncertainty estimation in deep neural networks. The key idea is to exploit scalable and structured posteriors of neural networks as informative priors with generalization guarantees. Our learned priors provide expressive probabilistic representations at large scale, like Bayesian counterparts of pre-trained models on ImageNet, and further produce non-vacuous generalization bounds. We also extend this idea to a continual learning framework, where the favorable properties of our priors are desirable. Major enablers are our technical contributions: (1) the sums-of-Kronecker-product computations, and (2) the derivations and optimizations of tractable objectives that lead to improved generalization bounds. Empirically, we exhaustively show the effectiveness of this method for uncertainty estimation and generalization.

This paper investigates the utility gain of using Iterative Bayesian Update (IBU) for private discrete distribution estimation using data obfuscated with Locally Differentially Private (LDP) mechanisms. We compare the performance of IBU to Matrix Inversion (MI), a standard estimation technique, for seven LDP mechanisms designed for one-time data collection and for other seven LDP mechanisms designed for multiple data collections (e.g., RAPPOR). To broaden the scope of our study, we also varied the utility metric, the number of users n, the domain size k, and the privacy parameter {\epsilon}, using both synthetic and real-world data. Our results suggest that IBU can be a useful post-processing tool for improving the utility of LDP mechanisms in different scenarios without any additional privacy cost. For instance, our experiments show that IBU can provide better utility than MI, especially in high privacy regimes (i.e., when {\epsilon} is small). Our paper provides insights for practitioners to use IBU in conjunction with existing LDP mechanisms for more accurate and privacy-preserving data analysis. Finally, we implemented IBU for all fourteen LDP mechanisms into the state-of-the-art multi-freq-ldpy Python package (https://pypi.org/project/multi-freq-ldpy/) and open-sourced all our code used for the experiments as tutorials.

Model-based disease mapping remains a fundamental policy-informing tool in the fields of public health and disease surveillance. Hierarchical Bayesian models have emerged as the state-of-the-art approach for disease mapping since they are able to both capture structure in the data and robustly characterise uncertainty. When working with areal data, e.g.~aggregates at the administrative unit level such as district or province, current models rely on the adjacency structure of areal units to account for spatial correlations and perform shrinkage. The goal of disease surveillance systems is to track disease outcomes over time. This task is especially challenging in crisis situations which often lead to redrawn administrative boundaries, meaning that data collected before and after the crisis are no longer directly comparable. Moreover, the adjacency-based approach ignores the continuous nature of spatial processes and cannot solve the change-of-support problem, i.e.~when estimates are required to be produced at different administrative levels or levels of aggregation. We present a novel, practical, and easy to implement solution to solve these problems relying on a methodology combining deep generative modelling and fully Bayesian inference: we build on the recently proposed PriorVAE method able to encode spatial priors over small areas with variational autoencoders by encoding aggregates over administrative units. We map malaria prevalence in Kenya, a country in which administrative boundaries changed in 2010.

This paper proposes a Bayesian model to compare multiple algorithms on multiple data sets, on any metric. The model is based on the Bradley-Terry model, that counts the number of times one algorithm performs better than another on different data sets. Because of its Bayesian foundations, the Bayesian Bradley Terry model (BBT) has different characteristics than frequentist approaches to comparing multiple algorithms on multiple data sets, such as Demsar (2006) tests on mean rank, and Benavoli et al. (2016) multiple pairwise Wilcoxon tests with p-adjustment procedures. In particular, a Bayesian approach allows for more nuanced statements regarding the algorithms beyond claiming that the difference is or it is not statistically significant. Bayesian approaches also allow to define when two algorithms are equivalent for practical purposes, or the region of practical equivalence (ROPE). Different than a Bayesian signed rank comparison procedure proposed by Benavoli et al. (2017), our approach can define a ROPE for any metric, since it is based on probability statements, and not on differences of that metric. This paper also proposes a local ROPE concept, that evaluates whether a positive difference between a mean measure across some cross validation to the mean of some other algorithms is should be really seen as the first algorithm being better than the second, based on effect sizes. This local ROPE proposal is independent of a Bayesian use, and can be used in frequentist approaches based on ranks. A R package and a Python program that implements the BBT is available.