Inverse problems are ubiquitous in the sciences and engineering. Two categories of inverse problems concerning a physical system are (1) estimate parameters in a model of the system from observed input-output pairs and (2) given a model of the system, reconstruct the input to it that caused some observed output. Applied inverse problems are challenging because a solution may (i) not exist, (ii) not be unique, or (iii) be sensitive to measurement noise contaminating the data. Bayesian statistical inversion (BSI) is an approach to tackle ill-posed and/or ill-conditioned inverse problems. Advantageously, BSI provides a "solution" that (i) quantifies uncertainty by assigning a probability to each possible value of the unknown parameter/input and (ii) incorporates prior information and beliefs about the parameter/input. Herein, we provide a tutorial of BSI for inverse problems, by way of illustrative examples dealing with heat transfer from ambient air to a cold lime fruit. First, we use BSI to infer a parameter in a dynamic model of the lime temperature from measurements of the lime temperature over time. Second, we use BSI to reconstruct the initial condition of the lime from a measurement of its temperature later in time. We demonstrate the incorporation of prior information, visualize the posterior distributions of the parameter/initial condition, and show posterior samples of lime temperature trajectories from the model. Our tutorial aims to reach a wide range of scientists and engineers.

The framework of causal models provides a principled approach to causal reasoning, applied today across many scientific domains. Here we present this framework in the language of string diagrams, interpreted formally using category theory. A class of string diagrams, called network diagrams, are in 1-to-1 correspondence with directed acyclic graphs. A causal model is given by such a diagram with its components interpreted as stochastic maps, functions, or general channels in a symmetric monoidal category with a 'copy-discard' structure (cd-category), turning a model into a single mathematical object that can be reasoned with intuitively and yet rigorously. Building on prior works by Fong and Jacobs, Kissinger and Zanasi, as well as Fritz and Klingler, we present diagrammatic definitions of causal models and functional causal models in a cd-category, generalising causal Bayesian networks and structural causal models, respectively. We formalise general interventions on a model, including but beyond do-interventions, and present the natural notion of an open causal model with inputs. We also give an approach to conditioning based on a normalisation box, allowing for causal inference calculations to be done fully diagrammatically. We define counterfactuals in this setup, and treat the problems of the identifiability of causal effects and counterfactuals fully diagrammatically. The benefits of such a presentation of causal models lie in foundational questions in causal reasoning and in their clarificatory role and pedagogical value. This work aims to be accessible to different communities, from causal model practitioners to researchers in applied category theory, and discusses many examples from the literature for illustration. Overall, we argue and demonstrate that causal reasoning according to the causal model framework is most naturally and intuitively done as diagrammatic reasoning.

Structural learning, which aims to learn directed acyclic graphs (DAGs) from observational data, is foundational to causal reasoning and scientific discovery. Recent advancements formulate structural learning into a continuous optimization problem; however, DAG learning remains a highly non-convex problem, and there has not been much work on leveraging well-developed convex optimization techniques for causal structural learning. We fill this gap by proposing a data-adaptive linear approach for causal structural learning from time series data, which can be conveniently cast into a convex optimization problem using a recently developed monotone operator variational inequality (VI) formulation. Furthermore, we establish non-asymptotic recovery guarantee of the VI-based approach and show the superior performance of our proposed method on structure recovery over existing methods via extensive numerical experiments.

Dialogue acts (DAs) can represent conversational actions of tutors or students that take place during tutoring dialogues. Automating the identification of DAs in tutoring dialogues is significant to the design of dialogue-based intelligent tutoring systems. Many prior studies employ machine learning models to classify DAs in tutoring dialogues and invest much effort to optimize the classification accuracy by using limited amounts of training data (i.e., low-resource data scenario). However, beyond the classification accuracy, the robustness of the classifier is also important, which can reflect the capability of the classifier on learning the patterns from different class distributions. We note that many prior studies on classifying educational DAs employ cross entropy (CE) loss to optimize DA classifiers on low-resource data with imbalanced DA distribution. The DA classifiers in these studies tend to prioritize accuracy on the majority class at the expense of the minority class which might not be robust to the data with imbalanced ratios of different DA classes. To optimize the robustness of classifiers on imbalanced class distributions, we propose to optimize the performance of the DA classifier by maximizing the area under the ROC curve (AUC) score (i.e., AUC maximization). Through extensive experiments, our study provides evidence that (i) by maximizing AUC in the training process, the DA classifier achieves significant performance improvement compared to the CE approach under low-resource data, and (ii) AUC maximization approaches can improve the robustness of the DA classifier under different class imbalance ratios.

We study mean-field variational Bayesian inference using the TAP approach, for Z2-synchronization as a prototypical example of a high-dimensional Bayesian model. We show that for any signal strength $\lambda > 1$ (the weak-recovery threshold), there exists a unique local minimizer of the TAP free energy functional near the mean of the Bayes posterior law. Furthermore, the TAP free energy in a local neighborhood of this minimizer is strongly convex. Consequently, a natural-gradient/mirror-descent algorithm achieves linear convergence to this minimizer from a local initialization, which may be obtained by a constant number of iterates of Approximate Message Passing (AMP). This provides a rigorous foundation for variational inference in high dimensions via minimization of the TAP free energy. We also analyze the finite-sample convergence of AMP, showing that AMP is asymptotically stable at the TAP minimizer for any $\lambda > 1$, and is linearly convergent to this minimizer from a spectral initialization for sufficiently large $\lambda$. Such a guarantee is stronger than results obtainable by state evolution analyses, which only describe a fixed number of AMP iterations in the infinite-sample limit. Our proofs combine the Kac-Rice formula and Sudakov-Fernique Gaussian comparison inequality to analyze the complexity of critical points that satisfy strong convexity and stability conditions within their local neighborhoods.

Physics-informed neural networks (PINNs) have proven a suitable mathematical scaffold for solving inverse ordinary (ODE) and partial differential equations (PDE). Typical inverse PINNs are formulated as soft-constrained multi-objective optimization problems with several hyperparameters. In this work, we demonstrate that inverse PINNs can be framed in terms of maximum-likelihood estimators (MLE) to allow explicit error propagation from interpolation to the physical model space through Taylor expansion, without the need of hyperparameter tuning. We explore its application to high-dimensional coupled ODEs constrained by differential algebraic equations that are common in transient chemical and biological kinetics. Furthermore, we show that singular-value decomposition (SVD) of the ODE coupling matrices (reaction stoichiometry matrix) provides reduced uncorrelated subspaces in which PINNs solutions can be represented and over which residuals can be projected. Finally, SVD bases serve as preconditioners for the inversion of covariance matrices in this hyperparameter-free robust application of MLE to ``kinetics-informed neural networks''.

In this paper, we propose a novel Fine-Tuned attribute Weighted Na\"ive Bayes (FT-WNB) classifier to identify the Line-of-Sight (LoS) and Non-Line-of-Sight (NLoS) for UltraWide Bandwidth (UWB) signals in an Indoor Positioning System (IPS). The FT-WNB classifier assigns each signal feature a specific weight and fine-tunes its probabilities to address the mismatch between the predicted and actual class. The performance of the FT-WNB classifier is compared with the state-of-the-art Machine Learning (ML) classifiers such as minimum Redundancy Maximum Relevance (mRMR)- $k$-Nearest Neighbour (KNN), Support Vector Machine (SVM), Decision Tree (DT), Na\"ive Bayes (NB), and Neural Network (NN). It is demonstrated that the proposed classifier outperforms other algorithms by achieving a high NLoS classification accuracy of $99.7\%$ with imbalanced data and $99.8\%$ with balanced data. The experimental results indicate that our proposed FT-WNB classifier significantly outperforms the existing state-of-the-art ML methods for LoS and NLoS signals in IPS in the considered scenario.

Performance monitoring of machine learning (ML)-based risk prediction models in healthcare is complicated by the issue of confounding medical interventions (CMI): when an algorithm predicts a patient to be at high risk for an adverse event, clinicians are more likely to administer prophylactic treatment and alter the very target that the algorithm aims to predict. A simple approach is to ignore CMI and monitor only the untreated patients, whose outcomes remain unaltered. In general, ignoring CMI may inflate Type I error because (i) untreated patients disproportionally represent those with low predicted risk and (ii) evolution in both the model and clinician trust in the model can induce complex dependencies that violate standard assumptions. Nevertheless, we show that valid inference is still possible if one monitors conditional performance and if either conditional exchangeability or time-constant selection bias hold. Specifically, we develop a new score-based cumulative sum (CUSUM) monitoring procedure with dynamic control limits. Through simulations, we demonstrate the benefits of combining model updating with monitoring and investigate how over-trust in a prediction model may delay detection of performance deterioration. Finally, we illustrate how these monitoring methods can be used to detect calibration decay of an ML-based risk calculator for postoperative nausea and vomiting during the COVID-19 pandemic.

Non-Line-of-Sight (NLoS) propagation condition is a crucial factor affecting the precision of the localization in the Ultra-Wideband (UWB) Indoor Positioning System (IPS). Numerous supervised Machine Learning (ML) approaches have been applied for NLoS identification to improve the accuracy of the IPS. However, it is difficult for existing ML approaches to maintain a high classification accuracy when the database contains a small number of NLoS signals and a large number of Line-of-Sight (LoS) signals. The inaccurate localization of the target node caused by this small number of NLoS signals can still be problematic. To solve this issue, we propose feature-based Gaussian Distribution (GD) and Generalized Gaussian Distribution (GGD) NLoS detection algorithms. By employing our detection algorithm for the imbalanced dataset, a classification accuracy of $96.7\%$ and $98.0\%$ can be achieved. We also compared the proposed algorithm with the existing cutting-edge such as Support-Vector-Machine (SVM), Decision Tree (DT), Naive Bayes (NB), and Neural Network (NN), which can achieve an accuracy of $92.6\%$, $92.8\%$, $93.2\%$, and $95.5\%$, respectively. The results demonstrate that the GGD algorithm can achieve high classification accuracy with the imbalanced dataset. Finally, the proposed algorithm can also achieve a higher classification accuracy for different ratios of LoS and NLoS signals which proves the robustness and effectiveness of the proposed method.

Considering a probability distribution over parameters is known as an efficient strategy to learn a neural network with non-differentiable activation functions. We study the expectation of a probabilistic neural network as a predictor by itself, focusing on the aggregation of binary activated neural networks with normal distributions over real-valued weights. Our work leverages a recent analysis derived from the PAC-Bayesian framework that derives tight generalization bounds and learning procedures for the expected output value of such an aggregation, which is given by an analytical expression. While the combinatorial nature of the latter has been circumvented by approximations in previous works, we show that the exact computation remains tractable for deep but narrow neural networks, thanks to a dynamic programming approach. This leads us to a peculiar bound minimization learning algorithm for binary activated neural networks, where the forward pass propagates probabilities over representations instead of activation values. A stochastic counterpart that scales to wide architectures is proposed.