Bayesian optimization (BO) methods choose sample points by optimizing an acquisition function derived from a statistical model of the objective. These acquisition functions are chosen to balance sampling regions with predicted good objective values against exploring regions where the objective is uncertain. Standard acquisition functions are myopic, considering only the impact of the next sample, but non-myopic acquisition functions may be more effective. In principle, one could model the sampling by a Markov decision process, and optimally choose the next sample by maximizing an expected reward computed by dynamic programming; however, this is infeasibly expensive. More practical approaches, such as rollout, consider a parametric family of sampling policies. In this paper, we show how to efficiently estimate rollout acquisition functions and their gradients, enabling stochastic gradient-based optimization of sampling policies.

Hyperspectral unmixing is the process of determining the presence of individual materials and their respective abundances from an observed pixel spectrum. Unmixing is a fundamental process in hyperspectral image analysis, and is growing in importance as increasingly large spectral libraries are created and used. Unmixing is typically done with ordinary least squares (OLS) regression. However, unmixing with large spectral libraries where the materials present in a pixel are not a priori known, solving for the coefficients in OLS requires inverting a non-invertible matrix from a large spectral library. A number of regression methods are available that can produce a numerical solution using regularization, but with considerably varied effectiveness. Also, simple methods that are unpopular in the statistics literature (i.e. step-wise regression) are used with some level of effectiveness in hyperspectral analysis. In this paper, we provide a thorough performance evaluation of the methods considered, evaluating methods based on how often they select the correct materials in the models. Investigated methods include ordinary least squares regression, non-negative least squares regression, ridge regression, lasso regression, step-wise regression and Bayesian model averaging. We evaluated these unmixing approaches using multiple criteria: incorporation of non-negative abundances, model size, accurate mineral detection and root mean squared error (RMSE). We provide a taxonomy of the regression methods, showing that most methods can be understood as Bayesian methods with specific priors. We conclude that methods that can be derived with priors that correspond to the phenomenology of hyperspectral imagery outperform those with priors that are optimal for prediction performance under the assumptions of ordinary least squares linear regression.

Causal inference in hybrid domains, characterized by a mixture of discrete and continuous variables, presents a formidable challenge. We take a step towards this direction and propose Characteristic Interventional Sum-Product Network ($\chi$SPN) that is capable of estimating interventional distributions in presence of random variables drawn from mixed distributions. $\chi$SPN uses characteristic functions in the leaves of an interventional SPN (iSPN) thereby providing a unified view for discrete and continuous random variables through the Fourier-Stieltjes transform of the probability measures. A neural network is used to estimate the parameters of the learned iSPN using the intervened data. Our experiments on 3 synthetic heterogeneous datasets suggest that $\chi$SPN can effectively capture the interventional distributions for both discrete and continuous variables while being expressive and causally adequate. We also show that $\chi$SPN generalize to multiple interventions while being trained only on a single intervention data.

Inverter-based volt-var control is studied in this paper. One key issue in DRL-based approaches is the limited measurement deployment in active distribution networks, which leads to problems of a partially observable state and unknown reward. To address those problems, this paper proposes a robust DRL approach with a conservative critic and a surrogate reward. The conservative critic utilizes the quantile regression technology to estimate conservative state-action value function based on the partially observable state, which helps to train a robust policy; the surrogate rewards of power loss and voltage violation are designed that can be calculated from the limited measurements. The proposed approach optimizes the power loss of the whole network and the voltage profile of buses with measurable voltages while indirectly improving the voltage profile of other buses. Extensive simulations verify the effectiveness of the robust DRL approach in different limited measurement conditions, even when only the active power injection of the root bus and less than 10% of bus voltages are measurable.

Learning from Label Proportion (LLP) is a weakly supervised learning scenario in which training data is organized into predefined bags of instances, disclosing only the class label proportions per bag. This paradigm is essential for user modeling and personalization, where user privacy is paramount, offering insights into user preferences without revealing individual data. LLP faces a unique difficulty: the misalignment between bag-level supervision and the objective of instance-level prediction, primarily due to the inherent ambiguity in label proportion matching. Previous studies have demonstrated deep representation learning can generate auxiliary signals to promote the supervision level in the image domain. However, applying these techniques to tabular data presents significant challenges: 1) they rely heavily on label-invariant augmentation to establish multi-view, which is not feasible with the heterogeneous nature of tabular datasets, and 2) tabular datasets often lack sufficient semantics for perfect class distinction, making them prone to suboptimality caused by the inherent ambiguity of label proportion matching. To address these challenges, we propose an augmentation-free contrastive framework TabLLP-BDC that introduces class-aware supervision (explicitly aware of class differences) at the instance level. Our solution features a two-stage Bag Difference Contrastive (BDC) learning mechanism that establishes robust class-aware instance-level supervision by disassembling the nuance between bag label proportions, without relying on augmentations. Concurrently, our model presents a pioneering multi-task pretraining pipeline tailored for tabular-based LLP, capturing intrinsic tabular feature correlations in alignment with label proportion distribution. Extensive experiments demonstrate that TabLLP-BDC achieves state-of-the-art performance for LLP in the tabular domain.

Monotonic Partially Observable Markov Decision Processes (POMDPs), where the system state progressively decreases until a restorative action is performed, can be used to model sequential repair problems effectively. This paper considers the problem of solving budget-constrained multi-component monotonic POMDPs, where a finite budget limits the maximal number of restorative actions. For a large number of components, solving such a POMDP using current methods is computationally intractable due to the exponential growth in the state space with an increasing number of components. To address this challenge, we propose a two-step approach. Since the individual components of a budget-constrained multi-component monotonic POMDP are only connected via the shared budget, we first approximate the optimal budget allocation among these components using an approximation of each component POMDP's optimal value function which is obtained through a random forest model. Subsequently, we introduce an oracle-guided meta-trained Proximal Policy Optimization (PPO) algorithm to solve each of the independent budget-constrained single-component monotonic POMDPs. The oracle policy is obtained by performing value iteration on the corresponding monotonic Markov Decision Process (MDP). This two-step method provides scalability in solving truly massive multi-component monotonic POMDPs. To demonstrate the efficacy of our approach, we consider a real-world maintenance scenario that involves inspection and repair of an administrative building by a team of agents within a maintenance budget. Finally, we perform a computational complexity analysis for a varying number of components to show the scalability of the proposed approach.

The Sleeping Beauty problem is a puzzle in probability theory that has gained much attention since Elga's discussion of it [Elga, Adam, Analysis 60 (2), p.143-147 (2000)]. Sleeping Beauty is put asleep, and a coin is tossed. If the outcome of the coin toss is Tails, Sleeping Beauty is woken up on Monday, put asleep again and woken up again on Tuesday (with no recollection of having woken up on Monday). If the outcome is Heads, Sleeping Beauty is woken up on Monday only. Each time Sleeping Beauty is woken up, she is asked what her belief is that the outcome was Heads. What should Sleeping Beauty reply? In literature arguments have been given for both 1/3 and 1/2 as the correct answer. In this short note we argue using simple Bayesian probability theory why 1/3 is the right answer, and not 1/2. Briefly, when Sleeping Beauty awakens, her being awake is nontrivial extra information that leads her to update her beliefs about Heads to 1/3. We strengthen our claim by considering an additional observer, Prince Probability, who may or may not meet Sleeping Beauty. If he meets Sleeping Beauty while she is awake, he lowers his credence in Heads to 1/3. We also briefly consider the credence in Heads of a Sleeping Beauty who knows that she is dreaming (and thus asleep).

Gaussian Process Latent Variable Models (GPLVMs) have become increasingly popular for unsupervised tasks such as dimensionality reduction and missing data recovery due to their flexibility and non-linear nature. An importance-weighted version of the Bayesian GPLVMs has been proposed to obtain a tighter variational bound. However, this version of the approach is primarily limited to analyzing simple data structures, as the generation of an effective proposal distribution can become quite challenging in high-dimensional spaces or with complex data sets. In this work, we propose an Annealed Importance Sampling (AIS) approach to address these issues. By transforming the posterior into a sequence of intermediate distributions using annealing, we combine the strengths of Sequential Monte Carlo samplers and VI to explore a wider range of posterior distributions and gradually approach the target distribution. We further propose an efficient algorithm by reparameterizing all variables in the evidence lower bound (ELBO). Experimental results on both toy and image datasets demonstrate that our method outperforms state-of-the-art methods in terms of tighter variational bounds, higher log-likelihoods, and more robust convergence.

"Artificial Neural Network and Deep Learning: Fundamentals and Theory" offers a comprehensive exploration of the foundational principles and advanced methodologies in neural networks and deep learning. This book begins with essential concepts in descriptive statistics and probability theory, laying a solid groundwork for understanding data and probability distributions. As the reader progresses, they are introduced to matrix calculus and gradient optimization, crucial for training and fine-tuning neural networks. The book delves into multilayer feed-forward neural networks, explaining their architecture, training processes, and the backpropagation algorithm. Key challenges in neural network optimization, such as activation function saturation, vanishing and exploding gradients, and weight initialization, are thoroughly discussed. The text covers various learning rate schedules and adaptive algorithms, providing strategies to optimize the training process. Techniques for generalization and hyperparameter tuning, including Bayesian optimization and Gaussian processes, are also presented to enhance model performance and prevent overfitting. Advanced activation functions are explored in detail, categorized into sigmoid-based, ReLU-based, ELU-based, miscellaneous, non-standard, and combined types. Each activation function is examined for its properties and applications, offering readers a deep understanding of their impact on neural network behavior. The final chapter introduces complex-valued neural networks, discussing complex numbers, functions, and visualizations, as well as complex calculus and backpropagation algorithms. This book equips readers with the knowledge and skills necessary to design, and optimize advanced neural network models, contributing to the ongoing advancements in artificial intelligence.

Traditional deep Gaussian processes model the data evolution using a discrete hierarchy, whereas differential Gaussian processes (DIFFGPs) represent the evolution as an infinitely deep Gaussian process. However, prior DIFFGP methods often overlook the uncertainty of kernel hyperparameters and assume them to be fixed and time-invariant, failing to leverage the unique synergy between continuous-time models and approximate inference. In this work, we propose a fully Bayesian approach that treats the kernel hyperparameters as random variables and constructs coupled stochastic differential equations (SDEs) to learn their posterior distribution and that of inducing points. By incorporating estimation uncertainty on hyperparameters, our method enhances the model's flexibility and adaptability to complex dynamics. Additionally, our approach provides a time-varying, comprehensive, and realistic posterior approximation through coupling variables using SDE methods. Experimental results demonstrate the advantages of our method over traditional approaches, showcasing its superior performance in terms of flexibility, accuracy, and other metrics. Our work opens up exciting research avenues for advancing Bayesian inference and offers a powerful modeling tool for continuous-time Gaussian processes.