Bayesian inference, while foundational to probabilistic reasoning, is often hampered by the computational intractability of posterior distributions, particularly through the challenging evidence integral. Conventional approaches like Markov Chain Monte Carlo (MCMC) and Variational Inference (VI) face significant scalability and efficiency limitations. This paper introduces a novel, unifying framework for fast Bayesian updates by leveraging harmonic analysis. We demonstrate that representing the prior and likelihood in a suitable orthogonal basis transforms the Bayesian update rule into a spectral convolution. Specifically, the Fourier coefficients of the posterior are shown to be the normalized convolution of the prior and likelihood coefficients. To achieve computational feasibility, we introduce a spectral truncation scheme, which, for smooth functions, yields an exceptionally accurate finite-dimensional approximation and reduces the update to a circular convolution. This formulation allows us to exploit the Fast Fourier Transform (FFT), resulting in a deterministic algorithm with O(N log N) complexity -- a substantial improvement over the O(N^2) cost of naive methods. We establish rigorous mathematical criteria for the applicability of our method, linking its efficiency to the smoothness and spectral decay of the involved distributions. The presented work offers a paradigm shift, connecting Bayesian computation to signal processing and opening avenues for real-time, sequential inference in a wide class of problems.

The rapid evolution of the gaming industry, driven by technological advancements and a burgeoning community, necessitates a deeper understanding of user sentiments, especially as expressed on popular social media platforms like YouTube. This study presents a sentiment analysis on video games based on YouTube comments, aiming to understand user sentiments within the gaming community. Utilizing YouTube API, comments related to various video games were collected and analyzed using the TextBlob sentiment analysis tool. The pre-processed data underwent classification using machine learning algorithms, including Naïve Bayes, Logistic Regression, and Support Vector Machine (SVM). Among these, SVM demonstrated superior performance, achieving the highest classification accuracy across different datasets. The analysis spanned multiple popular gaming videos, revealing trends and insights into user preferences and critiques. The findings underscore the importance of advanced sentiment analysis in capturing the nuanced emotions expressed in user comments, providing valuable feedback for game developers to enhance game design and user experience. Future research will focus on integrating more sophisticated natural language processing techniques and exploring additional data sources to further refine sentiment analysis in the gaming domain.

We study the problem of learning the topology of a directed Gaussian Graphical Model under the equal-variance assumption, where the graph has $n$ nodes and maximum in-degree $d$. Prior work has established that $O(d \log n)$ samples are sufficient for this task. However, an important factor that is often overlooked in these analyses is the dependence on the condition number of the covariance matrix of the model. Indeed, all algorithms from prior work require a number of samples that grows polynomially with this condition number. In many cases this is unsatisfactory, since the condition number could grow polynomially with $n$, rendering these prior approaches impractical in high-dimensional settings. In this work, we provide an algorithm that recovers the underlying graph and prove that the number of samples required is independent of the condition number. Furthermore, we establish lower bounds that nearly match the upper bound up to a $d$-factor, thus providing an almost tight characterization of the true sample complexity of the problem. Moreover, under a further assumption that all the variances of the variables are bounded, we design a polynomial-time algorithm that recovers the underlying graph, at the cost of an additional polynomial dependence of the sample complexity on $d$. We complement our theoretical findings with simulations on synthetic datasets that confirm our predictions.

Although Generative Flow Networks (GFlowNets) are designed to capture multiple modes of a reward function, they often suffer from mode collapse in practice, getting trapped in early-discovered modes and requiring prolonged training to find diverse solutions. Existing exploration techniques often rely on heuristic novelty signals. We propose Loss-Guided GFlowNets (LGGFN), a novel approach where an auxiliary GFlowNet's exploration is \textbf{directly driven by the main GFlowNet's training loss}. By prioritizing trajectories where the main model exhibits \textbf{high loss}, LGGFN focuses sampling on poorly understood regions of the state space. This targeted exploration significantly accelerates the discovery of diverse, high-reward samples. Empirically, across \textbf{diverse benchmarks} including grid environments, structured sequence generation, Bayesian structure learning, and biological sequence design, LGGFN consistently \textbf{outperforms} baselines in exploration efficiency and sample diversity. For instance, on a challenging sequence generation task, it discovered over 40 times more unique valid modes while simultaneously reducing the exploration error metric by approximately 99\%.

Large Language Models (LLMs) have demonstrated remarkable capabilities across numerous tasks, yet principled explanations for their underlying mechanisms and several phenomena, such as scaling laws, hallucinations, and related behaviors, remain elusive. In this work, we revisit the classical relationship between compression and prediction, grounded in Kolmogorov complexity and Shannon information theory, to provide deeper insights into LLM behaviors. By leveraging the Kolmogorov Structure Function and interpreting LLM compression as a two-part coding process, we offer a detailed view of how LLMs acquire and store information across increasing model and data scales -- from pervasive syntactic patterns to progressively rarer knowledge elements. Motivated by this theoretical perspective and natural assumptions inspired by Heap's and Zipf's laws, we introduce a simplified yet representative hierarchical data-generation framework called the Syntax-Knowledge model. Under the Bayesian setting, we show that prediction and compression within this model naturally lead to diverse learning and scaling behaviors observed in LLMs. In particular, our theoretical analysis offers intuitive and principled explanations for both data and model scaling laws, the dynamics of knowledge acquisition during training and fine-tuning, factual knowledge hallucinations in LLMs. The experimental results validate our theoretical predictions.

Ambient sensor-based human activity recognition (HAR) in smart homes remains challenging due to the need for real-time inference, spatially grounded reasoning, and context-aware temporal modeling. Existing approaches often rely on pre-segmented, within-activity data and overlook the physical layout of the environment, limiting their robustness in continuous, real-world deployments. In this paper, we propose MARAuder's Map, a novel framework for real-time activity recognition from raw, unsegmented sensor streams. Our method projects sensor activations onto the physical floorplan to generate trajectory-aware, image-like sequences that capture the spatial flow of human movement. These representations are processed by a hybrid deep learning model that jointly captures spatial structure and temporal dependencies. To enhance temporal awareness, we introduce a learnable time embedding module that encodes contextual cues such as hour-of-day and day-of-week. Additionally, an attention-based encoder selectively focuses on informative segments within each observation window, enabling accurate recognition even under cross-activity transitions and temporal ambiguity. Extensive experiments on multiple real-world smart home datasets demonstrate that our method outperforms strong baselines, offering a practical solution for real-time HAR in ambient sensor environments.

Abstract--An adaptive structural learning method of Restricted Boltzmann Machine (RBM) and Deep Belief Network (DBN) has been developed as one of prominent deep learning models. The neuron generation-annihilation algorithm in R BM and layer generation algorithm in DBN make an optimal networ k structure for given input during the learning. In this paper, our model is applied to an automatic recognition method of road network system, called RoadTracer . A novel method of RoadTracer using the T eacher-Student base d ensemble learning model of Adaptive DBN is proposed, since t he road maps contain many complicated features so that a model with high representation power to detect should be required . The experimental results showed the detection accuracy of t he proposed model was improved from 40.0% to 89.0% on average in the seven major cities among the test dataset. In addition, we challenged to apply our method to the detection of availab le roads when landslide by natural disaster is occurred, in ord er to rapidly obtain a way of transportation. For fast inferenc e, a small size of the trained model was implemented on a small embedded edge device as lightweight deep learning. Recently there have been more cases of extreme climate events including unexpected and unusual weather. The atten - tion of these events has been received in the last few years, d ue to the significant loss of human lives and escalating economi c costs, as well as the impacts on landslides and changes in ecosystems. In Japan, the Japan Meteorological Agency (JMA) has issued "Climate Change Monitoring Report" every year informing the latest status of climate change. According to [1 ], during the Heavy Rain Event of July 2018, Japan experienced unprecedented heavy rainfall. Overall precipitation obse rved at AMeDAS stations throughout Japan in July 2018 was extremely high in comparison with past heavy rainfall event s since 1982. A prominent characteristic of this rain event is that the record-breaking local precipitation, particularly wi thin 48 to 72 hours, was observed extensively over western Japan and Tokyo region, including the Seto Inland Sea side of Chugoku and Shikoku regions. S. Kamada is with Hiroshima City University, Hiroshima, Jap an T. Ichimura is with Prefectural University of Hiroshima, Hi roshima, Japan In addition, lifelines such as wat er supply and communications damaged, and traffic obstacles occurred over a wide area. Due to the disruption of major roads and railroads, the supply was also suspended.

Ordinal categorical data are routinely encountered in a wide range of practical applications. When the primary goal is to construct a regression model for ordinal outcomes, cumulative link models represent one of the most popular choices to link the cumulative probabilities of the response with a set of covariates through a parsimonious linear predictor, shared across response categories. When the number of observations grows, standard sampling algorithms for Bayesian inference scale poorly, making posterior computation increasingly challenging in large datasets. In this article, we propose three scalable algorithms for approximating the posterior distribution of the regression coefficients in cumulative probit models relying on Variational Bayes and Expectation Propagation. We compare the proposed approaches with inference based on Markov Chain Monte Carlo, demonstrating superior computational performance and remarkable accuracy; finally, we illustrate the utility of the proposed algorithms on a challenging case study to investigate the structure of a criminal network.

The quantity of interest in the classical Cramér-Rao theory of unbiased estimation (e.g., the Cramér-Rao lower bound, its exact attainment for exponential families, and asymptotic efficiency of maximum likelihood estimation) is the variance, which represents the instability of an estimator when its value is compared to the value for an independently-sampled data set from the same distribution. In this paper we are interested in a quantity which represents the instability of an estimator when its value is compared to the value for an infinitesimal additive perturbation of the original data set; we refer to this as the "sensitivity" of an estimator. The resulting theory of sensitivity is based on the Wasserstein geometry in the same way that the classical theory of variance is based on the Fisher-Rao (equivalently, Hellinger) geometry, and this insight allows us to determine a collection of results which are analogous to the classical case: a Wasserstein-Cramér-Rao lower bound for the sensitivity of any unbiased estimator, a characterization of models in which there exist unbiased estimators achieving the lower bound exactly, and some concrete results that show that the Wasserstein projection estimator achieves the lower bound asymptotically. We use these results to treat many statistical examples, sometimes revealing new optimality properties for existing estimators and other times revealing entirely new estimators.

Robust causal discovery from observational data under imperfect prior knowledge remains a significant and largely unresolved challenge. Existing methods typically presuppose perfect priors or can only handle specific, pre-identified error types. And their performance degrades substantially when confronted with flawed constraints of unknown location and type. This decline arises because most of them rely on inflexible and biased thresholding strategies that may conflict with the data distribution. To overcome these limitations, we propose to harmonizes knowledge and data through prior alignment and conflict resolution. First, we assess the credibility of imperfect structural constraints through a surrogate model, which then guides a sparse penalization term measuring the loss between the learned and constrained adjacency matrices. We theoretically prove that, under ideal assumption, the knowledge-driven objective aligns with the data-driven objective. Furthermore, to resolve conflicts when this assumption is violated, we introduce a multi-task learning framework optimized via multi-gradient descent, jointly minimizing both objectives. Our proposed method is robust to both linear and nonlinear settings. Extensive experiments, conducted under diverse noise conditions and structural equation model types, demonstrate the effectiveness and efficiency of our method under imperfect structural constraints.