Locally adapting parameters within Markov chain Monte Carlo methods while preserving reversibility is notoriously difficult. The success of the No-U-Turn Sampler (NUTS) largely stems from its clever local adaptation of the integration time in Hamiltonian Monte Carlo via a geometric U-turn condition. However, posterior distributions frequently exhibit multi-scale geometries with extreme variations in scale, making it necessary to also adapt the leapfrog integrator's step size locally and dynamically. Despite its practical importance, this problem has remained largely open since the introduction of NUTS by Hoffman and Gelman (2014). To address this issue, we introduce the Within-orbit Adaptive Leapfrog No-U-Turn Sampler (WALNUTS), a generalization of NUTS that adapts the leapfrog step size at fixed intervals of simulated time as the orbit evolves. At each interval, the algorithm selects the largest step size from a dyadic schedule that keeps the energy error below a user-specified threshold. Like NUTS, WALNUTS employs biased progressive state selection to favor states with positions that are further from the initial point along the orbit. Empirical evaluations on multiscale target distributions, including Neal's funnel and the Stock-Watson stochastic volatility time-series model, demonstrate that WALNUTS achieves substantial improvements in sampling efficiency and robustness compared to standard NUTS.

Recent literature has examined the relationship between the curvature of the loss function at minima and generalization, mainly in the context of overparameterized networks. A key observation is that "flat" minima tend to generalize better than "sharp" minima. While this idea is supported by empirical evidence, it has also been shown that deep networks can generalize even with arbitrary sharpness, as measured by either the trace or the spectral norm of the Hessian. In this paper, we argue that generalization could be assessed by measuring flatness using a soft rank measure of the Hessian. We show that when the common neural network model (neural network with exponential family negative log likelihood loss) is calibrated, and its prediction error and its confidence in the prediction are not correlated with the first and the second derivatives of the network's output, our measure accurately captures the asymptotic expected generalization gap. For non-calibrated models, we connect our flatness measure to the well-known Takeuchi Information Criterion and show that it still provides reliable estimates of generalization gaps for models that are not overly confident. Experimental results indicate that our approach offers a robust estimate of the generalization gap compared to baselines.

This paper develops a Bayesian Generalised Pareto Regression (GPR) model to forecast extreme losses in Indian equity markets, with a focus on the Nifty 50 index. Extreme negative returns, though rare, can cause significant financial disruption, and accurate modelling of such events is essential for effective risk management. Traditional Generalised Pareto Distribution (GPD) models often ignore market conditions; in contrast, our framework links the scale parameter to covariates using a log-linear function, allowing tail risk to respond dynamically to market volatility. We examine four prior choices for Bayesian regularisation of regression coefficients: Cauchy, Lasso (Laplace), Ridge (Gaussian), and Zellner's g-prior. Simulation results suggest that the Cauchy prior delivers the best trade-off between predictive accuracy and model simplicity, achieving the lowest RMSE, AIC, and BIC values. Empirically, we apply the model to large negative returns (exceeding 5%) in the Nifty 50 index. Volatility measures from the Nifty 50, S&P 500, and gold are used as covariates to capture both domestic and global risk drivers. Our findings show that tail risk increases significantly with higher market volatility. In particular, both S&P 500 and gold volatilities contribute meaningfully to crash prediction, highlighting global spillover and flight-to-safety effects. The proposed GPR model offers a robust and interpretable approach for tail risk forecasting in emerging markets. It improves upon traditional EVT-based models by incorporating real-time financial indicators, making it useful for practitioners, policymakers, and financial regulators concerned with systemic risk and stress testing.

Neural networks make accurate predictions but often fail to provide reliable uncertainty estimates, especially under covariate distribution shifts between training and testing. To address this problem, we propose a Bayesian framework for uncertainty estimation that explicitly accounts for covariate shifts. While conventional approaches rely on fixed priors, the key idea of our method is an adaptive prior, conditioned on both training and new covariates. This prior naturally increases uncertainty for inputs that lie far from the training distribution in regions where predictive performance is likely to degrade. To efficiently approximate the resulting posterior predictive distribution, we employ amortized variational inference. Finally, we construct synthetic environments by drawing small bootstrap samples from the training data, simulating a range of plausible covariate shift using only the original dataset. We evaluate our method on both synthetic and real-world data. It yields substantially improved uncertainty estimates under distribution shifts.

Latent stochastic differential equation (SDE) models are important tools for the unsupervised discovery of dynamical systems from data, with applications ranging from engineering to neuroscience. In these complex domains, exact posterior inference of the latent state path is typically intractable, motivating the use of approximate methods such as variational inference (VI). However, existing VI methods for inference in latent SDEs often suffer from slow convergence and numerical instability. Here, we propose SDE Inference via Natural Gradients (SING), a method that leverages natural gradient VI to efficiently exploit the underlying geometry of the model and variational posterior. SING enables fast and reliable inference in latent SDE models by approximating intractable integrals and parallelizing computations in time. We provide theoretical guarantees that SING will approximately optimize the intractable, continuous-time objective of interest. Moreover, we demonstrate that better state inference enables more accurate estimation of nonlinear drift functions using, for example, Gaussian process SDE models. SING outperforms prior methods in state inference and drift estimation on a variety of datasets, including a challenging application to modeling neural dynamics in freely behaving animals. Altogether, our results illustrate the potential of SING as a tool for accurate inference in complex dynamical systems, especially those characterized by limited prior knowledge and non-conjugate structure.

Longitudinal data is commonly utilised across various domains, such as health, biomedical, education and survey studies. This ubiquity has led to a rise in statistical, machine and deep learning-based methods for Longitudinal Data Classification (LDC). However, the intricate nature of the data, characterised by its multi-dimensionality, causes instance-level heterogeneity and temporal correlations that add to the complexity of longitudinal data analysis. Additionally, LDC accuracy is often hampered by the pervasiveness of missing values in longitudinal data. Despite ongoing research that draw on the generative power and utility of Generative Adversarial Networks (GANs) to address the missing data problem, critical considerations include statistical assumptions surrounding longitudinal data and missingness within it, as well as other data-level challenges like class imbalance and mixed data types that impact longitudinal data imputation (LDI) and the subsequent LDC process in GANs. This paper provides a comprehensive overview of how GANs have been applied in LDI, with a focus whether GANS have adequately addressed fundamental assumptions about the data from a LDC perspective. We propose a categorisation of main approaches to GAN-based LDI, highlight strengths and limitations of methods, identify key research trends, and provide promising future directions. Our findings indicate that while GANs show great potential for LDI to improve usability and quality of longitudinal data for tasks like LDC, there is need for more versatile approaches that can handle the wider spectrum of challenges presented by longitudinal data with missing values. By synthesising current knowledge and identifying critical research gaps, this survey aims to guide future research efforts in developing more effective GAN-based solutions to address LDC challenges.

As Large Language Models (LLMs) are integrated into safety-critical applications involving sequential decision-making in the real world, it is essential to know when to trust LLM decisions. Existing LLM Uncertainty Quantification (UQ) methods are primarily designed for single-turn question-answering formats, resulting in multi-step decision-making scenarios, e.g., LLM agentic system, being underexplored. In this paper, we introduce a principled, information-theoretic framework that decomposes LLM sequential decision uncertainty into two parts: (i) internal uncertainty intrinsic to the current decision, which is focused on existing UQ methods, and (ii) extrinsic uncertainty, a Mutual-Information (MI) quantity describing how much uncertainty should be inherited from preceding decisions. We then propose UProp, an efficient and effective extrinsic uncertainty estimator that converts the direct estimation of MI to the estimation of Pointwise Mutual Information (PMI) over multiple Trajectory-Dependent Decision Processes (TDPs). UProp is evaluated over extensive multi-step decision-making benchmarks, e.g., AgentBench and HotpotQA, with state-of-the-art LLMs, e.g., GPT-4.1 and DeepSeek-V3. Experimental results demonstrate that UProp significantly outperforms existing single-turn UQ baselines equipped with thoughtful aggregation strategies. Moreover, we provide a comprehensive analysis of UProp, including sampling efficiency, potential applications, and intermediate uncertainty propagation, to demonstrate its effectiveness. Codes will be available at https://github.com/jinhaoduan/UProp.

This monograph studies the relations between two approaches using positive definite kernels: probabilistic methods using Gaussian processes, and non-probabilistic methods using reproducing kernel Hilbert spaces (RKHS). They are widely studied and used in machine learning, statistics, and numerical analysis. Connections and equivalences between them are reviewed for fundamental topics such as regression, interpolation, numerical integration, distributional discrepancies, and statistical dependence, as well as for sample path properties of Gaussian processes. A unifying perspective for these equivalences is established, based on the equivalence between the Gaussian Hilbert space and the RKHS. The monograph serves as a basis to bridge many other methods based on Gaussian processes and reproducing kernels, which are developed in parallel by the two research communities.

Group structures are common in regression analysis. They can appear in the form of categorical predictors represented by groups of dummy variables or in the context of additive modeling, where each predictor can be expressed as a set of basis functions forming a group; in applications such as gene expression analysis and financial market modeling, groupings exist naturally in the data. For instance, genes that influence similar traits form groups in gene expression data, while stocks from the same sector form groups in financial data. In these scenarios, group shrinkage plays an important role: when there is insufficient evidence to suggest the significance of predictors within a group, the entire group of predictors is shrunk towards zero. This reduces the noise from individual "spurious predictors", which tend to appear more frequently in high-dimensional settings, and decreases model complexity, thereby reducing the risk of overfitting. 1 Within the Bayesian framework, there has been extensive research focusing on the application of continuous shrinkage priors for linear regression problems involving group predictor variables. Traditional approaches, such as the group lasso[31, 24], the group bridge [16], and the group horseshoe [29] primarily apply shrinkage at the group level and do not consider within-group shrinkage.

Parameter estimation is a foundational step in statistical modeling, enabling us to extract knowledge from data and apply it effectively. Bayesian estimation of parameters incorporates prior beliefs with observed data to infer distribution parameters probabilistically and robustly. Moreover, it provides full posterior distributions, allowing uncertainty quantification and regularization, especially useful in small or truncated samples. Utilizing the left-truncated log-logistic (LTLL) distribution is particularly well-suited for modeling time-to-event data where observations are subject to a known lower bound such as precipitation data and cancer survival times. In this paper, we propose a Bayesian approach for estimating the parameters of the LTLL distribution with a fixed truncation point \( x_L > 0 \). Given a random variable \( X \sim LL(α, β; x_L) \), where \( α> 0 \) is the scale parameter and \( β> 0 \) is the shape parameter, the likelihood function is derived based on a truncated sample \( X_1, X_2, \dots, X_N \) with \( X_i > x_L \). We assume independent prior distributions for the parameters, and the posterior inference is conducted via Markov Chain Monte Carlo sampling, specifically using the Metropolis-Hastings algorithm to obtain posterior estimates \( \hatα \) and \( \hatβ \). Through simulation studies and real-world applications, we demonstrate that Bayesian estimation provides more stable and reliable parameter estimates, particularly when the likelihood surface is irregular due to left truncation. The results highlight the advantages of Bayesian inference outperform the estimation of parameter uncertainty in truncated distributions for time to event data analysis.