Timely assessment of current conditions is essential especially for small, open economies such as Singapore, where external shocks transmit rapidly to domestic activity. We develop a real-time nowcasting framework for quarterly GDP growth using a high-dimensional panel of approximately 70 indicators, encompassing economic and financial indicators over 1990Q1-2023Q2. The analysis covers penalized regressions, dimensionality-reduction methods, ensemble learning algorithms, and neural architectures, benchmarked against a Random Walk, an AR(3), and a Dynamic Factor Model. The pipeline preserves temporal ordering through an expanding-window walk-forward design with Bayesian hyperparameter optimization, and uses moving block-bootstrap procedures both to construct prediction intervals and to obtain confidence bands for feature-importance measures. It adopts model-specific and XAI-based explainability tools. A Model Confidence Set procedure identifies statistically superior learners, which are then combined through simple, weighted, and exponentially weighted schemes; the resulting time-varying weights provide an interpretable representation of model contributions. Predictive ability is assessed via Giacomini-White tests. Empirical results show that penalized regressions, dimensionality-reduction models, and GRU networks consistently outperform all benchmarks, with RMSFE reductions of roughly 40-60%; aggregation delivers further gains. Feature-attribution methods highlight industrial production, external trade, and labor-market indicators as dominant drivers of Singapore's short-run growth dynamics.

Reinforcement Learning (RL) following Supervised Fine-Tuning (SFT) has become the standard paradigm for post-training Large Language Models (LLMs). However, the mechanism by which RL contributes to reasoning capabilities-- whether it incentivizes the synthesis of new skills or merely amplifies existing behaviors--remains a subject of intense debate. In this work, we investigate this question through the lens of Complementary Reasoning, a complex task that requires integrating internal parametric knowledge with external contextual information. Using a controlled synthetic dataset of human biographies, we strictly decouple this ability into two atomic skills: Parametric Reasoning (relying on internal knowledge encoded in model parameters) and Contextual Reasoning (depending on novel information provided in the context window). To rigorously assess capability boundaries, we evaluate generalization across three distinct levels of difficulty: I.I.D., Composition, and Zero-shot settings. We find that while SFT is sufficient for in-distribution performance, it struggles with out-of-distribution generalization, particularly in Zero-shot settings where relational combinations are novel. Crucially, we identify the SFT Generalization Paradox: Models supervised solely on the composite task achieve near-perfect in-distribution accuracy (90%) but collapse on out-of-distribution generalization (18%), indicating their reliance on rote memorization of path shortcuts. In contrast, we find that RL acts as a reasoning synthesizer rather than a probability amplifier. However, we uncover a strict atomic prerequisite: RL can only synthesize these complex strategies if the base model has first mastered the independent atomic skills (Parametric and Contextual) via SFT. These findings challenge the view of RL as a mere amplifier, suggesting that given sufficient atomic foundations, RL can actively synthesize complex reasoning strategies from learned primitives without explicit supervision on such complex strategies. This indicates that decoupled atomic training followed by RL offers a scalable path to generalization for complex reasoning tasks. Code and data will be at https://github.com/sitaocheng/from The rapid evolution of Large Language Models (LLMs) has been fundamentally driven by advanced post-training strategies, specifically an initial Supervised Fine-Tuning (SFT) stage followed by a Reinforcement Learning (RL) stage (Achiam et al., 2023; Team et al., 2024; Guo et al., 2025). While SFT is effective at establishing behavioral norms and imparting foundational knowledge, it fundamentally relies on maximum likelihood estimation, which tends to favor the memorization of the training distribution.

Learning the structure of a Bayesian network from decentralized data poses two major challenges: (i) ensuring rigorous privacy guarantees for participants, and (ii) avoiding communication costs that scale poorly with dimensionality. In this work, we introduce Fed-Sparse-BNSL, a novel federated method for learning linear Gaussian Bayesian network structures that addresses both challenges. By combining differential privacy with greedy updates that target only a few relevant edges per participant, Fed-Sparse-BNSL efficiently uses the privacy budget while keeping communication costs low. Our careful algorithmic design preserves model identifiability and enables accurate structure estimation. Experiments on synthetic and real datasets demonstrate that Fed-Sparse-BNSL achieves utility close to non-private baselines while offering substantially stronger privacy and communication efficiency.

Foundation models, and in particular large language models, can generate highly informative responses, prompting growing interest in using these ''synthetic'' outputs as data in empirical research and decision-making. This paper introduces the idea of a foundation prior, which shows that model-generated outputs are not as real observations, but draws from the foundation prior induced prior predictive distribution. As such synthetic data reflects both the model's learned patterns and the user's subjective priors, expectations, and biases. We model the subjectivity of the generative process by making explicit the dependence of synthetic outputs on the user's anticipated data distribution, the prompt-engineering process, and the trust placed in the foundation model. We derive the foundation prior as an exponential-tilted, generalized Bayesian update of the user's primitive prior, where a trust parameter governs the weight assigned to synthetic data. We then show how synthetic data and the associated foundation prior can be incorporated into standard statistical and econometric workflows, and discuss their use in applications such as refining complex models, informing latent constructs, guiding experimental design, and augmenting random-coefficient and partially linear specifications. By treating generative outputs as structured, explicitly subjective priors rather than as empirical observations, the framework offers a principled way to harness foundation models in empirical work while avoiding the conflation of synthetic ''facts'' with real data.

Differential privacy is increasingly formalized through the lens of hypothesis testing via the robust and interpretable $f$-DP framework, where privacy guarantees are encoded by a baseline Blackwell trade-off function $f_{\infty} = T(P_{\infty}, Q_{\infty})$ involving a pair of distributions $(P_{\infty}, Q_{\infty})$. The problem of choosing the right privacy metric in practice leads to a central question: what is a statistically appropriate baseline $f_{\infty}$ given some prior modeling assumptions? The special case of Gaussian differential privacy (GDP) showed that, under compositions of nearly perfect mechanisms, these trade-off functions exhibit a central limit behavior with a Gaussian limit experiment. Inspired by Le Cam's theory of limits of statistical experiments, we answer this question in full generality in an infinitely divisible setting. We show that suitable composition experiments $(P_n^{\otimes n}, Q_n^{\otimes n})$ converge to a binary limit experiment $(P_{\infty}, Q_{\infty})$ whose log-likelihood ratio $L = \log(dQ_{\infty} / dP_{\infty})$ is infinitely divisible under $P_{\infty}$. Thus any limiting trade-off function $f_{\infty}$ is determined by an infinitely divisible law $P_{\infty}$, characterized by its Levy--Khintchine triplet, and its Esscher tilt defined by $dQ_{\infty}(x) = e^{x} dP_{\infty}(x)$. This characterizes all limiting baseline trade-off functions $f_{\infty}$ arising from compositions of nearly perfect differentially private mechanisms. Our framework recovers GDP as the purely Gaussian case and yields explicit non-Gaussian limits, including Poisson examples. It also positively resolves the empirical $s^2 = 2k$ phenomenon observed in the GDP paper and provides an optimal mechanism for count statistics achieving asymmetric Poisson differential privacy.

We study the problem of learning disentangled signals from data using non-linear Independent Component Analysis (ICA). Motivated by advances in self-supervised learning, we propose to learn self-sufficient signals: A recovered signal should be able to reconstruct a missing value of its own from all remaining components without relying on any other signals. We formulate this problem as the minimization of a conditional KL divergence. Compared to traditional maximum likelihood estimation, our algorithm is prior-free and likelihood-free, meaning that we do not need to impose any prior on the original signals or any observational model, which often restricts the model's flexibility. To tackle the KL divergence minimization problem, we propose a sequential algorithm that reduces the KL divergence and learns an optimal de-mixing flow model at each iteration. This approach completely avoids the unstable adversarial training, a common issue in minimizing the KL divergence. Experiments on toy and real-world datasets show the effectiveness of our method.

Many tasks require mapping continuous input data (e.g. images) to discrete task outputs (e.g. class labels). Yet, how neural networks learn to perform such discrete computations on continuous data manifolds remains poorly understood. Here, we show that signatures of such computations emerge in the representational geometry of neural networks as they learn. By analysing the Riemannian pullback metric across layers of a neural network, we find that network computation can be decomposed into two functions: discretising continuous input features and performing logical operations on these discretised variables. Furthermore, we demonstrate how different learning regimes (rich vs. lazy) have contrasting metric and curvature structures, affecting the ability of the networks to generalise to unseen inputs. Overall, our work provides a geometric framework for understanding how neural networks learn to perform discrete computations on continuous manifolds.

We study streaming data with categorical features where the vocabulary of categorical feature values is changing and can even grow unboundedly over time. Feature hashing is commonly used as a pre-processing step to map these categorical values into a feature space of fixed size before learning their embeddings. While these methods have been developed and evaluated for offline or batch settings, in this paper we consider online settings. We show that deterministic embeddings are sensitive to the arrival order of categories and suffer from forgetting in online learning, leading to performance deterioration. To mitigate this issue, we propose a probabilistic hash embedding (PHE) model that treats hash embeddings as stochastic and applies Bayesian online learning to learn incrementally from data. Based on the structure of PHE, we derive a scalable inference algorithm to learn model parameters and infer/update the posteriors of hash embeddings and other latent variables. Our algorithm (i) can handle an evolving vocabulary of categorical items, (ii) is adaptive to new items without forgetting old items, (iii) is implementable with a bounded set of parameters that does not grow with the number of distinct observed values on the stream, and (iv) is invariant to the item arrival order. Experiments in classification, sequence modeling, and recommendation systems in online learning setups demonstrate the superior performance of PHE while maintaining high memory efficiency (consumes as low as 2~4 memory of a one-hot embedding table). Supplementary materials are at https://github.com/aodongli/probabilistic-hash-embeddings

Stochastic Variance Reduced Gradient (SVRG) and its variants aim to speed-up training by using gradient corrections, but have seen limited success in deep learning. Here, we show surprising new foundational connections of SVRG to a recently proposed Bayesian method called posterior correction. Specifically, we show that SVRG is recovered as a special case of posterior correction over the isotropic-Gaussian family, while novel extensions are automatically obtained by using more flexible exponential families. We derive two new SVRG variants by using Gaussian families: First, a Newton-like variant that employs novel Hessian corrections, and second, an Adam-like extension that improves pretraining and finetuning of Transformer language models. This is the first work to connect SVRG to Bayes and use it to boost variational training for deep networks.

Modeling historical events poses fundamental challenges for machine learning: extreme data scarcity (N << 100), heterogeneous and noisy measurements, missing counterfactuals, and the requirement for human interpretable explanations. We present HistoricalML, a probabilistic neuro-symbolic framework that addresses these challenges through principled integration of (1) Bayesian uncertainty quantification to separate epistemic from aleatoric uncertainty, (2) structural causal models for counterfactual reasoning under confounding, (3) cooperative game theory (Shapley values) for fair allocation modeling, and (4) attention based neural architectures for context dependent factor weighting. We provide theoretical analysis showing that our approach achieves consistent estimation in the sparse data regime when strong priors from domain knowledge are available, and that Shapley based allocation satisfies axiomatic fairness guarantees that pure regression approaches cannot provide. We instantiate the framework on two historical case studies: the 19th century partition of Africa (N = 7 colonial powers) and the Second Punic War (N = 2 factions). Our model identifies Germany's +107.9 percent discrepancy as a quantifiable structural tension preceding World War I, with tension factor 36.43 and 0.79 naval arms race correlation. For the Punic Wars, Monte Carlo battle simulations achieve a 57.3 percent win probability for Carthage at Cannae and 57.8 percent for Rome at Zama, aligning with historical outcomes. Counterfactual analysis reveals that Carthaginian political support (support score 6.4 vs Napoleon's 7.1), rather than military capability, was the decisive factor.