A different approach, Reparameter-ized Discrete diffusion Models (RDMs) [62], establishes an alternative formulation for the reverse process, which simplifies the training objective to a weighted cross-entropy loss. This enables more flexible and adaptive decoding strategies, leading to significant performance gains over previous discrete diffusion models. Similarly, MD4 [63] derives a simple weighted integral of cross-entropy losses as the continuous-time variational objective of masked diffusion models, providing a simple and generalized framework for training DLMs. Another analogous approach is MDLM [64], which introduces a simplified, Rao-Blackwellized objective that takes the form of a weighted average of masked language modeling losses. Diffusion-LLM [65] demonstrates the scalability of DLMs by adapting pre-trained masked language models to diffusion paradigm and further task-specific finetuning and instruction finetuning, unlocking their versatility in solving general language tasks. Diffusion-NAT [66] unifies a discrete diffusion model with a PLM by reformulating the denoising process as a non-autoregressive masked token recovery task, allowing BART to act as an effective denoiser. Plaid [67] is the first diffusion language model trained to maximize data likelihood, demonstrating through scaling laws that it can outperform autoregressive models like GPT-2 on standard benchmarks. T o improve the training objective, SEDD [68] introduces a score entropy loss to directly learn the ratios of the data distribution, which serves as a discrete extension of score matching. Reparameterized Absorbing Discrete Diffusion (RADD) [69] reveals that the concrete score in absorbing diffusion can be expressed as a time-independent conditional probability of the clean data, multiplied by an analytic, time-dependent scalar.

This thesis develops methods for causal inference and causal representation learning (CRL) in high-dimensional, time-varying data. The first contribution introduces the Causal Dynamic Variational Autoencoder (CDVAE), a model for estimating Individual Treatment Effects (ITEs) by capturing unobserved heterogeneity in treatment response driven by latent risk factors that affect only outcomes. CDVAE comes with theoretical guarantees on valid latent adjustment and generalization bounds for ITE error. Experiments on synthetic and real datasets show that CDVAE outperforms baselines, and that state-of-the-art models greatly improve when augmented with its latent substitutes, approaching oracle performance without access to true adjustment variables. The second contribution proposes an efficient framework for long-term counterfactual regression based on RNNs enhanced with Contrastive Predictive Coding (CPC) and InfoMax. It captures long-range dependencies under time-varying confounding while avoiding the computational cost of transformers, achieving state-of-the-art results and introducing CPC into causal inference. The third contribution advances CRL by addressing how latent causes manifest in observed variables. We introduce a model-agnostic interpretability layer based on the geometry of the decoder Jacobian. A sparse self-expression prior induces modular, possibly overlapping groups of observed features aligned with shared latent influences. We provide recovery guarantees in both disjoint and overlapping settings and show that meaningful latent-to-observed structure can be recovered without anchor features or single-parent assumptions. Scalable Jacobian-based regularization techniques are also developed.

Across diverse domains--from complex systems and finance to high-energy physics and astrophysics--scientific inquiry often relies on deriving theoretical parameters from observational data [1]. At the core of this challenge lies the inverse problem: inferring the posterior distribution of theoretical parameters given a set of observables [2]. Traditional approaches for posterior estimation rely on sampling algorithms such as Markov Chain Monte Carlo (MCMC) [3, 4] and Nested Sampling (NS) [5]. In astrophysics and cosmology, implementations like emcee [6] and dynesty [7] have become standard tools. While these frameworks are statistically robust, they suffer significantly from the curse of dimensionality. In real-world scenarios, where the parameter space is high-dimensional and the likelihood function relies on computationally expensive simulators (e.g., in particle physics phenomenology [8]), convergence can take weeks or even months. Recent advances in machine learning have introduced Normalizing Flows (NFs) as a powerful alternative for probabilistic modelling [9, 10]. By learning a bijective mapping between a simple base distribution (e.g., a Gaussian) and the complex target distribution, NFs allow for exact density estimation and efficient sampling [11] from the target distribution. Modern architectures, such as RealNVP [12] and Neural Spline Flows [13], offer enough expressivity to model highly complex distributions.

Score matching estimators have garnered significant attention in recent years because they eliminate the need to compute normalizing constants, thereby mitigating the computational challenges associated with maximum likelihood estimation (MLE).While several studies have proposed score matching estimators for point processes, this work highlights the limitations of these existing methods, which stem primarily from the lack of a mathematically rigorous analysis of how score matching behaves on finite point processes -- special random configurations on bounded spaces where many of the usual assumptions and properties of score matching no longer hold. To this end, we develop a formal framework for score matching on finite point processes via Janossy measures and, within this framework, introduce an (autoregressive) weighted score-matching estimator, whose statistical properties we analyze in classical parametric settings. For general nonparametric (e.g., deep) point process models, we show that score matching alone does not uniquely identify the ground-truth distribution due to subtle normalization issues, and we propose a simple survival-classification augmentation that yields a complete, integration-free training objective for any intensity-based point process model for spatio-temporal case. Experiments on synthetic and real-world temporal and spatio-temporal datasets, demonstrate that our method accurately recovers intensities and achieves performance comparable to MLE with better efficiency.

Real-world tasks and environments exhibit differences from the static datasets that large language models (LLMs) are typically evaluated on. Such tasks can involve sequential interaction, requiring coherent updating of beliefs in light of new evidence, and making appropriate decisions based on those beliefs. Predicting how LLMs will perform in such dynamic environments is important, but can be tricky to determine from measurements in static settings. In this work, we examine two critical components of LLM performance: the ability of LLMs to coherently update their beliefs, and the extent to which the actions they take are consistent with those beliefs. First, we find that LLMs are largely inconsistent in how they update their beliefs; models can exhibit up to a 30% average difference between the directly elicited posterior, and the correct update of their prior. Second, we find that LLMs also often take actions which are inconsistent with the beliefs they hold. On a betting market, for example, LLMs often do not even bet in the same direction as their internally held beliefs over the underlying outcomes. We also find they have moderate self-inconsistency in how they respond to challenges by users to given answers. Finally, we show that the above properties hold even for strong models that obtain high accuracy or that are well-calibrated on the tasks at hand. Our results highlight the difficulties of predicting LLM behavior in complex real-world settings.

This article presents the first systematic review of unsupervised and semi-supervised computational text-based ideal point estimation (CT-IPE) algorithms, methods designed to infer latent political positions from textual data. These algorithms are widely used in political science, communication, computational social science, and computer science to estimate ideological preferences from parliamentary speeches, party manifestos, and social media. Over the past two decades, their development has closely followed broader NLP trends -- beginning with word-frequency models and most recently turning to large language models (LLMs). While this trajectory has greatly expanded the methodological toolkit, it has also produced a fragmented field that lacks systematic comparison and clear guidance for applied use. To address this gap, we identified 25 CT-IPE algorithms through a systematic literature review and conducted a manual content analysis of their modeling assumptions and development contexts. To compare them meaningfully, we introduce a conceptual framework that distinguishes how algorithms generate, capture, and aggregate textual variance. On this basis, we identify four methodological families -- word-frequency, topic modeling, word embedding, and LLM-based approaches -- and critically assess their assumptions, interpretability, scalability, and limitations. Our review offers three contributions. First, it provides a structured synthesis of two decades of algorithm development, clarifying how diverse methods relate to one another. Second, it translates these insights into practical guidance for applied researchers, highlighting trade-offs in transparency, technical requirements, and validation strategies that shape algorithm choice. Third, it emphasizes that differences in estimation outcomes across algorithms are themselves informative, underscoring the need for systematic benchmarking.

Gaussian process regression (GPR) is a popular nonparametric Bayesian method that provides predictive uncertainty estimates and is widely used in safety-critical applications. While prior research has introduced various uncertainty bounds, most existing approaches require access to specific input features, and rely on posterior mean and variance estimates or the tuning of hyperparameters. These limitations hinder robustness and fail to capture the model's global behavior in expectation. To address these limitations, we propose a chaining-based framework for estimating upper and lower bounds on the expected extreme values over unseen data, without requiring access to specific input features. We provide kernel-specific refinements for commonly used kernels such as RBF and Matérn, in which our bounds are tighter than generic constructions. We further improve numerical tightness by avoiding analytical relaxations. In addition to global estimation, we also develop a novel method for local uncertainty quantification at specified inputs. This approach leverages chaining geometry through partition diameters, adapting to local structures without relying on posterior variance scaling. Our experimental results validate the theoretical findings and demonstrate that our method outperforms existing approaches on both synthetic and real-world datasets.

The LogSumExp function, also known as the free energy, plays a central role in many important optimization problems, including entropy-regularized optimal transport and distributionally robust optimization (DRO). It is also the dual to the Kullback-Leibler (KL) divergence, which is widely used in machine learning. In practice, when the number of exponential terms inside the logarithm is large or infinite, optimization becomes challenging since computing the gradient requires differentiating every term. Previous approaches that replace the full sum with a small batch introduce significant bias. We propose a novel approximation to LogSumExp that can be efficiently optimized using stochastic gradient methods. This approximation is rooted in a sound modification of the KL divergence in the dual, resulting in a new $f$-divergence called the safe KL divergence. The accuracy of the approximation is controlled by a tunable parameter and can be made arbitrarily small. Like the LogSumExp, our approximation preserves convexity. Moreover, when applied to an $L$-smooth function bounded from below, the smoothness constant of the resulting objective scales linearly with $L$. Experiments in DRO and continuous optimal transport demonstrate the advantages of our approach over state-of-the-art baselines and the effective treatment of numerical issues associated with the standard LogSumExp and KL.

We introduce Sketch Tomography, an efficient procedure for quantum state tomography based on the classical shadow protocol used for quantum observable estimations. The procedure applies to the case where the ground truth quantum state is a matrix product state (MPS). The density matrix of the ground truth state admits a tensor train ansatz as a result of the MPS assumption, and we estimate the tensor components of the ansatz through a series of observable estimations, thus outputting an approximation of the density matrix. The procedure is provably convergent with a sample complexity that scales quadratically in the system size. We conduct extensive numerical experiments to show that the procedure outputs an accurate approximation to the quantum state. For observable estimation tasks involving moderately large subsystems, we show that our procedure gives rise to a more accurate estimation than the classical shadow protocol. We also show that sketch tomography is more accurate in observable estimation than quantum states trained from the maximum likelihood estimation formulation.

High quality data is needed to unlock the full potential of AI for end users. However finding new sources of such data is getting harder: most publicly-available human generated data will soon have been used. Additionally, publicly available data often is not representative of users of a particular system -- for example, a research speech dataset of contractors interacting with an AI assistant will likely be more homogeneous, well articulated and self-censored than real world commands that end users will issue. Therefore unlocking high-quality data grounded in real user interactions is of vital interest. However, the direct use of user data comes with significant privacy risks. Differential Privacy (DP) is a well established framework for reasoning about and limiting information leakage, and is a gold standard for protecting user privacy. The focus of this work, \emph{Differentially Private Synthetic data}, refers to synthetic data that preserves the overall trends of source data,, while providing strong privacy guarantees to individuals that contributed to the source dataset. DP synthetic data can unlock the value of datasets that have previously been inaccessible due to privacy concerns and can replace the use of sensitive datasets that previously have only had rudimentary protections like ad-hoc rule-based anonymization. In this paper we explore the full suite of techniques surrounding DP synthetic data, the types of privacy protections they offer and the state-of-the-art for various modalities (image, tabular, text and decentralized). We outline all the components needed in a system that generates DP synthetic data, from sensitive data handling and preparation, to tracking the use and empirical privacy testing. We hope that work will result in increased adoption of DP synthetic data, spur additional research and increase trust in DP synthetic data approaches.