Differential privacy (DP) is the standard for privacy-preserving analysis, and introduces a fundamental trade-off between privacy guarantees and model performance. Selecting the optimal balance is a critical challenge that can be framed as a multi-objective optimization (MOO) problem where one first discovers the set of optimal trade-offs (the Pareto front) and then learns a decision-maker's preference over them. While a rich body of work on interactive MOO exists, the standard approach -- modeling the objective functions with generic surrogates and learning preferences from simple pairwise feedback -- is inefficient for DP because it fails to leverage the problem's unique structure: a point on the Pareto front can be generated directly by maximizing accuracy for a fixed privacy level. Motivated by this property, we first derive the shape of the trade-off theoretically, which allows us to model the Pareto front directly and efficiently. To address inefficiency in preference learning, we replace pairwise comparisons with a more informative interaction. In particular, we present the user with hypothetical trade-off curves and ask them to pick their preferred trade-off. Our experiments on differentially private logistic regression and deep transfer learning across six real-world datasets show that our method converges to the optimal privacy-accuracy trade-off with significantly less computational cost and user interaction than baselines.

We formulate the inverse problem in a Bayesian framework and aim to train a generative model that allows us to simulate (i.e., sample from the likelihood) and do inference (i.e., sample from the posterior). We review the use of triangular normalizing flows for conditional sampling in this context and show how to combine two such triangular maps (an upper and a lower one) in to one invertible mapping that can be used for simulation and inference. We work out several useful properties of this invertible generative model and propose a possible training loss for training the map directly. We illustrate the workings of this new approach to conditional generative modeling numerically on a few stylized examples.

Stein variational gradient descent (SVGD) is a kernel-based and non-parametric particle method for sampling from a target distribution, such as in Bayesian inference and other machine learning tasks. Different from other particle methods, SVGD does not require estimating the score, which is the gradient of the log-density. However, in practice, SVGD can be slow compared to score-estimation-based sampling algorithms. To design a fast and efficient high-dimensional sampling algorithm with the advantages of SVGD, we introduce accelerated SVGD (ASVGD), based on an accelerated gradient flow in a metric space of probability densities following Nesterov's method. We then derive a momentum-based discrete-time sampling algorithm, which evolves a set of particles deterministically. To stabilize the particles' position update, we also include a Wasserstein metric regularization. This paper extends the conference version \cite{SL2025}. For the bilinear kernel and Gaussian target distributions, we study the kernel parameter and damping parameters with an optimal convergence rate of the proposed dynamics. This is achieved by analyzing the linearized accelerated gradient flows at the equilibrium. Interestingly, the optimal parameter is a constant, which does not depend on the covariance of the target distribution. For the generalized kernel functions, such as the Gaussian kernel, numerical examples with varied target distributions demonstrate the effectiveness of ASVGD compared to SVGD and other popular sampling methods. Furthermore, we show that in the setting of Bayesian neural networks, ASVGD outperforms SVGD significantly in terms of log-likelihood and total iteration times.

Simulation-based inference (SBI) enables Bayesian analysis when the likelihood is intractable but model simulations are available. Recent advances in statistics and machine learning, including Approximate Bayesian Computation and deep generative models, have expanded the applicability of SBI, yet these methods often face challenges in moderate to high-dimensional parameter spaces. Motivated by the success of gradient-based Monte Carlo methods in Bayesian sampling, we propose a novel SBI method that integrates score matching with Langevin dynamics to explore complex posterior landscapes more efficiently in such settings. Our approach introduces tailored score-matching procedures for SBI, including a localization scheme that reduces simulation costs and an architectural regularization that embeds the statistical structure of log-likelihood scores to improve score-matching accuracy. We provide theoretical analysis of the method and illustrate its practical benefits on benchmark tasks and on more challenging problems in moderate to high dimensions, where it performs favorably compared to existing approaches.

We investigate partially identifiable queries in a class of causal models. We focus on acyclic Structural Causal Models that are quasi-Markovian (that is, each endogenous variable is connected with at most one exogenous confounder). We look into scenarios where endogenous variables are observed (and a distribution over them is known), while exogenous variables are not fully specified. This leads to a representation that is in essence a Bayesian network where the distribution of root variables is not uniquely determined. In such circumstances, it may not be possible to precisely compute a probability value of interest. We thus study the computation of tight probability bounds, a problem that has been solved by multilinear programming in general, and by linear programming when a single confounded component is intervened upon. We present a new algorithm to simplify the construction of such programs by exploiting input probabilities over endogenous variables. For scenarios with a single intervention, we apply column generation to compute a probability bound through a sequence of auxiliary linear integer programs, thus showing that a representation with polynomial cardinality for exogenous variables is possible. Experiments show column generation techniques to be superior to existing methods.

While calibration of probabilistic predictions has been widely studied, this paper rather addresses calibration of likelihood functions. This has been discussed, especially in biometrics, in cases with only two exhaustive and mutually exclusive hypotheses (classes) where likelihood functions can be written as log-likelihood-ratios (LLRs). After defining calibration for LLRs and its connection with the concept of weight-of-evidence, we present the idempotence property and its associated constraint on the distribution of the LLRs. Although these results have been known for decades, they have been limited to the binary case. Here, we extend them to cases with more than two hypotheses by using the Aitchison geometry of the simplex, which allows us to recover, in a vector form, the additive form of the Bayes' rule; extending therefore the LLR and the weight-of-evidence to any number of hypotheses. Especially, we extend the definition of calibration, the idempotence, and the constraint on the distribution of likelihood functions to this multiple hypotheses and multiclass counterpart of the LLR: the isometric-log-ratio transformed likelihood function. This work is mainly conceptual, but we still provide one application to machine learning by presenting a non-linear discriminant analysis where the discriminant components form a calibrated likelihood function over the classes, improving therefore the interpretability and the reliability of the method.

This article focuses on covariance estimation for multi-view data. Popular approaches rely on factor-analytic decompositions that have shared and view-specific latent factors. Posterior computation is conducted via expensive and brittle Markov chain Monte Carlo (MCMC) sampling or variational approximations that underestimate uncertainty and lack theoretical guarantees. Our proposed methodology employs spectral decompositions to estimate and align latent factors that are active in at least one view. Conditionally on these factors, we choose jointly conjugate prior distributions for factor loadings and residual variances. The resulting posterior is a simple product of normal-inverse gamma distributions for each variable, bypassing MCMC and facilitating posterior computation. We prove favorable increasing-dimension asymptotic properties, including posterior contraction and central limit theorems for point estimators. We show excellent performance in simulations, including accurate uncertainty quantification, and apply the methodology to integrate four high-dimensional views from a multi-omics dataset of cancer cell samples.

Generating synthetic datasets that accurately reflect real-world observational data is critical for evaluating causal estimators, but remains a challenging task. Existing generative methods offer a solution by producing synthetic datasets anchored in the observed data (source data) while allowing variation in key parameters such as the treatment effect and amount of confounding bias. However, existing methods typically require users to provide point estimates of such parameters (rather than distributions) and fixed estimates (rather than estimates that can be improved with reference to the source data). This denies users the ability to express uncertainty over parameter values and removes the potential for posterior inference, potentially leading to unreliable estimator comparisons. We introduce simulation-based inference for causal evaluation (SBICE), a framework that models generative parameters as uncertain and infers their posterior distribution given a source dataset. Leveraging techniques in simulation-based inference, SBICE identifies parameter configurations that produce synthetic datasets closely aligned with the source data distribution. Empirical results demonstrate that SBICE improves the reliability of estimator evaluations by generating more realistic datasets, which supports a robust and data-consistent approach to causal benchmarking under uncertainty.

Human cognition is profoundly shaped by the environments in which it unfolds. Yet, it remains an open question whether learning and decision making can be explained as a principled adaptation to the statistical structure of real-world tasks. We introduce ecologically rational analysis, a computational framework that unifies the normative foundations of rational analysis with ecological grounding. Leveraging large language models to generate ecologically valid cognitive tasks at scale, and using meta-learning to derive rational models optimized for these environments, we develop a new class of learning algorithms: Ecologically Rational Meta-learned Inference (ERMI). ERMI internalizes the statistical regularities of naturalistic problem spaces and adapts flexibly to novel situations, without requiring hand-crafted heuristics or explicit parameter updates. We show that ERMI captures human behavior across 15 experiments spanning function learning, category learning, and decision making, outperforming several established cognitive models in trial-by-trial prediction. Our results suggest that much of human cognition may reflect adaptive alignment to the ecological structure of the problems we encounter in everyday life.

Diffusion models have shown remarkable promise for image restoration by leveraging powerful priors. Prominent methods typically frame the restoration problem within a Bayesian inference framework, which iteratively combines a denoising step with a likelihood guidance step. However, the interactions between these two components in the generation process remain underexplored. In this paper, we analyze the underlying gradient dynamics of these components and identify significant instabilities. Specifically, we demonstrate conflicts between the prior and likelihood gradient directions, alongside temporal fluctuations in the likelihood gradient itself. We show that these instabilities disrupt the generative process and compromise restoration performance. To address these issues, we propose Stabilized Progressive Gradient Diffusion (SPGD), a novel gradient management technique. SPGD integrates two synergistic components: (1) a progressive likelihood warm-up strategy to mitigate gradient conflicts; and (2) adaptive directional momentum (ADM) smoothing to reduce fluctuations in the likelihood gradient. Extensive experiments across diverse restoration tasks demonstrate that SPGD significantly enhances generation stability, leading to state-of-the-art performance in quantitative metrics and visually superior results. Code is available at https://github.com/74587887/SPGD.