W e present the Object-Based Sub-Environment Recognition (OBSER) framework, a novel Bayesian framework that infers three fundamental relationships between sub-environments and their constituent objects. In the OBSER framework, metric and self-supervised learning models estimate the object distributions of sub-environments on the latent space to compute these measures. Both theoretically and empirically, we validate the proposed framework by introducing the ( ϵ, δ) statistically separable (EDS) function which indicates the alignment of the representation. Our framework reliably performs inference in open-world and photorealistic environments and outperforms scene-based methods in chained retrieval tasks. The OBSER framework enables zero-shot recognition of environments to achieve autonomous environment understanding.

Bayesian optimisation in the latent space of a Variational AutoEncoder (VAE) is a powerful framework for optimisation tasks over complex structured domains, such as the space of scientifically interesting molecules. However, existing approaches tightly couple the surrogate and generative models, which can lead to suboptimal performance when the latent space is not tailored to specific tasks, which in turn has led to the proposal of increasingly sophisticated algorithms. In this work, we explore a new direction, instead proposing a decoupled approach that trains a generative model and a Gaussian Process (GP) surrogate separately, then combines them via a simple yet principled Bayesian update rule. This separation allows each component to focus on its strengths -- structure generation from the VAE and predictive modelling by the GP. We show that our decoupled approach improves our ability to identify high-potential candidates in molecular optimisation problems under constrained evaluation budgets.

The conditional density characterizes the distribution of a response variable $y$ given other predictor $x$, and plays a key role in many statistical tasks, including classification and outlier detection. Although there has been abundant work on the problem of Conditional Density Estimation (CDE) for a low-dimensional response in the presence of a high-dimensional predictor, little work has been done for a high-dimensional response such as images. The promising performance of normalizing flow (NF) neural networks in unconditional density estimation acts a motivating starting point. In this work, we extend NF neural networks when external $x$ is present. Specifically, they use the NF to parameterize a one-to-one transform between a high-dimensional $y$ and a latent $z$ that comprises two components \([z_P,z_N]\). The $z_P$ component is a low-dimensional subvector obtained from the posterior distribution of an elementary predictive model for $x$, such as logistic/linear regression. The $z_N$ component is a high-dimensional independent Gaussian vector, which explains the variations in $y$ not or less related to $x$. Unlike existing CDE methods, the proposed approach, coined Augmented Posterior CDE (AP-CDE), only requires a simple modification on the common normalizing flow framework, while significantly improving the interpretation of the latent component, since $z_P$ represents a supervised dimension reduction. In image analytics applications, AP-CDE shows good separation of $x$-related variations due to factors such as lighting condition and subject id, from the other random variations. Further, the experiments show that an unconditional NF neural network, based on an unsupervised model of $z$, such as Gaussian mixture, fails to generate interpretable results.

Implicit Quantile BART (IQ-BART) posits a non-parametric Bayesian model on the conditional quantile function, acting as a model over a conditional model for $Y$ given $X$. One of the key ingredients is augmenting the observed data $\{(Y_i,X_i)\}_{i=1}^n$ with uniformly sampled values $τ_i$ for $1\leq i\leq n$ which serve as training data for quantile function estimation. Using the fact that the location parameter $μ$ in a $τ$-tilted asymmetric Laplace distribution corresponds to the $τ^{th}$ quantile, we build a check-loss likelihood targeting $μ$ as the parameter of interest. We equip the check-loss likelihood parametrized by $μ=f(X,τ)$ with a BART prior on $f(\cdot)$, allowing the conditional quantile function to vary both in $X$ and $τ$. The posterior distribution over $μ(τ,X)$ can be then distilled for estimation of the {\em entire quantile function} as well as for assessing uncertainty through the variation of posterior draws. Simulation-based predictive inference is immediately available through inverse transform sampling using the learned quantile function. The sum-of-trees structure over the conditional quantile function enables flexible distribution-free regression with theoretical guarantees. As a byproduct, we investigate posterior mean quantile estimator as an alternative to the routine sample (posterior mode) quantile estimator. We demonstrate the power of IQ-BART on time series forecasting datasets where IQ-BART can capture multimodality in predictive distributions that might be otherwise missed using traditional parametric approaches.

This thesis focuses on the discovery of stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) from noisy and discrete time series. A major challenge is selecting the simplest possible correct model from vast libraries of candidate models, where standard information criteria (AIC, BIC) are often limited. We introduce PASTIS (Parsimonious Stochastic Inference), a new information criterion derived from extreme value theory. Its penalty term, $n_\mathcal{B} \ln(n_0/p)$, explicitly incorporates the size of the initial library of candidate parameters ($n_0$), the number of parameters in the considered model ($n_\mathcal{B}$), and a significance threshold ($p$). This significance threshold represents the probability of selecting a model containing more parameters than necessary when comparing many models. Benchmarks on various systems (Lorenz, Ornstein-Uhlenbeck, Lotka-Volterra for SDEs; Gray-Scott for SPDEs) demonstrate that PASTIS outperforms AIC, BIC, cross-validation (CV), and SINDy (a competing method) in terms of exact model identification and predictive capability. Furthermore, real-world data can be subject to large sampling intervals ($Δt$) or measurement noise ($σ$), which can impair model learning and selection capabilities. To address this, we have developed robust variants of PASTIS, PASTIS-$Δt$ and PASTIS-$σ$, thus extending the applicability of the approach to imperfect experimental data. PASTIS thus provides a statistically grounded, validated, and practical methodological framework for discovering simple models for processes with stochastic dynamics.

We develop a theory of transfer learning in infinitely wide neural networks where both the pretraining (source) and downstream (target) task can operate in a feature learning regime. We analyze both the Bayesian framework, where learning is described by a posterior distribution over the weights, and gradient flow training of randomly initialized networks trained with weight decay. Both settings track how representations evolve in both source and target tasks. The summary statistics of these theories are adapted feature kernels which, after transfer learning, depend on data and labels from both source and target tasks. Reuse of features during transfer learning is controlled by an elastic weight coupling which controls the reliance of the network on features learned during training on the source task. We apply our theory to linear and polynomial regression tasks as well as real datasets. Our theory and experiments reveal interesting interplays between elastic weight coupling, feature learning strength, dataset size, and source and target task alignment on the utility of transfer learning.

Reliability analysis typically relies on deterministic simulators, which yield repeatable outputs for identical inputs. However, many real-world systems display intrinsic randomness, requiring stochastic simulators whose outputs are random variables. This inherent variability must be accounted for in reliability analysis. While Monte Carlo methods can handle this, their high computational cost is often prohibitive. To address this, stochastic emulators have emerged as efficient surrogate models capable of capturing the random response of simulators at reduced cost. Although promising, current methods still require large training sets to produce accurate reliability estimates, which limits their practicality for expensive simulations. This work introduces an active learning framework to further reduce the computational burden of reliability analysis using stochastic emulators. We focus on stochastic polynomial chaos expansions (SPCE) and propose a novel learning function that targets regions of high predictive uncertainty relevant to failure probability estimation. To quantify this uncertainty, we exploit the asymptotic normality of the maximum likelihood estimator. The resulting method, named active learning stochastic polynomial chaos expansions (AL-SPCE), is applied to three test cases. Results demonstrate that AL-SPCE maintains high accuracy in reliability estimates while significantly improving efficiency compared to conventional surrogate-based methods and direct Monte Carlo simulation. This confirms the potential of active learning in enhancing the practicality of stochastic reliability analysis for complex, computationally expensive models.

In designing generative models, it is commonly believed that in order to learn useful latent structure, we face a fundamental tension between expressivity and structure. In this paper we challenge this view by proposing a new approach to training arbitrarily expressive generative models that simultaneously learn disentangled latent structure. This is accomplished by adding a simple decoder-only module to the head of an existing decoder block that can be arbitrarily complex. The module learns to process concept information by implicitly inverting linear representations from an encoder. Inspired by the notion of intervention in causal graphical models, our module selectively modifies its architecture during training, allowing it to learn a compact joint model over different contexts. We show how adding this module leads to disentangled representations that can be composed for out-of-distribution generation. To further validate our proposed approach, we prove a new identifiability result that extends existing work on identifying structured representations in nonlinear models.

Machine-learned systems are in widespread use for making decisions about humans, and it is important that they are fair, i.e., not biased against individuals based on sensitive attributes. We present a general framework of runtime verification of algorithmic fairness for systems whose models are unknown, but are assumed to have a Markov chain structure, with or without full observation of the state space. We introduce a specification language that can model many common algorithmic fairness properties, such as demographic parity, equal opportunity, and social burden. We build monitors that observe a long sequence of events as generated by a given system, and output, after each observation, a quantitative estimate of how fair or biased the system was on that run until that point in time. The estimate is proven to be correct modulo a variable error bound and a given confidence level, where the error bound gets tighter as the observed sequence gets longer. We present two categories of monitoring algorithms, namely ones with a uniform error bound across all time points, and ones with weaker non-uniform, pointwise error bounds at different time points. Our monitoring algorithms use statistical tools that are adapted to suit the dynamic requirements of monitoring and the special needs of the fairness specifications. Using a prototype implementation, we show how we can monitor if a bank is fair in giving loans to applicants from different social backgrounds, and if a college is fair in admitting students while maintaining a reasonable financial burden on the society. In these experiments, our monitors took less than a millisecond to update their verdicts after each observation.

The proliferation of ride-hailing aggregator platforms presents significant growth opportunities for ride-service providers by increasing order volume and gross merchandise value (GMV). On most ride-hailing aggregator platforms, service providers that offer lower fares are ranked higher in listings and, consequently, are more likely to be selected by passengers. This competitive ranking mechanism creates a strong incentive for service providers to adopt coupon strategies that lower prices to secure a greater number of orders, as order volume directly influences their long-term viability and sustainability. Thus, designing an effective coupon strategy that can dynamically adapt to market fluctuations while optimizing order acquisition under budget constraints is a critical research challenge. However, existing studies in this area remain scarce. To bridge this gap, we propose FCA-RL, a novel reinforcement learning-based subsidy strategy framework designed to rapidly adapt to competitors' pricing adjustments. Our approach integrates two key techniques: Fast Competition Adaptation (FCA), which enables swift responses to dynamic price changes, and Reinforced Lagrangian Adjustment (RLA), which ensures adherence to budget constraints while optimizing coupon decisions on new price landscape. Furthermore, we introduce RideGym, the first dedicated simulation environment tailored for ride-hailing aggregators, facilitating comprehensive evaluation and benchmarking of different pricing strategies without compromising real-world operational efficiency. Experimental results demonstrate that our proposed method consistently outperforms baseline approaches across diverse market conditions, highlighting its effectiveness in subsidy optimization for ride-hailing service providers.