The problem of the priors is well known: it concerns the challenge of identifying norms that govern one's prior credences. I argue that a key to addressing this problem lies in considering what I call the problem of the posteriors -- the challenge of identifying norms that directly govern one's posterior credences, which then induce constraints on the priors via the diachronic requirement of conditionalization. This forward-looking approach can be summarized as: Think ahead, work backward. Although this idea can be traced to Freedman (1963), Carnap (1963), and Shimony (1970), it has received little attention in philosophy. In this paper, I initiate a systematic defense of forward-looking Bayesianism, addressing potential objections from more traditional views (both subjectivist and objectivist) and arguing for its advantages. In particular, I develop a specific approach to forward-looking Bayesianism -- one that treats the convergence of posterior credences to the truth as a fundamental rather than derived normative requirement. This approach, called convergentist Bayesianism, is argued to be crucial for a Bayesian foundation of Ockham's razor and related inference methods in statistics and machine learning.

Cognition swiftly breaks high-dimensional sensory streams into familiar parts and uncovers their relations. Why do structures emerge, and how do they enable learning, generalization, and prediction? What computational principles underlie this core aspect of perception and intelligence? A sensory stream, simplified, is a one-dimensional sequence. In learning such sequences, we naturally segment them into parts -- a process known as chunking. In the first project, I investigated factors influencing chunking in a serial reaction time task and showed that humans adapt to underlying chunks while balancing speed and accuracy. Building on this, I developed models that learn chunks and parse sequences chunk by chunk. Normatively, I proposed chunking as a rational strategy for discovering recurring patterns and nested hierarchies, enabling efficient sequence factorization. Learned chunks serve as reusable primitives for transfer, composition, and mental simulation -- letting the model compose the new from the known. I demonstrated this model's ability to learn hierarchies in single and multi-dimensional sequences and highlighted its utility for unsupervised pattern discovery. The second part moves from concrete to abstract sequences. I taxonomized abstract motifs and examined their role in sequence memory. Behavioral evidence suggests that humans exploit pattern redundancies for compression and transfer. I proposed a non-parametric hierarchical variable model that learns both chunks and abstract variables, uncovering invariant symbolic patterns. I showed its similarity to human learning and compared it to large language models. Taken together, this thesis suggests that chunking and abstraction as simple computational principles enable structured knowledge acquisition in hierarchically organized sequences, from simple to complex, concrete to abstract.

This article proposes numerically robust algorithms for Gaussian state estimation with singular observation noise. Our approach combines a series of basis changes with Bayes' rule, transforming the singular estimation problem into a nonsingular one with reduced state dimension. In addition to ensuring low runtime and numerical stability, our proposal facilitates marginal-likelihood computations and Gauss-Markov representations of the posterior process. We analyse the proposed method's computational savings and numerical robustness and validate our findings in a series of simulations.

An important step in understanding how children acquire languages is studying how infants learn word segmentation. It has been established in previous research that infants may use statistical regularities in speech to learn word segmentation. The research of Goldwater et al., demonstrated that incorporating context in models improves their ability to learn word segmentation. We implemented two of their models, a unigram and bigram model, to examine how context can improve statistical word segmentation. The results are consistent with our hypothesis that the bigram model outperforms the unigram model at predicting word segmentation. Extending the work of Goldwater et al., we also explored basic ways to model how young children might use previously learned words to segment new utterances.

Heterogeneous tabular data poses unique challenges in generative modelling due to its fundamentally different underlying data structure compared to homogeneous modalities, such as images and text. Although previous research has sought to adapt the successes of generative modelling in homogeneous modalities to the tabular domain, defining an effective generator for tabular data remains an open problem. One major reason is that the evaluation criteria inherited from other modalities often fail to adequately assess whether tabular generative models effectively capture or utilise the unique structural information encoded in tabular data. In this paper, we carefully examine the limitations of the prevailing evaluation framework and introduce $\textbf{TabStruct}$, a novel evaluation benchmark that positions structural fidelity as a core evaluation dimension. Specifically, TabStruct evaluates the alignment of causal structures in real and synthetic data, providing a direct measure of how effectively tabular generative models learn the structure of tabular data. Through extensive experiments using generators from eight categories on seven datasets with expert-validated causal graphical structures, we show that structural fidelity offers a task-independent, domain-agnostic evaluation dimension. Our findings highlight the importance of tabular data structure and offer practical guidance for developing more effective and robust tabular generative models. Code is available at https://github.com/SilenceX12138/TabStruct.

In this paper, we address the problem of human trajectory forecasting, which aims to predict the inherently multi-modal future movements of humans based on their past trajectories and other contextual cues. We propose a novel motion prediction conditional flow matching model, termed MoFlow, to predict K-shot future trajectories for all agents in a given scene. We design a novel flow matching loss function that not only ensures at least one of the $K$ sets of future trajectories is accurate but also encourages all $K$ sets of future trajectories to be diverse and plausible. Furthermore, by leveraging the implicit maximum likelihood estimation (IMLE), we propose a novel distillation method for flow models that only requires samples from the teacher model. Extensive experiments on the real-world datasets, including SportVU NBA games, ETH-UCY, and SDD, demonstrate that both our teacher flow model and the IMLE-distilled student model achieve state-of-the-art performance. These models can generate diverse trajectories that are physically and socially plausible. Moreover, our one-step student model is $\textbf{100}$ times faster than the teacher flow model during sampling. The code, model, and data are available at our project page: https://moflow-imle.github.io

Quantification learning is the task of predicting the label distribution of a set of instances. We study this problem in the context of graph-structured data, where the instances are vertices. Previously, this problem has only been addressed via node clustering methods. In this paper, we extend the popular Adjusted Classify & Count (ACC) method to graphs. We show that the prior probability shift assumption upon which ACC relies is often not fulfilled and propose two novel graph quantification techniques: Structural importance sampling (SIS) makes ACC applicable in graph domains with covariate shift. Neighborhood-aware ACC improves quantification in the presence of non-homophilic edges. We show the effectiveness of our techniques on multiple graph quantification tasks.

Research on learned cardinality estimation has achieved significant progress in recent years. However, existing methods still face distinct challenges that hinder their practical deployment in production environments. We conceptualize these challenges as the "Trilemma of Cardinality Estimation", where learned cardinality estimation methods struggle to balance generality, accuracy, and updatability. To address these challenges, we introduce DistJoin, a join cardinality estimator based on efficient distribution prediction using multi-autoregressive models. Our contributions are threefold: (1) We propose a method for estimating both equi and non-equi join cardinality by leveraging the conditional probability distributions of individual tables in a decoupled manner. (2) To meet the requirements of efficient training and inference for DistJoin, we develop Adaptive Neural Predicate Modulation (ANPM), a high-throughput conditional probability distribution estimation model. (3) We formally analyze the variance of existing similar methods and demonstrate that such approaches suffer from variance accumulation issues. To mitigate this problem, DistJoin employs a selectivity-based approach rather than a count-based approach to infer join cardinality, effectively reducing variance. In summary, DistJoin not only represents the first data-driven method to effectively support both equi and non-equi joins but also demonstrates superior accuracy while enabling fast and flexible updates. We evaluate DistJoin on JOB-light and JOB-light-ranges, extending the evaluation to non-equi join conditions. The results demonstrate that our approach achieves the highest accuracy, robustness to data updates, generality, and comparable update and inference speed relative to existing methods.

Bayesian inference for hierarchical models can be very challenging. MCMC methods have difficulty scaling to large models with many observations and latent variables. While variational inference (VI) and reweighted wake-sleep (RWS) can be more scalable, they are gradient-based methods and so often require many iterations to converge. Our key insight was that modern massively parallel importance weighting methods (Bowyer et al., 2024) give fast and accurate posterior moment estimates, and we can use these moment estimates to rapidly learn an approximate posterior. Specifically, we propose using expectation maximization to fit the approximate posterior, which we call QEM. The expectation step involves computing the posterior moments using high-quality massively parallel estimates from Bowyer et al. (2024). The maximization step involves fitting the approximate posterior using these moments, which can be done straightforwardly for simple approximate posteriors such as Gaussian, Gamma, Beta, Dirichlet, Binomial, Multinomial, Categorical, etc. (or combinations thereof). We show that QEM is faster than state-of-the-art, massively parallel variants of RWS and VI, and is invariant to reparameterizations of the model that dramatically slow down gradient based methods.

Traditional Federated Learning (FL) necessitates numerous rounds of communication between the server and clients, posing significant challenges including high communication costs, connection drop risks and susceptibility to privacy attacks. One-shot FL has become a compelling learning paradigm to overcome above drawbacks by enabling the training of a global server model via a single communication round. However, existing one-shot FL methods suffer from expensive computation cost on the server or clients and cannot deal with non-IID (Independent and Identically Distributed) data stably and effectively. To address these challenges, this paper proposes FedCGS, a novel Federated learning algorithm that Capture Global feature Statistics leveraging pre-trained models. With global feature statistics, we achieve training-free and heterogeneity-resistant one-shot FL. Furthermore, we extend its application to personalization scenario, where clients only need execute one extra communication round with server to download global statistics. Extensive experimental results demonstrate the effectiveness of our methods across diverse data heterogeneity settings. Code is available at https://github.com/Yuqin-G/FedCGS.