We study the problem of learning mixtures of $k$ Gaussians in $d$ dimensions. We make no separation assumptions on the underlying mixture components: we only require that the covariance matrices have bounded condition number and that the means and covariances lie in a ball of bounded radius. We give an algorithm that draws $d^{\mathrm{poly}(k/\varepsilon)}$ samples from the target mixture, runs in sample-polynomial time, and constructs a sampler whose output distribution is $\varepsilon$-far from the unknown mixture in total variation. Prior works for this problem either (i) required exponential runtime in the dimension $d$, (ii) placed strong assumptions on the instance (e.g., spherical covariances or clusterability), or (iii) had doubly exponential dependence on the number of components $k$. Our approach departs from commonly used techniques for this problem like the method of moments. Instead, we leverage a recently developed reduction, based on diffusion models, from distribution learning to a supervised learning task called score matching. We give an algorithm for the latter by proving a structural result showing that the score function of a Gaussian mixture can be approximated by a piecewise-polynomial function, and there is an efficient algorithm for finding it. To our knowledge, this is the first example of diffusion models achieving a state-of-the-art theoretical guarantee for an unsupervised learning task.

Why is language vague? Vagueness may be explained and rationalized if it can be shown that vague language is more useful to speaker and hearer than precise language. In a well-known paper, Lipman proposes a game-theoretic account of vagueness in terms of mixed strategy that leads to a puzzle: vagueness cannot be strictly better than precision at equilibrium. More recently, \'Egr\'e, Spector, Mortier and Verheyen have put forward a Bayesian account of vagueness establishing that using vague words can be strictly more informative than using precise words. This paper proposes to compare both results and to explain why they are not in contradiction. Lipman's definition of vagueness relies exclusively on a property of signaling strategies, without making any assumptions about the lexicon, whereas \'Egr\'e et al.'s involves a layer of semantic content. We argue that the semantic account of vagueness is needed, and more adequate and explanatory of vagueness.

This paper investigates the consequences of encoding a $K$-valued categorical variable incorrectly as $K$ bits via one-hot encoding, when using a Na\"{\i}ve Bayes classifier. This gives rise to a product-of-Bernoullis (PoB) assumption, rather than the correct categorical Na\"{\i}ve Bayes classifier. The differences between the two classifiers are analysed mathematically and experimentally. In our experiments using probability vectors drawn from a Dirichlet distribution, the two classifiers are found to agree on the maximum a posteriori class label for most cases, although the posterior probabilities are usually greater for the PoB case.

Predictive uncertainty quantification is crucial for reliable decision-making in various applied domains. Bayesian neural networks offer a powerful framework for this task. However, defining meaningful priors and ensuring computational efficiency remain significant challenges, especially for complex real-world applications. This paper addresses these challenges by proposing a novel neural adaptive empirical Bayes (NA-EB) framework. NA-EB leverages a class of implicit generative priors derived from low-dimensional distributions. This allows for efficient handling of complex data structures and effective capture of underlying relationships in real-world datasets. The proposed NA-EB framework combines variational inference with a gradient ascent algorithm. This enables simultaneous hyperparameter selection and approximation of the posterior distribution, leading to improved computational efficiency. We establish the theoretical foundation of the framework through posterior and classification consistency. We demonstrate the practical applications of our framework through extensive evaluations on a variety of tasks, including the two-spiral problem, regression, 10 UCI datasets, and image classification tasks on both MNIST and CIFAR-10 datasets. The results of our experiments highlight the superiority of our proposed framework over existing methods, such as sparse variational Bayesian and generative models, in terms of prediction accuracy and uncertainty quantification.

There is growing recognition among financial institutions, financial regulators and policy makers of the importance of addressing nature-related risks and opportunities. Evaluating and assessing nature-related risks for financial institutions is challenging due to the large volume of heterogeneous data available on nature and the complexity of investment value chains and the various components' relationship to nature. The dual problem of scaling data analytics and analysing complex systems can be addressed using Artificial Intelligence (AI). We address issues such as plugging existing data gaps with discovered data, data estimation under uncertainty, time series analysis and (near) real-time updates. This report presents potential AI solutions for models of two distinct use cases, the Brazil Beef Supply Use Case and the Water Utility Use Case. Our two use cases cover a broad perspective within sustainable finance. The Brazilian cattle farming use case is an example of greening finance - integrating nature-related considerations into mainstream financial decision-making to transition investments away from sectors with poor historical track records and unsustainable operations. The deployment of nature-based solutions in the UK water utility use case is an example of financing green - driving investment to nature-positive outcomes. The two use cases also cover different sectors, geographies, financial assets and AI modelling techniques, providing an overview on how AI could be applied to different challenges relating to nature's integration into finance. This report is primarily aimed at financial institutions but is also of interest to ESG data providers, TNFD, systems modellers, and, of course, AI practitioners.

Machines that can replicate human intelligence with type 2 reasoning capabilities should be able to reason at multiple levels of spatio-temporal abstractions and scales using internal world models. Devising formalisms to develop such internal world models, which accurately reflect the causal hierarchies inherent in the dynamics of the real world, is a critical research challenge in the domains of artificial intelligence and machine learning. This thesis identifies several limitations with the prevalent use of state space models (SSMs) as internal world models and propose two new probabilistic formalisms namely Hidden-Parameter SSMs and Multi-Time Scale SSMs to address these drawbacks. The structure of graphical models in both formalisms facilitates scalable exact probabilistic inference using belief propagation, as well as end-to-end learning via backpropagation through time. This approach permits the development of scalable, adaptive hierarchical world models capable of representing nonstationary dynamics across multiple temporal abstractions and scales. Moreover, these probabilistic formalisms integrate the concept of uncertainty in world states, thus improving the system's capacity to emulate the stochastic nature of the real world and quantify the confidence in its predictions. The thesis also discuss how these formalisms are in line with related neuroscience literature on Bayesian brain hypothesis and predicitive processing. Our experiments on various real and simulated robots demonstrate that our formalisms can match and in many cases exceed the performance of contemporary transformer variants in making long-range future predictions. We conclude the thesis by reflecting on the limitations of our current models and suggesting directions for future research.

Language processing is at the heart of current developments in artificial intelligence, and quantum computers are becoming available at the same time. This has led to great interest in quantum natural language processing, and several early proposals and experiments. This paper surveys the state of this area, showing how NLP-related techniques have been used in quantum language processing. We examine the art of word embeddings and sequential models, proposing some avenues for future investigation and discussing the tradeoffs present in these directions. We also highlight some recent methods to compute attention in transformer models, and perform grammatical parsing. We also introduce a new quantum design for the basic task of text encoding (representing a string of characters in memory), which has not been addressed in detail before. Quantum theory has contributed toward quantifying uncertainty and explaining "What is intelligence?" In this context, we argue that "hallucinations" in modern artificial intelligence systems are a misunderstanding of the way facts are conceptualized: language can express many plausible hypotheses, of which only a few become actual.

Adversarial examples have been shown to cause neural networks to fail on a wide range of vision and language tasks, but recent work has claimed that Bayesian neural networks (bnns) are inherently robust to adversarial perturbations. In this work, we examine this claim. To study the adversarial robustness of bnns, we investigate whether it is possible to successfully break state-of-the-art bnn inference methods and prediction pipelines using even relatively unsophisticated attacks for three tasks: (1) label prediction under the posterior predictive mean, (2) adversarial example detection with Bayesian predictive uncertainty, and (3) semantic shift detection. We find that bnns trained with state-of-the-art approximate inference methods, and even bnns trained with Hamiltonian Monte Carlo, are highly susceptible to adversarial attacks. We also identify various conceptual and experimental errors in previous works that claimed inherent adversarial robustness of bnns and conclusively demonstrate that bnns and uncertainty-aware Bayesian prediction pipelines are not inherently robust against adversarial attacks.

Fully supervised models are predominant in Bayesian active learning. We argue that their neglect of the information present in unlabelled data harms not just predictive performance but also decisions about what data to acquire. Our proposed solution is a simple framework for semi-supervised Bayesian active learning. We find it produces better-performing models than either conventional Bayesian active learning or semi-supervised learning with randomly acquired data. It is also easier to scale up than the conventional approach. As well as supporting a shift towards semi-supervised models, our findings highlight the importance of studying models and acquisition methods in conjunction.

We propose data-dependent uniform generalization bounds by approaching the problem from a PAC-Bayesian perspective. We first apply the PAC-Bayesian framework on `random sets' in a rigorous way, where the training algorithm is assumed to output a data-dependent hypothesis set after observing the training data. This approach allows us to prove data-dependent bounds, which can be applicable in numerous contexts. To highlight the power of our approach, we consider two main applications. First, we propose a PAC-Bayesian formulation of the recently developed fractal-dimension-based generalization bounds. The derived results are shown to be tighter and they unify the existing results around one simple proof technique. Second, we prove uniform bounds over the trajectories of continuous Langevin dynamics and stochastic gradient Langevin dynamics. These results provide novel information about the generalization properties of noisy algorithms.