In Bayesian structure learning, we are interested in inferring a distribution over the directed acyclic graph (DAG) structure of Bayesian networks, from data. Defining such a distribution is very challenging, due to the combinatorially large sample space, and approximations based on MCMC are often required. Recently, a novel class of probabilistic models, called Generative Flow Networks (GFlowNets), have been introduced as a general framework for generative modeling of discrete and composite objects, such as graphs. In this work, we propose to use a GFlowNet as an alternative to MCMC for approximating the posterior distribution over the structure of Bayesian networks, given a dataset of observations. Generating a sample DAG from this approximate distribution is viewed as a sequential decision problem, where the graph is constructed one edge at a time, based on learned transition probabilities. Through evaluation on both simulated and real data, we show that our approach, called DAG-GFlowNet, provides an accurate approximation of the posterior over DAGs, and it compares favorably against other methods based on MCMC or variational inference.

Real-world applications of reinforcement learning for recommendation and experimentation faces a practical challenge: the relative reward of different bandit arms can evolve over the lifetime of the learning agent. To deal with these non-stationary cases, the agent must forget some historical knowledge, as it may no longer be relevant to minimise regret. We present a solution to handling non-stationarity that is suitable for deployment at scale, to provide business operators with automated adaptive optimisation. Our solution aims to provide interpretable learning that can be trusted by humans, whilst responding to non-stationarity to minimise regret. To this end, we develop an adaptive Bayesian learning agent that employs a novel form of dynamic memory. It enables interpretability through statistical hypothesis testing, by targeting a set point of statistical power when comparing rewards and adjusting its memory dynamically to achieve this power. By design, the agent is agnostic to different kinds of non-stationarity. Using numerical simulations, we compare its performance against an existing proposal and show that, under multiple non-stationary scenarios, our agent correctly adapts to real changes in the true rewards. In all bandit solutions, there is an explicit trade-off between learning and achieving maximal performance. Our solution sits on a different point on this trade-off when compared to another similarly robust approach: we prioritise interpretability, which relies on more learning, at the cost of some regret. We describe the architecture of a large-scale deployment of automatic optimisation-as-a-service where our agent achieves interpretability whilst adapting to changing circumstances.

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Recent performance breakthroughs in Artificial intelligence (AI) and Machine learning (ML), especially advances in Deep learning (DL), the availability of powerful, easy-to-use ML libraries (e.g., scikit-learn, TensorFlow, PyTorch.), and increasing computational power have led to unprecedented interest in AI/ML among nuclear engineers. For physics-based computational models, Verification, Validation and Uncertainty Quantification (VVUQ) have been very widely investigated and a lot of methodologies have been developed. However, VVUQ of ML models has been relatively less studied, especially in nuclear engineering. In this work, we focus on UQ of ML models as a preliminary step of ML VVUQ, more specifically, Deep Neural Networks (DNNs) because they are the most widely used supervised ML algorithm for both regression and classification tasks. This work aims at quantifying the prediction, or approximation uncertainties of DNNs when they are used as surrogate models for expensive physical models. Three techniques for UQ of DNNs are compared, namely Monte Carlo Dropout (MCD), Deep Ensembles (DE) and Bayesian Neural Networks (BNNs). Two nuclear engineering examples are used to benchmark these methods, (1) time-dependent fission gas release data using the Bison code, and (2) void fraction simulation based on the BFBT benchmark using the TRACE code. It was found that the three methods typically require different DNN architectures and hyperparameters to optimize their performance. The UQ results also depend on the amount of training data available and the nature of the data. Overall, all these three methods can provide reasonable estimations of the approximation uncertainties. The uncertainties are generally smaller when the mean predictions are close to the test data, while the BNN methods usually produce larger uncertainties than MCD and DE.

Score-based generative models have excellent performance in terms of generation quality and likelihood. They model the data distribution by matching a parameterized score network with first-order data score functions. The score network can be used to define an ODE ("score-based diffusion ODE") for exact likelihood evaluation. However, the relationship between the likelihood of the ODE and the score matching objective is unclear. In this work, we prove that matching the first-order score is not sufficient to maximize the likelihood of the ODE, by showing a gap between the maximum likelihood and score matching objectives. To fill up this gap, we show that the negative likelihood of the ODE can be bounded by controlling the first, second, and third-order score matching errors; and we further present a novel high-order denoising score matching method to enable maximum likelihood training of score-based diffusion ODEs. Our algorithm guarantees that the higher-order matching error is bounded by the training error and the lower-order errors. We empirically observe that by high-order score matching, score-based diffusion ODEs achieve better likelihood on both synthetic data and CIFAR-10, while retaining the high generation quality.

In most data-scientific approaches, the principle of Maximum Entropy (MaxEnt) is used to a posteriori justify some parametric model which has been already chosen based on experience, prior knowledge or computational simplicity. In a perpendicular formulation to conventional model building, we start from the linear system of phenomenological constraints and asymptotically derive the distribution over all viable distributions that satisfy the provided set of constraints. The MaxEnt distribution plays a special role, as it is the most typical among all phenomenologically viable distributions representing a good expansion point for large-N techniques. This enables us to consistently formulate hypothesis testing in a fully-data driven manner. The appropriate parametric model which is supported by the data can be always deduced at the end of model selection. In the MaxEnt framework, we recover major scores and selection procedures used in multiple applications and assess their ability to capture associations in the data-generating process and identify the most generalizable model. This data-driven counterpart of standard model selection demonstrates the unifying prospective of the deductive logic advocated by MaxEnt principle, while potentially shedding new insights to the inverse problem.

This is a collection of (mostly) pen-and-paper exercises in machine learning. The exercises are on the following topics: linear algebra, optimisation, directed graphical models, undirected graphical models, expressive power of graphical models, factor graphs and message passing, inference for hidden Markov models, model-based learning (including ICA and unnormalised models), sampling and Monte-Carlo integration, and variational inference.

Machine learning models deployed on medical imaging tasks must be equipped with out-of-distribution detection capabilities in order to avoid erroneous predictions. It is unsure whether out-of-distribution detection models reliant on deep neural networks are suitable for detecting domain shifts in medical imaging. Gaussian Processes can reliably separate in-distribution data points from out-of-distribution data points via their mathematical construction. Hence, we propose a parameter efficient Bayesian layer for hierarchical convolutional Gaussian Processes that incorporates Gaussian Processes operating in Wasserstein-2 space to reliably propagate uncertainty. This directly replaces convolving Gaussian Processes with a distance-preserving affine operator on distributions. Our experiments on brain tissue-segmentation show that the resulting architecture approaches the performance of well-established deterministic segmentation algorithms (U-Net), which has not been achieved with previous hierarchical Gaussian Processes. Moreover, by applying the same segmentation model to out-of-distribution data (i.e., images with pathology such as brain tumors), we show that our uncertainty estimates result in out-of-distribution detection that outperforms the capabilities of previous Bayesian networks and reconstruction-based approaches that learn normative distributions. To facilitate future work our code is publicly available.

Deep learning[133] uses several layers of neurons between the network's inputs and outputs. The multiple layers can progressively extract higher-level features from the raw input. For example, in image processing, lower layers may identify edges, while higher layers may identify the concepts relevant to a human such as digits or letters or faces.[134] Deep learning has drastically improved the performance of programs in many important subfields of artificial intelligence, including computer vision, speech recognition, image classification[135] and others. Deep learning often uses convolutional neural networks for many or all of its layers.

Spike-and-slab priors are commonly used for Bayesian variable selection, due to their interpretability and favorable statistical properties. However, existing samplers for spike-and-slab posteriors incur prohibitive computational costs when the number of variables is large. In this article, we propose Scalable Spike-and-Slab ($S^3$), a scalable Gibbs sampling implementation for high-dimensional Bayesian regression with the continuous spike-and-slab prior of George and McCulloch (1993). For a dataset with $n$ observations and $p$ covariates, $S^3$ has order $\max\{ n^2 p_t, np \}$ computational cost at iteration $t$ where $p_t$ never exceeds the number of covariates switching spike-and-slab states between iterations $t$ and $t-1$ of the Markov chain. This improves upon the order $n^2 p$ per-iteration cost of state-of-the-art implementations as, typically, $p_t$ is substantially smaller than $p$. We apply $S^3$ on synthetic and real-world datasets, demonstrating orders of magnitude speed-ups over existing exact samplers and significant gains in inferential quality over approximate samplers with comparable cost.