The conventional approach to Bayesian decision-theoretic experiment design involves searching over possible experiments to select a design that maximizes the expected value of a specified utility function. The expectation is over the joint distribution of all unknown variables implied by the statistical model that will be used to analyze the collected data. The utility function defines the objective of the experiment where a common utility function is the information gain. This article introduces an expanded framework for this process, where we go beyond the traditional Expected Information Gain criteria and introduce the Expected General Information Gain which measures robustness to the model discrepancy and Expected Discriminatory Information as a criterion to quantify how well an experiment can detect model discrepancy. The functionality of the framework is showcased through its application to a scenario involving a linearized spring mass damper system and an F-16 model where the model discrepancy is taken into account while doing Bayesian optimal experiment design.

We propose the NeRF-LEBM, a likelihood-based top-down 3D-aware 2D image generative model that incorporates 3D representation via Neural Radiance Fields (NeRF) and 2D imaging process via differentiable volume rendering. The model represents an image as a rendering process from 3D object to 2D image and is conditioned on some latent variables that account for object characteristics and are assumed to follow informative trainable energy-based prior models. We propose two likelihood-based learning frameworks to train the NeRF-LEBM: (i) maximum likelihood estimation with Markov chain Monte Carlo-based inference and (ii) variational inference with the reparameterization trick. We study our models in the scenarios with both known and unknown camera poses. Experiments on several benchmark datasets demonstrate that the NeRF-LEBM can infer 3D object structures from 2D images, generate 2D images with novel views and objects, learn from incomplete 2D images, and learn from 2D images with known or unknown camera poses.

We study offline reinforcement learning (RL) in partially observable Markov decision processes. In particular, we aim to learn an optimal policy from a dataset collected by a behavior policy which possibly depends on the latent state. Such a dataset is confounded in the sense that the latent state simultaneously affects the action and the observation, which is prohibitive for existing offline RL algorithms. To this end, we propose the \underline{P}roxy variable \underline{P}essimistic \underline{P}olicy \underline{O}ptimization (\texttt{P3O}) algorithm, which addresses the confounding bias and the distributional shift between the optimal and behavior policies in the context of general function approximation. At the core of \texttt{P3O} is a coupled sequence of pessimistic confidence regions constructed via proximal causal inference, which is formulated as minimax estimation. Under a partial coverage assumption on the confounded dataset, we prove that \texttt{P3O} achieves a $n^{-1/2}$-suboptimality, where $n$ is the number of trajectories in the dataset. To our best knowledge, \texttt{P3O} is the first provably efficient offline RL algorithm for POMDPs with a confounded dataset.

In today's rapidly advancing world of Artificial Intelligence (AI), the need for explainable AI has become more critical than ever. As AI systems are being increasingly integrated into various aspects of our daily lives, it is crucial to understand how these systems make decisions and provide explanations for their actions. Bayesian networks, a powerful and versatile graphical modeling technique, are gaining prominence as a tool for building explainable AI models. In this blog, we will demystify Bayesian networks and explore their relevance in the field of AI. We will delve into the fundamentals of Bayesian networks, their applications in AI, and how they enable explainable AI.

Inverse problems are ubiquitous in the sciences and engineering. Two categories of inverse problems concerning a physical system are (1) estimate parameters in a model of the system from observed input-output pairs and (2) given a model of the system, reconstruct the input to it that caused some observed output. Applied inverse problems are challenging because a solution may (i) not exist, (ii) not be unique, or (iii) be sensitive to measurement noise contaminating the data. Bayesian statistical inversion (BSI) is an approach to tackle ill-posed and/or ill-conditioned inverse problems. Advantageously, BSI provides a "solution" that (i) quantifies uncertainty by assigning a probability to each possible value of the unknown parameter/input and (ii) incorporates prior information and beliefs about the parameter/input. Herein, we provide a tutorial of BSI for inverse problems, by way of illustrative examples dealing with heat transfer from ambient air to a cold lime fruit. First, we use BSI to infer a parameter in a dynamic model of the lime temperature from measurements of the lime temperature over time. Second, we use BSI to reconstruct the initial condition of the lime from a measurement of its temperature later in time. We demonstrate the incorporation of prior information, visualize the posterior distributions of the parameter/initial condition, and show posterior samples of lime temperature trajectories from the model. Our tutorial aims to reach a wide range of scientists and engineers.

The framework of causal models provides a principled approach to causal reasoning, applied today across many scientific domains. Here we present this framework in the language of string diagrams, interpreted formally using category theory. A class of string diagrams, called network diagrams, are in 1-to-1 correspondence with directed acyclic graphs. A causal model is given by such a diagram with its components interpreted as stochastic maps, functions, or general channels in a symmetric monoidal category with a 'copy-discard' structure (cd-category), turning a model into a single mathematical object that can be reasoned with intuitively and yet rigorously. Building on prior works by Fong and Jacobs, Kissinger and Zanasi, as well as Fritz and Klingler, we present diagrammatic definitions of causal models and functional causal models in a cd-category, generalising causal Bayesian networks and structural causal models, respectively. We formalise general interventions on a model, including but beyond do-interventions, and present the natural notion of an open causal model with inputs. We also give an approach to conditioning based on a normalisation box, allowing for causal inference calculations to be done fully diagrammatically. We define counterfactuals in this setup, and treat the problems of the identifiability of causal effects and counterfactuals fully diagrammatically. The benefits of such a presentation of causal models lie in foundational questions in causal reasoning and in their clarificatory role and pedagogical value. This work aims to be accessible to different communities, from causal model practitioners to researchers in applied category theory, and discusses many examples from the literature for illustration. Overall, we argue and demonstrate that causal reasoning according to the causal model framework is most naturally and intuitively done as diagrammatic reasoning.

Structural learning, which aims to learn directed acyclic graphs (DAGs) from observational data, is foundational to causal reasoning and scientific discovery. Recent advancements formulate structural learning into a continuous optimization problem; however, DAG learning remains a highly non-convex problem, and there has not been much work on leveraging well-developed convex optimization techniques for causal structural learning. We fill this gap by proposing a data-adaptive linear approach for causal structural learning from time series data, which can be conveniently cast into a convex optimization problem using a recently developed monotone operator variational inequality (VI) formulation. Furthermore, we establish non-asymptotic recovery guarantee of the VI-based approach and show the superior performance of our proposed method on structure recovery over existing methods via extensive numerical experiments.

Dialogue acts (DAs) can represent conversational actions of tutors or students that take place during tutoring dialogues. Automating the identification of DAs in tutoring dialogues is significant to the design of dialogue-based intelligent tutoring systems. Many prior studies employ machine learning models to classify DAs in tutoring dialogues and invest much effort to optimize the classification accuracy by using limited amounts of training data (i.e., low-resource data scenario). However, beyond the classification accuracy, the robustness of the classifier is also important, which can reflect the capability of the classifier on learning the patterns from different class distributions. We note that many prior studies on classifying educational DAs employ cross entropy (CE) loss to optimize DA classifiers on low-resource data with imbalanced DA distribution. The DA classifiers in these studies tend to prioritize accuracy on the majority class at the expense of the minority class which might not be robust to the data with imbalanced ratios of different DA classes. To optimize the robustness of classifiers on imbalanced class distributions, we propose to optimize the performance of the DA classifier by maximizing the area under the ROC curve (AUC) score (i.e., AUC maximization). Through extensive experiments, our study provides evidence that (i) by maximizing AUC in the training process, the DA classifier achieves significant performance improvement compared to the CE approach under low-resource data, and (ii) AUC maximization approaches can improve the robustness of the DA classifier under different class imbalance ratios.

We study mean-field variational Bayesian inference using the TAP approach, for Z2-synchronization as a prototypical example of a high-dimensional Bayesian model. We show that for any signal strength $\lambda > 1$ (the weak-recovery threshold), there exists a unique local minimizer of the TAP free energy functional near the mean of the Bayes posterior law. Furthermore, the TAP free energy in a local neighborhood of this minimizer is strongly convex. Consequently, a natural-gradient/mirror-descent algorithm achieves linear convergence to this minimizer from a local initialization, which may be obtained by a constant number of iterates of Approximate Message Passing (AMP). This provides a rigorous foundation for variational inference in high dimensions via minimization of the TAP free energy. We also analyze the finite-sample convergence of AMP, showing that AMP is asymptotically stable at the TAP minimizer for any $\lambda > 1$, and is linearly convergent to this minimizer from a spectral initialization for sufficiently large $\lambda$. Such a guarantee is stronger than results obtainable by state evolution analyses, which only describe a fixed number of AMP iterations in the infinite-sample limit. Our proofs combine the Kac-Rice formula and Sudakov-Fernique Gaussian comparison inequality to analyze the complexity of critical points that satisfy strong convexity and stability conditions within their local neighborhoods.

Although Buffon's needle problem [1] may be considered as the earliest documented use of Monte Carlo sampling (18th century), the method was developed at the end of the second world war and dates from the early days of computer use [2, 3]. With the increase in computational power, it has become a pervasive and versatile technique in basic sciences and engineering. It uses random sampling for solving both deterministic and stochastic problems, as found in physics, biology, chemistry, or artificial intelligence [4-10]. Monte Carlo techniques also allow to assess risk in quantitative analysis and decision making [11, 12], and their methodological developments provide tools for economy, epidemiology or archaeology [13]. It is then crucial to understand the type of errors which can be introduced as a consequence of the incomplete convergence of such algorithms. Our interest goes to Markov Chain Monte Carlo techniques [14, 15], that create correlated random samples from a target distribution; a special emphasis is put on the relaxation rate of these methods. From the target probability distribution, a sequence of samples is obtained by a random walk, with appropriate transition probabilities. The walker's density evolves at long times towards the target distribution, and quantities of interest follow from the law of large numbers and other methods of statistical inference [11-19]. The Monte-Carlo method and its modern developments [20-30] have now been adopted in many topics in and outside physics.