In this paper, we study error bounds for {\em Bayesian quadrature} (BQ), with an emphasis on noisy settings, randomized algorithms, and average-case performance measures. We seek to approximate the integral of functions in a {\em Reproducing Kernel Hilbert Space} (RKHS), particularly focusing on the Mat\'ern-$\nu$ and squared exponential (SE) kernels, with samples from the function potentially being corrupted by Gaussian noise. We provide a two-step meta-algorithm that serves as a general tool for relating the average-case quadrature error with the $L^2$-function approximation error. When specialized to the Mat\'ern kernel, we recover an existing near-optimal error rate while avoiding the existing method of repeatedly sampling points. When specialized to other settings, we obtain new average-case results for settings including the SE kernel with noise and the Mat\'ern kernel with misspecification. Finally, we present algorithm-independent lower bounds that have greater generality and/or give distinct proofs compared to existing ones.

In statistical inference, uncertainty is unknown and all models are wrong. That is to say, a person who makes a statistical model and a prior distribution is simultaneously aware that both are fictional candidates. To study such cases, statistical measures have been constructed, such as cross validation, information criteria, and marginal likelihood, however, their mathematical properties have not yet been completely clarified when statistical models are under- and over- parametrized. We introduce a place of mathematical theory of Bayesian statistics for unknown uncertainty, which clarifies general properties of cross validation, information criteria, and marginal likelihood, even if an unknown data-generating process is unrealizable by a model or even if the posterior distribution cannot be approximated by any normal distribution. Hence it gives a helpful standpoint for a person who cannot believe in any specific model and prior. This paper consists of three parts. The first is a new result, whereas the second and third are well-known previous results with new experiments. We show there exists a more precise estimator of the generalization loss than leave-one-out cross validation, there exists a more accurate approximation of marginal likelihood than BIC, and the optimal hyperparameters for generalization loss and marginal likelihood are different.

ABRIDGED. The analysis of spectral energy distributions (SEDs) of protoplanetary disks to determine their physical properties is known to be highly degenerate. Hence, a Bayesian analysis is required to obtain parameter uncertainties and degeneracies. The challenge here is computational speed, as one radiative transfer model requires a couple of minutes to compute. We performed a Bayesian analysis for 30 well-known protoplanetary disks to determine their physical disk properties, including uncertainties and degeneracies. To circumvent the computational cost problem, we created neural networks (NNs) to emulate the SED generation process. We created two sets of radiative transfer disk models to train and test two NNs that predict SEDs for continuous and discontinuous disks. A Bayesian analysis was then performed on 30 protoplanetary disks with SED data collected by the DIANA project to determine the posterior distributions of all parameters. We ran this analysis twice, (i) with old distances and additional parameter constraints as used in a previous study, to compare results, and (ii) with updated distances and free choice of parameters to obtain homogeneous and unbiased model parameters. We evaluated the uncertainties in the determination of physical disk parameters from SED analysis, and detected and quantified the strongest degeneracies. The NNs are able to predict SEDs within 1ms with uncertainties of about 5% compared to the true SEDs obtained by the radiative transfer code. We find parameter values and uncertainties that are significantly different from previous values obtained by $\chi^2$ fitting. Comparing the global evidence for continuous and discontinuous disks, we find that 26 out of 30 objects are better described by disks that have two distinct radial zones. Also, we created an interactive tool that instantly returns the SED predicted by our NNs for any parameter combination.

Bayesian posterior distributions arising in modern applications, including inverse problems in partial differential equation models in tomography and subsurface flow, are often computationally intractable due to the large computational cost of evaluating the data likelihood. To alleviate this problem, we consider using Gaussian process regression to build a surrogate model for the likelihood, resulting in an approximate posterior distribution that is amenable to computations in practice. This work serves as an introduction to Gaussian process regression, in particular in the context of building surrogate models for inverse problems, and presents new insights into a suitable choice of training points. We show that the error between the true and approximate posterior distribution can be bounded by the error between the true and approximate likelihood, measured in the $L^2$-norm weighted by the true posterior, and that efficiently bounding the error between the true and approximate likelihood in this norm suggests choosing the training points in the Gaussian process surrogate model based on the true posterior.

We introduce a framework that enables efficient sampling from learned probability distributions for MRI reconstruction. Different from conventional deep learning-based MRI reconstruction techniques, samples are drawn from the posterior distribution given the measured k-space using the Markov chain Monte Carlo (MCMC) method. In addition to the maximum a posteriori (MAP) estimate for the image, which can be obtained with conventional methods, the minimum mean square error (MMSE) estimate and uncertainty maps can also be computed. The data-driven Markov chains are constructed from the generative model learned from a given image database and are independent of the forward operator that is used to model the k-space measurement. This provides flexibility because the method can be applied to k-space acquired with different sampling schemes or receive coils using the same pre-trained models. Furthermore, we use a framework based on a reverse diffusion process to be able to utilize advanced generative models. The performance of the method is evaluated on an open dataset using 10-fold undersampling in k-space.

This paper proposes an unsupervised classification method that partitions a set of files into non-overlapping dialects based upon their behaviors, determined by messages produced by a collection of programs that consume them. The pattern of messages can be used as the signature of a particular kind of behavior, with the understanding that some messages are likely to co-occur, while others are not. Patterns of messages can be used to classify files into dialects. A dialect is defined by a subset of messages, called the required messages. Once files are conditioned upon dialect and its required messages, the remaining messages are statistically independent. With this definition of dialect in hand, we present a greedy algorithm that deduces candidate dialects from a dataset consisting of a matrix of file-message data, demonstrate its performance on several file formats, and prove conditions under which it is optimal. We show that an analyst needs to consider fewer dialects than distinct message patterns, which reduces their cognitive load when studying a complex format.

Intention prediction has become a relevant field of research in Human-Machine and Human-Robot Interaction. Indeed, any artificial system (co)-operating with and along humans, designed to assist and coordinate its actions with a human partner, would benefit from first inferring the human's current intention. To spare the user the cognitive burden of explicitly uttering their goals, this inference relies mostly on behavioral cues deemed indicative of the current action. It has been long known that eye movements are highly anticipatory of the single steps unfolding during a task, hence they can serve as a very early and reliable behavioural cue for intention recognition. This review aims to draw a line between insights in the psychological literature on visuomotor control and relevant applications of gaze-based intention recognition in technical domains, with a focus on teleoperated and assistive robotic systems. Starting from the cognitive principles underlying the relationship between intentions, eye movements, and action, the use of eye tracking and gaze-based models for intent recognition in Human-Robot Interaction is considered, with prevalent methodologies and their diverse applications. Finally, special consideration is given to relevant human factors issues and current limitations to be factored in when designing such systems.

Autoencoders and their variants are among the most widely used models in representation learning and generative modeling. However, autoencoder-based models usually assume that the learned representations are i.i.d. and fail to capture the correlations between the data samples. To address this issue, we propose a novel Sparse Gaussian Process Bayesian Autoencoder (SGPBAE) model in which we impose fully Bayesian sparse Gaussian Process priors on the latent space of a Bayesian Autoencoder. We perform posterior estimation for this model via stochastic gradient Hamiltonian Monte Carlo. We evaluate our approach qualitatively and quantitatively on a wide range of representation learning and generative modeling tasks and show that our approach consistently outperforms multiple alternatives relying on Variational Autoencoders.

Existing models to extract temporal relations between events lack a principled method to incorporate external knowledge. In this study, we introduce Bayesian-Trans, a Bayesian learning-based method that models the temporal relation representations as latent variables and infers their values via Bayesian inference and translational functions. Compared to conventional neural approaches, instead of performing point estimation to find the best set parameters, the proposed model infers the parameters' posterior distribution directly, enhancing the model's capability to encode and express uncertainty about the predictions. Experimental results on the three widely used datasets show that Bayesian-Trans outperforms existing approaches for event temporal relation extraction. We additionally present detailed analyses on uncertainty quantification, comparison of priors, and ablation studies, illustrating the benefits of the proposed approach.

Variational Autoencoders (VAEs) were originally motivated (Kingma & Welling, 2014) as probabilistic generative models in which one performs approximate Bayesian inference. The proposal of $\beta$-VAEs (Higgins et al., 2017) breaks this interpretation and generalizes VAEs to application domains beyond generative modeling (e.g., representation learning, clustering, or lossy data compression) by introducing an objective function that allows practitioners to trade off between the information content ("bit rate") of the latent representation and the distortion of reconstructed data (Alemi et al., 2018). In this paper, we reconsider this rate/distortion trade-off in the context of hierarchical VAEs, i.e., VAEs with more than one layer of latent variables. We identify a general class of inference models for which one can split the rate into contributions from each layer, which can then be tuned independently. We derive theoretical bounds on the performance of downstream tasks as functions of the individual layers' rates and verify our theoretical findings in large-scale experiments. Our results provide guidance for practitioners on which region in rate-space to target for a given application.