This article describes an approach to incorporate expert opinion on observable quantities through the use of a loss function which updates a prior belief as opposed to specifying parameters on the priors. Eliciting information on observable quantities allows experts to provide meaningful information on a quantity familiar to them, in contrast to elicitation on model parameters, which may be subject to interactions with other parameters or non-linear transformations before obtaining an observable quantity. The approach to incorporating expert opinion described in this paper is distinctive in that we do not specify a prior to match an expert's opinion on observed quantity, rather we obtain a posterior by updating the model parameters through a loss function. This loss function contains the observable quantity, expressed a function of the parameters, and is related to the expert's opinion which is typically operationalized as a statistical distribution. Parameters which generate observable quantities which are further from the expert's opinion incur a higher loss, allowing for the model parameters to be estimated based on their fidelity to both the data and expert opinion, with the relative strength determined by the number of observations and precision of the elicited belief. Including expert opinion in this fashion allows for a flexible specification of the opinion and in many situations is straightforward to implement with commonly used probabilistic programming software. We highlight this using three worked examples of varying model complexity including survival models, a multivariate normal distribution and a regression problem.

Mixture models of Plackett-Luce (PL) -- one of the most fundamental ranking models -- are an active research area of both theoretical and practical significance. Most previously proposed parameter estimation algorithms instantiate the EM algorithm, often with random initialization. However, such an initialization scheme may not yield a good initial estimate and the algorithms require multiple restarts, incurring a large time complexity. As for the EM procedure, while the E-step can be performed efficiently, maximizing the log-likelihood in the M-step is difficult due to the combinatorial nature of the PL likelihood function (Gormley and Murphy 2008). Therefore, previous authors favor algorithms that maximize surrogate likelihood functions (Zhao et al. 2018, 2020). However, the final estimate may deviate from the true maximum likelihood estimate as a consequence. In this paper, we address these known limitations. We propose an initialization algorithm that can provide a provably accurate initial estimate and an EM algorithm that maximizes the true log-likelihood function efficiently. Experiments on both synthetic and real datasets show that our algorithm is competitive in terms of accuracy and speed to baseline algorithms, especially on datasets with a large number of items.

We propose a continuous optimization framework for discovering a latent directed acyclic graph (DAG) from observational data. Our approach optimizes over the polytope of permutation vectors, the so-called Permutahedron, to learn a topological ordering. Edges can be optimized jointly, or learned conditional on the ordering via a non-differentiable subroutine. Compared to existing continuous optimization approaches our formulation has a number of advantages including: 1. validity: optimizes over exact DAGs as opposed to other relaxations optimizing approximate DAGs; 2. modularity: accommodates any edge-optimization procedure, edge structural parameterization, and optimization loss; 3. end-to-end: either alternately iterates between node-ordering and edge-optimization, or optimizes them jointly. We demonstrate, on real-world data problems in protein-signaling and transcriptional network discovery, that our approach lies on the Pareto frontier of two key metrics, the SID and SHD. In many domains, including cell biology (Sachs et al., 2005), finance (Sanford & Moosa, 2012), and genetics (Zhang et al., 2013), the data generating process is thought to be represented by an underlying directed acylic graph (DAG). Many models rely on DAG assumptions, e.g., causal modeling uses DAGs to model distribution shifts, ensure predictor fairness among subpopulations, or learn agents more sample-efficiently (Kaddour et al., 2022). A key question, with implications ranging from better modeling to causal discovery, is how to recover this unknown DAG from observed data alone. Learning DAGs from observational data alone is fundamentally difficult for two reasons. This riddles the search space with local minima; (ii) Computation: DAG discovery is a costly combinatorial optimization problem over an exponentially large solution space and subject to global acyclicity constraints. To address issue (ii), recent work has proposed continuous relaxations of the DAG learning problem.

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.

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.

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.