Measurement error (ME) and missing values in covariates are often unavoidable in disciplines that deal with data, and both problems have separately received considerable attention during the past decades. However, while most researchers are familiar with methods for treating missing data, accounting for ME in covariates of regression models is less common. In addition, ME and missing data are typically treated as two separate problems, despite practical and theoretical similarities. Here, we exploit the fact that missing data in a continuous covariate is an extreme case of classical ME, allowing us to use existing methodology that accounts for ME via a Bayesian framework that employs integrated nested Laplace approximations (INLA), and thus to simultaneously account for both ME and missing data in the same covariate. As a useful by-product, we present an approach to handle missing data in INLA, since this corresponds to the special case when no ME is present. In addition, we show how to account for Berkson ME in the same framework. In its broadest generality, the proposed joint Bayesian framework can thus account for Berkson ME, classical ME, and missing data, or for any combination of these in the same or different continuous covariates of the family of regression models that are feasible with INLA. The approach is exemplified using both simulated and real data. We provide extensive and fully reproducible Supplementary Material with thoroughly documented examples using {R-INLA} and {inlabru}.
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Mixed Laplace approximation for marginal posterior and Bayesian inference in error-in-operator model
Laplace approximation is a very useful tool in Bayesian inference and it claims a nearly Gaussian behavior of the posterior. \cite{SpLaplace2022} established some rather accurate finite sample results about the quality of Laplace approximation in terms of the so called effective dimension $p$ under the critical dimension constraint $p^{3} \ll n$. However, this condition can be too restrictive for many applications like error-in-operator problem or Deep Neuronal Networks. This paper addresses the question whether the dimensionality condition can be relaxed and the accuracy of approximation can be improved if the target of estimation is low dimensional while the nuisance parameter is high or infinite dimensional. Under mild conditions, the marginal posterior can be approximated by a Gaussian mixture and the accuracy of the approximation only depends on the target dimension. Under the condition $p^{2} \ll n$ or in some special situation like semi-orthogonality, the Gaussian mixture can be replaced by one Gaussian distribution leading to a classical Laplace result. The second result greatly benefits from the recent advances in Gaussian comparison from \cite{GNSUl2017}. The results are illustrated and specified for the case of error-in-operator model.
2D texture maps and 3D voxel arrays are widely used to add rich detail to the surfaces and volumes of rendered scenes, and filtered texture lookups are integral to producing high-quality imagery. We show that filtering textures after evaluating lighting, rather than before BSDF evaluation as is current practice, gives a more accurate solution to the rendering equation. These benefits are not merely theoretical, but are apparent in common cases. We further show that stochastically sampling texture filters is crucial for enabling this approach, which has not been possible previously except in limited cases. Stochastic texture filtering offers additional benefits, including efficient implementation of high-quality texture filters and efficient filtering of textures stored in compressed and sparse data structures, including neural representations. We demonstrate applications in both real-time and offline rendering and show that the additional stochastic error is minimal. Furthermore, this error is handled well by either spatiotemporal denoising or moderate pixel sampling rates.
Bayesian inference is a powerful tool for combining information in complex settings, a task of increasing importance in modern applications. However, Bayesian inference with a flawed model can produce unreliable conclusions. This review discusses approaches to performing Bayesian inference when the model is misspecified, where by misspecified we mean that the analyst is unwilling to act as if the model is correct. Much has been written about this topic, and in most cases we do not believe that a conventional Bayesian analysis is meaningful when there is serious model misspecification. Nevertheless, in some cases it is possible to use a well-specified model to give meaning to a Bayesian analysis of a misspecified model and we will focus on such cases. Three main classes of methods are discussed - restricted likelihood methods, which use a model based on a non-sufficient summary of the original data; modular inference methods which use a model constructed from coupled submodels and some of the submodels are correctly specified; and the use of a reference model to construct a projected posterior or predictive distribution for a simplified model considered to be useful for prediction or interpretation.
Single-cell RNA-sequencing technologies may provide valuable insights to the understanding of the composition of different cell types and their functions within a tissue. Recent technologies such as spatial transcriptomics, enable the measurement of gene expressions at the single cell level along with the spatial locations of these cells in the tissue. Dimension-reduction and spatial clustering are two of the most common exploratory analysis strategies for spatial transcriptomic data. However, existing dimension reduction methods may lead to a loss of inherent dependency structure among genes at any spatial location in the tissue and hence do not provide insights of gene co-expression pattern. In spatial transcriptomics, the matrix-variate gene expression data, along with spatial co-ordinates of the single cells, provides information on both gene expression dependencies and cell spatial dependencies through its row and column covariances. In this work, we propose a flexible Bayesian approach to simultaneously estimate the row and column covariances for the matrix-variate spatial transcriptomic data. The posterior estimates of the row and column covariances provide data summaries for downstream exploratory analysis. We illustrate our method with simulations and two analyses of real data generated from a recent spatial transcriptomic platform. Our work elucidates gene co-expression networks as well as clear spatial clustering patterns of the cells.
Modern biomedical datasets are increasingly high dimensional and exhibit complex correlation structures. Generalized Linear Mixed Models (GLMMs) have long been employed to account for such dependencies. However, proper specification of the fixed and random effects in GLMMs is increasingly difficult in high dimensions, and computational complexity grows with increasing dimension of the random effects. We present a novel reformulation of the GLMM using a factor model decomposition of the random effects, enabling scalable computation of GLMMs in high dimensions by reducing the latent space from a large number of random effects to a smaller set of latent factors. We also extend our prior work to estimate model parameters using a modified Monte Carlo Expectation Conditional Minimization algorithm, allowing us to perform variable selection on both the fixed and random effects simultaneously. We show through simulation that through this factor model decomposition, our method can fit high dimensional penalized GLMMs faster than comparable methods and more easily scale to larger dimensions not previously seen in existing approaches.
We consider four main goals when fitting spatial linear models: 1) estimating covariance parameters, 2) estimating fixed effects, 3) kriging (making point predictions), and 4) block-kriging (predicting the average value over a region). Each of these goals can present different challenges when analyzing large spatial data sets. Current research uses a variety of methods, including spatial basis functions (reduced rank), covariance tapering, etc, to achieve these goals. However, spatial indexing, which is very similar to composite likelihood, offers some advantages. We develop a simple framework for all four goals listed above by using indexing to create a block covariance structure and nearest-neighbor predictions while maintaining a coherent linear model. We show exact inference for fixed effects under this block covariance construction. Spatial indexing is very fast, and simulations are used to validate methods and compare to another popular method. We study various sample designs for indexing and our simulations showed that indexing leading to spatially compact partitions are best over a range of sample sizes, autocorrelation values, and generating processes. Partitions can be kept small, on the order of 50 samples per partition. We use nearest-neighbors for kriging and block kriging, finding that 50 nearest-neighbors is sufficient. In all cases, confidence intervals for fixed effects, and prediction intervals for (block) kriging, have appropriate coverage. Some advantages of spatial indexing are that it is available for any valid covariance matrix, can take advantage of parallel computing, and easily extends to non-Euclidean topologies, such as stream networks. We use stream networks to show how spatial indexing can achieve all four goals, listed above, for very large data sets, in a matter of minutes, rather than days, for an example data set.
Deep learning models, including modern systems like large language models, are well known to offer unreliable estimates of the uncertainty of their decisions. In order to improve the quality of the confidence levels, also known as calibration, of a model, common approaches entail the addition of either data-dependent or data-independent regularization terms to the training loss. Data-dependent regularizers have been recently introduced in the context of conventional frequentist learning to penalize deviations between confidence and accuracy. In contrast, data-independent regularizers are at the core of Bayesian learning, enforcing adherence of the variational distribution in the model parameter space to a prior density. The former approach is unable to quantify epistemic uncertainty, while the latter is severely affected by model misspecification. In light of the limitations of both methods, this paper proposes an integrated framework, referred to as calibration-aware Bayesian neural networks (CA-BNNs), that applies both regularizers while optimizing over a variational distribution as in Bayesian learning. Numerical results validate the advantages of the proposed approach in terms of expected calibration error (ECE) and reliability diagrams.
High-dimensional variable selection, with many more covariates than observations, is widely documented in standard regression models, but there are still few tools to address it in non-linear mixed-effects models where data are collected repeatedly on several individuals. In this work, variable selection is approached from a Bayesian perspective and a selection procedure is proposed, combining the use of a spike-and-slab prior and the SAEM algorithm. Similarly to Lasso regression, the set of relevant covariates is selected by exploring a grid of values for the penalisation parameter. The SAEM approach is much faster than a classical MCMC algorithm and our method shows very good selection performances on simulated data. Its flexibility is demonstrated by implementing it for a variety of nonlinear mixed effects models. The usefulness of the proposed method is illustrated on a problem of genetic markers identification, relevant for genomic-assisted selection in plant breeding.
Recent work has focused on the potential and pitfalls of causal identification in observational studies with multiple simultaneous treatments. Building on previous work, we show that even if the conditional distribution of unmeasured confounders given treatments were known exactly, the causal effects would not in general be identifiable, although they may be partially identified. Given these results, we propose a sensitivity analysis method for characterizing the effects of potential unmeasured confounding, tailored to the multiple treatment setting, that can be used to characterize a range of causal effects that are compatible with the observed data. Our method is based on a copula factorization of the joint distribution of outcomes, treatments, and confounders, and can be layered on top of arbitrary observed data models. We propose a practical implementation of this approach making use of the Gaussian copula, and establish conditions under which causal effects can be bounded. We also describe approaches for reasoning about effects, including calibrating sensitivity parameters, quantifying robustness of effect estimates, and selecting models that are most consistent with prior hypotheses.
Sentiment analysis is a widely studied NLP task where the goal is to determine opinions, emotions, and evaluations of users towards a product, an entity or a service that they are reviewing. One of the biggest challenges for sentiment analysis is that it is highly language dependent. Word embeddings, sentiment lexicons, and even annotated data are language specific. Further, optimizing models for each language is very time consuming and labor intensive especially for recurrent neural network models. From a resource perspective, it is very challenging to collect data for different languages. In this paper, we look for an answer to the following research question: can a sentiment analysis model trained on a language be reused for sentiment analysis in other languages, Russian, Spanish, Turkish, and Dutch, where the data is more limited? Our goal is to build a single model in the language with the largest dataset available for the task, and reuse it for languages that have limited resources. For this purpose, we train a sentiment analysis model using recurrent neural networks with reviews in English. We then translate reviews in other languages and reuse this model to evaluate the sentiments. Experimental results show that our robust approach of single model trained on English reviews statistically significantly outperforms the baselines in several different languages.
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