This work considers Bayesian inference under misspecification for complex statistical models comprised of simpler submodels, referred to as modules, that are coupled together. Such ``multi-modular" models often arise when combining information from different data sources, where there is a module for each data source. When some of the modules are misspecified, the challenges of Bayesian inference under misspecification can sometimes be addressed by using ``cutting feedback" methods, which modify conventional Bayesian inference by limiting the influence of unreliable modules. Here we investigate cutting feedback methods in the context of generalized posterior distributions, which are built from arbitrary loss functions, and present novel findings on their behaviour. We make three main contributions. First, we describe how cutting feedback methods can be defined in the generalized Bayes setting, and discuss the appropriate scaling of the loss functions for different modules to each other and the prior. Second, we derive a novel result about the large sample behaviour of the posterior for a given module's parameters conditional on the parameters of other modules. This formally justifies the use of conditional Laplace approximations, which provide better approximations of conditional posterior distributions compared to conditional distributions from a Laplace approximation of the joint posterior. Our final contribution leverages the large sample approximations of our second contribution to provide convenient diagnostics for understanding the sensitivity of inference to the coupling of the modules, and to implement a new semi-modular posterior approach for conducting robust Bayesian modular inference. The usefulness of the methodology is illustrated in several benchmark examples from the literature on cut model inference.
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Ordinary state-based peridynamic (OSB-PD) models have an unparalleled capability to simulate crack propagation phenomena in solids with arbitrary Poisson's ratio. However, their non-locality also leads to prohibitively high computational cost. In this paper, a fast solution scheme for OSB-PD models based on matrix operation is introduced, with which, the graphics processing units (GPUs) are used to accelerate the computation. For the purpose of comparison and verification, a commonly used solution scheme based on loop operation is also presented. An in-house software is developed in MATLAB. Firstly, the vibration of a cantilever beam is solved for validating the loop- and matrix-based schemes by comparing the numerical solutions to those produced by a FEM software. Subsequently, two typical dynamic crack propagation problems are simulated to illustrate the effectiveness of the proposed schemes in solving dynamic fracture problems. Finally, the simulation of the Brokenshire torsion experiment is carried out by using the matrix-based scheme, and the similarity in the shapes of the experimental and numerical broken specimens further demonstrates the ability of the proposed approach to deal with 3D non-planar fracture problems. In addition, the speed-up of the matrix-based scheme with respect to the loop-based scheme and the performance of the GPU acceleration are investigated. The results emphasize the high computational efficiency of the matrix-based implementation scheme.
The study further explores randomized QMC (RQMC), which maintains the QMC convergence rate and facilitates computational efficiency analysis. Emphasis is laid on integrating randomly shifted lattice rules, a distinct RQMC quadrature, with IS,a classic variance reduction technique. The study underscores the intricacies of establishing a theoretical convergence rate for IS in QMC compared to MC, given the influence of problem dimensions and smoothness on QMC. The research also touches on the significance of IS density selection and its potential implications. The study culminates in examining the error bound of IS with a randomly shifted lattice rule, drawing inspiration from the reproducing kernel Hilbert space (RKHS). In the realm of finance and statistics, many problems boil down to computing expectations, predominantly integrals concerning a Gaussian measure. This study considers optimal drift importance sampling (ODIS) and Laplace importance sampling (LapIS) as common importance densities. Conclusively, the paper establishes that under certain conditions, the IS-randomly shifted lattice rule can achieve a near $O(N^{-1})$ error bound.
In recent years, the development of technologies for causal inference with privacy preservation of distributed data has gained considerable attention. Many existing methods for distributed data focus on resolving the lack of subjects (samples) and can only reduce random errors in estimating treatment effects. In this study, we propose a data collaboration quasi-experiment (DC-QE) that resolves the lack of both subjects and covariates, reducing random errors and biases in the estimation. Our method involves constructing dimensionality-reduced intermediate representations from private data from local parties, sharing intermediate representations instead of private data for privacy preservation, estimating propensity scores from the shared intermediate representations, and finally, estimating the treatment effects from propensity scores. Through numerical experiments on both artificial and real-world data, we confirm that our method leads to better estimation results than individual analyses. While dimensionality reduction loses some information in the private data and causes performance degradation, we observe that sharing intermediate representations with many parties to resolve the lack of subjects and covariates sufficiently improves performance to overcome the degradation caused by dimensionality reduction. Although external validity is not necessarily guaranteed, our results suggest that DC-QE is a promising method. With the widespread use of our method, intermediate representations can be published as open data to help researchers find causalities and accumulate a knowledge base.
Quantum neural networks (QNNs) and quantum kernels stand as prominent figures in the realm of quantum machine learning, poised to leverage the nascent capabilities of near-term quantum computers to surmount classical machine learning challenges. Nonetheless, the training efficiency challenge poses a limitation on both QNNs and quantum kernels, curbing their efficacy when applied to extensive datasets. To confront this concern, we present a unified approach: coreset selection, aimed at expediting the training of QNNs and quantum kernels by distilling a judicious subset from the original training dataset. Furthermore, we analyze the generalization error bounds of QNNs and quantum kernels when trained on such coresets, unveiling the comparable performance with those training on the complete original dataset. Through systematic numerical simulations, we illuminate the potential of coreset selection in expediting tasks encompassing synthetic data classification, identification of quantum correlations, and quantum compiling. Our work offers a useful way to improve diverse quantum machine learning models with a theoretical guarantee while reducing the training cost.
Scalable high-dimensional Bayesian varying coefficient models with unknown within-subject covariance
Nonparametric varying coefficient (NVC) models are useful for modeling time-varying effects on responses that are measured repeatedly for the same subjects. When the number of covariates is moderate or large, it is desirable to perform variable selection from the varying coefficient functions. However, existing methods for variable selection in NVC models either fail to account for within-subject correlations or require the practitioner to specify a parametric form for the correlation structure. In this paper, we introduce the nonparametric varying coefficient spike-and-slab lasso (NVC-SSL) for Bayesian high-dimensional NVC models. Through the introduction of functional random effects, our method allows for flexible modeling of within-subject correlations without needing to specify a parametric covariance function. We further propose several scalable optimization and Markov chain Monte Carlo (MCMC) algorithms. For variable selection, we propose an Expectation Conditional Maximization (ECM) algorithm to rapidly obtain maximum a posteriori (MAP) estimates. Our ECM algorithm scales linearly in the total number of observations $N$ and the number of covariates $p$. For uncertainty quantification, we introduce an approximate MCMC algorithm that also scales linearly in both $N$ and $p$. We demonstrate the scalability, variable selection performance, and inferential capabilities of our method through simulations and a real data application. These algorithms are implemented in the publicly available R package NVCSSL on the Comprehensive R Archive Network.
We present a robust deep incremental learning framework for regression tasks on financial temporal tabular datasets which is built upon the incremental use of commonly available tabular and time series prediction models to adapt to distributional shifts typical of financial datasets. The framework uses a simple basic building block (decision trees) to build self-similar models of any required complexity to deliver robust performance under adverse situations such as regime changes, fat-tailed distributions, and low signal-to-noise ratios. As a detailed study, we demonstrate our scheme using XGBoost models trained on the Numerai dataset and show that a two layer deep ensemble of XGBoost models over different model snapshots delivers high quality predictions under different market regimes. We also show that the performance of XGBoost models with different number of boosting rounds in three scenarios (small, standard and large) is monotonically increasing with respect to model size and converges towards the generalisation upper bound. We also evaluate the robustness of the model under variability of different hyperparameters, such as model complexity and data sampling settings. Our model has low hardware requirements as no specialised neural architectures are used and each base model can be independently trained in parallel.
We consider the problem of sequential change detection, where the goal is to design a scheme for detecting any changes in a parameter or functional $\theta$ of the data stream distribution that has small detection delay, but guarantees control on the frequency of false alarms in the absence of changes. In this paper, we describe a simple reduction from sequential change detection to sequential estimation using confidence sequences: we begin a new $(1-\alpha)$-confidence sequence at each time step, and proclaim a change when the intersection of all active confidence sequences becomes empty. We prove that the average run length is at least $1/\alpha$, resulting in a change detection scheme with minimal structural assumptions~(thus allowing for possibly dependent observations, and nonparametric distribution classes), but strong guarantees. Our approach bears an interesting parallel with the reduction from change detection to sequential testing of Lorden (1971) and the e-detector of Shin et al. (2022).
This paper introduces an innovative method for conducting conditional independence testing in high-dimensional data, facilitating the automated discovery of significant associations within distinct subgroups of a population, all while controlling the false discovery rate. This is achieved by expanding upon the model-X knockoff filter to provide more informative inferences. Our enhanced inferences can help explain sample heterogeneity and uncover interactions, making better use of the capabilities offered by modern machine learning models. Specifically, our method is able to leverage any model for the identification of data-driven hypotheses pertaining to interesting population subgroups. Then, it rigorously test these hypotheses without succumbing to selection bias. Importantly, our approach is efficient and does not require sample splitting. We demonstrate the effectiveness of our method through simulations and numerical experiments, using data derived from a randomized experiment featuring multiple treatment variables.
A change point detection (CPD) framework assisted by a predictive machine learning model called "Predict and Compare" is introduced and characterised in relation to other state-of-the-art online CPD routines which it outperforms in terms of false positive rate and out-of-control average run length. The method's focus is on improving standard methods from sequential analysis such as the CUSUM rule in terms of these quality measures. This is achieved by replacing typically used trend estimation functionals such as the running mean with more sophisticated predictive models (Predict step), and comparing their prognosis with actual data (Compare step). The two models used in the Predict step are the ARIMA model and the LSTM recursive neural network. However, the framework is formulated in general terms, so as to allow the use of other prediction or comparison methods than those tested here. The power of the method is demonstrated in a tribological case study in which change points separating the run-in, steady-state, and divergent wear phases are detected in the regime of very few false positives.
This work uses the entropy-regularised relaxed stochastic control perspective as a principled framework for designing reinforcement learning (RL) algorithms. Herein agent interacts with the environment by generating noisy controls distributed according to the optimal relaxed policy. The noisy policies on the one hand, explore the space and hence facilitate learning but, on the other hand, introduce bias by assigning a positive probability to non-optimal actions. This exploration-exploitation trade-off is determined by the strength of entropy regularisation. We study algorithms resulting from two entropy regularisation formulations: the exploratory control approach, where entropy is added to the cost objective, and the proximal policy update approach, where entropy penalises policy divergence between consecutive episodes. We focus on the finite horizon continuous-time linear-quadratic (LQ) RL problem, where a linear dynamics with unknown drift coefficients is controlled subject to quadratic costs. In this setting, both algorithms yield a Gaussian relaxed policy. We quantify the precise difference between the value functions of a Gaussian policy and its noisy evaluation and show that the execution noise must be independent across time. By tuning the frequency of sampling from relaxed policies and the parameter governing the strength of entropy regularisation, we prove that the regret, for both learning algorithms, is of the order $\mathcal{O}(\sqrt{N}) $ (up to a logarithmic factor) over $N$ episodes, matching the best known result from the literature.
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