We consider likelihood score-based methods for causal discovery in structural causal models. In particular, we focus on Gaussian scoring and analyze the effect of model misspecification in terms of non-Gaussian error distribution. We present a surprising negative result for Gaussian likelihood scoring in combination with nonparametric regression methods.
相關內容
Learning the kernel parameters for Gaussian processes is often the computational bottleneck in applications such as online learning, Bayesian optimization, or active learning. Amortizing parameter inference over different datasets is a promising approach to dramatically speed up training time. However, existing methods restrict the amortized inference procedure to a fixed kernel structure. The amortization network must be redesigned manually and trained again in case a different kernel is employed, which leads to a large overhead in design time and training time. We propose amortizing kernel parameter inference over a complete kernel-structure-family rather than a fixed kernel structure. We do that via defining an amortization network over pairs of datasets and kernel structures. This enables fast kernel inference for each element in the kernel family without retraining the amortization network. As a by-product, our amortization network is able to do fast ensembling over kernel structures. In our experiments, we show drastically reduced inference time combined with competitive test performance for a large set of kernels and datasets.
We introduce a Loss Discounting Framework for model and forecast combination which generalises and combines Bayesian model synthesis and generalized Bayes methodologies. We use a loss function to score the performance of different models and introduce a multilevel discounting scheme which allows a flexible specification of the dynamics of the model weights. This novel and simple model combination approach can be easily applied to large scale model averaging/selection, can handle unusual features such as sudden regime changes, and can be tailored to different forecasting problems. We compare our method to both established methodologies and state of the art methods for a number of macroeconomic forecasting examples. We find that the proposed method offers an attractive, computationally efficient alternative to the benchmark methodologies and often outperforms more complex techniques.
This article provides a comprehensive understanding of optimization in deep learning, with a primary focus on the challenges of gradient vanishing and gradient exploding, which normally lead to diminished model representational ability and training instability, respectively. We analyze these two challenges through several strategic measures, including the improvement of gradient flow and the imposition of constraints on a network's Lipschitz constant. To help understand the current optimization methodologies, we categorize them into two classes: explicit optimization and implicit optimization. Explicit optimization methods involve direct manipulation of optimizer parameters, including weight, gradient, learning rate, and weight decay. Implicit optimization methods, by contrast, focus on improving the overall landscape of a network by enhancing its modules, such as residual shortcuts, normalization methods, attention mechanisms, and activations. In this article, we provide an in-depth analysis of these two optimization classes and undertake a thorough examination of the Jacobian matrices and the Lipschitz constants of many widely used deep learning modules, highlighting existing issues as well as potential improvements. Moreover, we also conduct a series of analytical experiments to substantiate our theoretical discussions. This article does not aim to propose a new optimizer or network. Rather, our intention is to present a comprehensive understanding of optimization in deep learning. We hope that this article will assist readers in gaining a deeper insight in this field and encourages the development of more robust, efficient, and high-performing models.
This paper presents a novel approach to Bayesian nonparametric spectral analysis of stationary multivariate time series. Starting with a parametric vector-autoregressive model, the parametric likelihood is nonparametrically adjusted in the frequency domain to account for potential deviations from parametric assumptions. We show mutual contiguity of the nonparametrically corrected likelihood, the multivariate Whittle likelihood approximation and the exact likelihood for Gaussian time series. A multivariate extension of the nonparametric Bernstein-Dirichlet process prior for univariate spectral densities to the space of Hermitian positive definite spectral density matrices is specified directly on the correction matrices. An infinite series representation of this prior is then used to develop a Markov chain Monte Carlo algorithm to sample from the posterior distribution. The code is made publicly available for ease of use and reproducibility. With this novel approach we provide a generalization of the multivariate Whittle-likelihood-based method of Meier et al. (2020) as well as an extension of the nonparametrically corrected likelihood for univariate stationary time series of Kirch et al. (2019) to the multivariate case. We demonstrate that the nonparametrically corrected likelihood combines the efficiencies of a parametric with the robustness of a nonparametric model. Its numerical accuracy is illustrated in a comprehensive simulation study. We illustrate its practical advantages by a spectral analysis of two environmental time series data sets: a bivariate time series of the Southern Oscillation Index and fish recruitment and time series of windspeed data at six locations in California.
Privacy-utility tradeoff remains as one of the fundamental issues of differentially private machine learning. This paper introduces a geometrically inspired kernel-based approach to mitigate the accuracy-loss issue in classification. In this approach, a representation of the affine hull of given data points is learned in Reproducing Kernel Hilbert Spaces (RKHS). This leads to a novel distance measure that hides privacy-sensitive information about individual data points and improves the privacy-utility tradeoff via significantly reducing the risk of membership inference attacks. The effectiveness of the approach is demonstrated through experiments on MNIST dataset, Freiburg groceries dataset, and a real biomedical dataset. It is verified that the approach remains computationally practical. The application of the approach to federated learning is considered and it is observed that the accuracy-loss due to data being distributed is either marginal or not significantly high.
Generative AI (Gen-AI) methods are developed for Bayesian Computation. Gen-AI naturally applies to Bayesian models which can be easily simulated. First, we generate a large training dataset of data and parameters from the joint probability model. Secondly, we find a summary/sufficient statistic for dimensionality reduction. Thirdly, we use a deep neural network to uncover the inverse Bayes map between parameters and data. This finds the inverse posterior cumulative distribution function. Bayesian computation then is equivalent to high dimensional regression with dimensionality reduction (a.k.a feature selection) and nonlnearity (a.k.a. deep learning). The main advantage of Gen-AI is the ability to be density-free and hence avoids MCMC simulation of the posterior. Architecture design is important and we propose deep quantile NNs as a general framework for inference and decision making. To illustrate our methodology, we provide three examples: a stylized synthetic example, a traffic flow prediction problem and a satellite data-set. Finally, we conclude with directions for future research.
Understanding causality helps to structure interventions to achieve specific goals and enables predictions under interventions. With the growing importance of learning causal relationships, causal discovery tasks have transitioned from using traditional methods to infer potential causal structures from observational data to the field of pattern recognition involved in deep learning. The rapid accumulation of massive data promotes the emergence of causal search methods with brilliant scalability. Existing summaries of causal discovery methods mainly focus on traditional methods based on constraints, scores and FCMs, there is a lack of perfect sorting and elaboration for deep learning-based methods, also lacking some considers and exploration of causal discovery methods from the perspective of variable paradigms. Therefore, we divide the possible causal discovery tasks into three types according to the variable paradigm and give the definitions of the three tasks respectively, define and instantiate the relevant datasets for each task and the final causal model constructed at the same time, then reviews the main existing causal discovery methods for different tasks. Finally, we propose some roadmaps from different perspectives for the current research gaps in the field of causal discovery and point out future research directions.
We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
Causal inference is a critical research topic across many domains, such as statistics, computer science, education, public policy and economics, for decades. Nowadays, estimating causal effect from observational data has become an appealing research direction owing to the large amount of available data and low budget requirement, compared with randomized controlled trials. Embraced with the rapidly developed machine learning area, various causal effect estimation methods for observational data have sprung up. In this survey, we provide a comprehensive review of causal inference methods under the potential outcome framework, one of the well known causal inference framework. The methods are divided into two categories depending on whether they require all three assumptions of the potential outcome framework or not. For each category, both the traditional statistical methods and the recent machine learning enhanced methods are discussed and compared. The plausible applications of these methods are also presented, including the applications in advertising, recommendation, medicine and so on. Moreover, the commonly used benchmark datasets as well as the open-source codes are also summarized, which facilitate researchers and practitioners to explore, evaluate and apply the causal inference methods.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191