Background and Objective: Histograms and Pearson's coefficient of variation are among the most popular summary statistics. Researchers use them to judge the shape of quantitative data distribution by visual inspection of histograms. The coefficient of variation is taken as an estimator of relative variability of these data. We explore properties of histograms and coefficient of variation by examples in R, thus offering better alternatives: density plots and Eisenhauer's relative dispersion coefficient. Methods: Hypothetical examples developed in R are applied to create histograms and density and to compute coefficient of variation and relative dispersion coefficient. Results: These hypothetical examples clearly show that these two traditional approaches are flawed. Histograms are incapable of reflecting the distribution of probabilities and the coefficient of variation has issues with negative and positive values in the same dataset, it is sensible to outliers, and it is severely affected by mean value of a distribution. Potential replacements are explained and applied for contrast. Conclusions: With the use of modern computers and R language it is easy to replace histograms by density plots, which are able to approximate the theoretical probability distribution. In addition, Eisenhauer's relative dispersion coefficient is suggested as a suitable estimator of relative variability, including corrections for lower and upper bounds.
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Many scientific datasets are compositional in nature. Important examples include species abundances in ecology, rock compositions in geology, topic compositions in large-scale text corpora, and sequencing count data in molecular biology. Here, we provide a causal view on compositional data in an instrumental variable setting where the composition acts as the cause. First, we crisply articulate potential pitfalls for practitioners regarding the interpretation of compositional causes from the viewpoint of interventions and warn against attributing causal meaning to common summary statistics such as diversity indices. We then advocate for and develop multivariate methods using statistical data transformations and regression techniques that take the special structure of the compositional sample space into account. In a comparative analysis on synthetic and real data we show the advantages and limitations of our proposal. We posit that our framework provides a useful starting point and guidance for valid and informative cause-effect estimation in the context of compositional data.
When are inferences (whether Direct-Likelihood, Bayesian, or Frequentist) obtained from partial data valid? This paper answers this question by offering a new asymptotic theory about inference with missing data that is more general than existing theories. By using more powerful tools from real analysis and probability theory than those used in previous research, it proves that as the sample size increases and the extent of missingness decreases, the mean-loglikelihood function generated by partial data and that ignores the missingness mechanism will almost surely converge uniformly to that which would have been generated by complete data; and if the data are Missing at Random, this convergence depends only on sample size. Thus, inferences from partial data, such as posterior modes, uncertainty estimates, confidence intervals, likelihood ratios, test statistics, and indeed, all quantities or features derived from the partial-data loglikelihood function, will be consistently estimated. They will approximate their complete-data analogues. This adds to previous research which has only proved the consistency and asymptotic normality of the posterior mode, and developed separate theories for Direct-Likelihood, Bayesian, and Frequentist inference. Practical implications of this result are discussed, and the theory is verified using a previous study of International Human Rights Law.
The discrepant posterior phenomenon (DPP) is a counter-intuitive phenomenon that can frequently occur in a Bayesian analysis of multivariate parameters. It refers to the phenomenon that a parameter estimate based on a posterior is more extreme than both of those inferred based on either the prior or the likelihood alone. Inferential claims that exhibit DPP defy the common intuition that the posterior is a prior-data compromise, and the phenomenon can be surprisingly ubiquitous in well-behaved Bayesian models. In this paper we revisit this phenomenon and, using point estimation as an example, derive conditions under which the DPP occurs in Bayesian models with exponential quadratic likelihoods and conjugate multivariate Gaussian priors. The family of exponential quadratic likelihood models includes Gaussian models and those models with local asymptotic normality property. We provide an intuitive geometric interpretation of the phenomenon and show that there exists a nontrivial space of marginal directions such that the DPP occurs. We further relate the phenomenon to the Simpson's paradox and discover their deep-rooted connection that is associated with marginalization. We also draw connections with Bayesian computational algorithms when difficult geometry exists. Our discovery demonstrates that DPP is more prevalent than previously understood and anticipated. Theoretical results are complemented by numerical illustrations. Scenarios covered in this study have implications for parameterization, sensitivity analysis, and prior choice for Bayesian modeling.
The dominating NLP paradigm of training a strong neural predictor to perform one task on a specific dataset has led to state-of-the-art performance in a variety of applications (eg. sentiment classification, span-prediction based question answering or machine translation). However, it builds upon the assumption that the data distribution is stationary, ie. that the data is sampled from a fixed distribution both at training and test time. This way of training is inconsistent with how we as humans are able to learn from and operate within a constantly changing stream of information. Moreover, it is ill-adapted to real-world use cases where the data distribution is expected to shift over the course of a model's lifetime. The first goal of this thesis is to characterize the different forms this shift can take in the context of natural language processing, and propose benchmarks and evaluation metrics to measure its effect on current deep learning architectures. We then proceed to take steps to mitigate the effect of distributional shift on NLP models. To this end, we develop methods based on parametric reformulations of the distributionally robust optimization framework. Empirically, we demonstrate that these approaches yield more robust models as demonstrated on a selection of realistic problems. In the third and final part of this thesis, we explore ways of efficiently adapting existing models to new domains or tasks. Our contribution to this topic takes inspiration from information geometry to derive a new gradient update rule which alleviate catastrophic forgetting issues during adaptation.
Classic machine learning methods are built on the $i.i.d.$ assumption that training and testing data are independent and identically distributed. However, in real scenarios, the $i.i.d.$ assumption can hardly be satisfied, rendering the sharp drop of classic machine learning algorithms' performances under distributional shifts, which indicates the significance of investigating the Out-of-Distribution generalization problem. Out-of-Distribution (OOD) generalization problem addresses the challenging setting where the testing distribution is unknown and different from the training. This paper serves as the first effort to systematically and comprehensively discuss the OOD generalization problem, from the definition, methodology, evaluation to the implications and future directions. Firstly, we provide the formal definition of the OOD generalization problem. Secondly, existing methods are categorized into three parts based on their positions in the whole learning pipeline, namely unsupervised representation learning, supervised model learning and optimization, and typical methods for each category are discussed in detail. We then demonstrate the theoretical connections of different categories, and introduce the commonly used datasets and evaluation metrics. Finally, we summarize the whole literature and raise some future directions for OOD generalization problem. The summary of OOD generalization methods reviewed in this survey can be found at //out-of-distribution-generalization.com.
The focus of disentanglement approaches has been on identifying independent factors of variation in data. However, the causal variables underlying real-world observations are often not statistically independent. In this work, we bridge the gap to real-world scenarios by analyzing the behavior of the most prominent disentanglement approaches on correlated data in a large-scale empirical study (including 4260 models). We show and quantify that systematically induced correlations in the dataset are being learned and reflected in the latent representations, which has implications for downstream applications of disentanglement such as fairness. We also demonstrate how to resolve these latent correlations, either using weak supervision during training or by post-hoc correcting a pre-trained model with a small number of labels.
Conventional supervised learning methods, especially deep ones, are found to be sensitive to out-of-distribution (OOD) examples, largely because the learned representation mixes the semantic factor with the variation factor due to their domain-specific correlation, while only the semantic factor causes the output. To address the problem, we propose a Causal Semantic Generative model (CSG) based on a causal reasoning so that the two factors are modeled separately, and develop methods for OOD prediction from a single training domain, which is common and challenging. The methods are based on the causal invariance principle, with a novel design for both efficient learning and easy prediction. Theoretically, we prove that under certain conditions, CSG can identify the semantic factor by fitting training data, and this semantic-identification guarantees the boundedness of OOD generalization error and the success of adaptation. Empirical study shows improved OOD performance over prevailing baselines.
Feature attribution is often loosely presented as the process of selecting a subset of relevant features as a rationale of a prediction. This lack of clarity stems from the fact that we usually do not have access to any notion of ground-truth attribution and from a more general debate on what good interpretations are. In this paper we propose to formalise feature selection/attribution based on the concept of relaxed functional dependence. In particular, we extend our notions to the instance-wise setting and derive necessary properties for candidate selection solutions, while leaving room for task-dependence. By computing ground-truth attributions on synthetic datasets, we evaluate many state-of-the-art attribution methods and show that, even when optimised, some fail to verify the proposed properties and provide wrong solutions.
Optimal transport distances have found many applications in machine learning for their capacity to compare non-parametric probability distributions. Yet their algorithmic complexity generally prevents their direct use on large scale datasets. Among the possible strategies to alleviate this issue, practitioners can rely on computing estimates of these distances over subsets of data, {\em i.e.} minibatches. While computationally appealing, we highlight in this paper some limits of this strategy, arguing it can lead to undesirable smoothing effects. As an alternative, we suggest that the same minibatch strategy coupled with unbalanced optimal transport can yield more robust behavior. We discuss the associated theoretical properties, such as unbiased estimators, existence of gradients and concentration bounds. Our experimental study shows that in challenging problems associated to domain adaptation, the use of unbalanced optimal transport leads to significantly better results, competing with or surpassing recent baselines.
Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.
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