It is common practice in using regression type models for inferring causal effects, that inferring the correct causal relationship requires extra covariates are included or ``adjusted for''. Without performing this adjustment erroneous causal effects can be inferred. Given this phenomenon it is common practice to include as many covariates as possible, however such advice comes unstuck in the presence of M-bias. M-Bias is a problem in causal inference where the correct estimation of treatment effects requires that certain variables are not adjusted for i.e. are simply neglected from inclusion in the model. This issue caused a storm of controversy in 2009 when Rubin, Pearl and others disagreed about if it could be problematic to include additional variables in models when inferring causal effects. This paper makes two contributions to this issue. Firstly we provide a Bayesian solution to the M-Bias problem. The solution replicates Pearl's solution, but consistent with Rubin's advice we condition on all variables. Secondly the fact that we are able to offer a solution to this problem in Bayesian terms shows that it is indeed possible to represent causal relationships within the Bayesian paradigm, albeit in an extended space. We make several remarks on the similarities and differences between causal graphical models which implement the do-calculus and probabilistic graphical models which enable Bayesian statistics. We hope this work will stimulate more research on unifying Pearl's causal calculus using causal graphical models with traditional Bayesian statistics and probabilistic graphical models.
We consider Bayesian optimization in settings where observations can be adversarially biased, for example by an uncontrolled hidden confounder. Our first contribution is a reduction of the confounded setting to the dueling bandit model. Then we propose a novel approach for dueling bandits based on information-directed sampling (IDS). Thereby, we obtain the first efficient kernelized algorithm for dueling bandits that comes with cumulative regret guarantees. Our analysis further generalizes a previously proposed semi-parametric linear bandit model to non-linear reward functions, and uncovers in…
Given a supervised machine learning problem where the training set has been subject to a known sampling bias, how can a model be trained to fit the original dataset? We achieve this through the Bayesian inference framework by altering the posterior distribution to account for the sampling function. We then apply this solution to binary logistic regression, and discuss scenarios where a dataset might be subject to intentional sample bias such as label imbalance. This technique is widely applicable for statistical inference on big data, from the medical sciences to image recognition to marketing…
Kyoto University（京都大学）；National Institute of Informatics（国立信息学研究所）
A common strategy for sparse linear regression is to introduce regularization, which eliminates irrelevant features by letting the corresponding weights be zeros. However, regularization often shrinks the estimator for relevant features, which leads to incorrect feature selection. Motivated by the above-mentioned issue, we propose Bayesian masking (BM), a sparse estimation method which imposes no regularization on the weights. The key concept of BM is to introduce binary latent variables that randomly mask features. Estimating the masking rates determines the relevance of the features automati…
University of Tübingen（蒂宾根大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Full Bayesian posteriors are rarely analytically tractable, which is why real-world Bayesian inference heavily relies on approximate techniques. Approximations generally differ from the true posterior and require diagnostic tools to assess whether the inference can still be trusted. We investigate a new approach to diagnosing approximate inference: the approximation mismatch is attributed to a change in the inductive bias by treating the approximations as exact and reverse-engineering the corresponding prior. We show that the problem is more complicated than it appears to be at first glance, b…
London School of Economics（伦敦政治经济学院）；Royal Holloway College（皇家霍洛威学院）；University of London（伦敦大学）
In this paper the problem of learning appropriate bias for an environment of related tasks is examined from a Bayesian perspective. The environment of related tasks is shown to be naturally modelled by the concept of an {\em objective} prior distribution. Sampling from the objective prior corresponds to sampling different learning tasks from the environment. It is argued that for many common machine learning problems, although we don't know the true (objective) prior for the problem, we do have some idea of a set of possible priors to which the true prior belongs. It is shown that under these …
Imperial College London（帝国理工学院）；King’s College London（伦敦国王学院）；University College London（伦敦大学学院）
We derive a family of loss functions to train models in the presence of sampling bias. Examples are when the prevalence of a pathology differs from its sampling rate in the training dataset, or when a machine learning practioner rebalances their training dataset. Sampling bias causes large discrepancies between model performance in the lab and in more realistic settings. It is omnipresent in medical imaging applications, yet is often overlooked at training time or addressed on an ad-hoc basis. Our approach is based on Bayesian risk minimization. For arbitrary likelihood models we derive the as…
CERMICS, Ecole des Ponts（巴黎桥路学校 CERMICS 实验室）；Inria Paris（法国国家信息与自动化研究所巴黎中心）
Bayesian inference allows to obtain useful information on the parameters of models, either in computational statistics or more recently in the context of Bayesian Neural Networks. The computational cost of usual Monte Carlo methods for sampling posterior laws in Bayesian inference scales linearly with the number of data points. One option to reduce it to a fraction of this cost is to resort to mini-batching in conjunction with unadjusted discretizations of Langevin dynamics, in which case only a random fraction of the data is used to estimate the gradient. However, this leads to an additional …
Imperial College London（帝国理工学院）；King’s College London（伦敦国王学院）；The Royal Brompton & Harefield NHS Foundation Trust（皇家布朗普顿与哈菲尔德国民保健服务基金会信托）；University College London（伦敦大学学院）
Datasets are rarely a realistic approximation of the target population. Say, prevalence is misrepresented, image quality is above clinical standards, etc. This mismatch is known as sampling bias. Sampling biases are a major hindrance for machine learning models. They cause significant gaps between model performance in the lab and in the real world. Our work is a solution to prevalence bias. Prevalence bias is the discrepancy between the prevalence of a pathology and its sampling rate in the training dataset, introduced upon collecting data or due to the practioner rebalancing the training batc…
Correcting Boundary Bias and Observation Independence in Bayesian Experimental Design
Department of Information Technology, Uppsala University, Uppsala, Sweden（信息科技系，乌普萨拉大学）；Department of Solar Cell Technology, Uppsala University, Uppsala, Sweden（太阳能电池技术系，乌普萨拉大学）；Department of Computer Science, RISE Research Institute of Sweden, Stockholm, Sweden（计算机科学系，瑞典RISE研究所）
A density estimation method in a Bayesian nonparametric framework is presented when recorded data are not coming directly from the distribution of interest, but from a length biased version. From a Bayesian perspective, efforts to computationally evaluate posterior quantities conditionally on length biased data were hindered by the inability to circumvent the problem of a normalizing constant. In this paper we present a novel Bayesian nonparametric approach to the length bias sampling problem which circumvents the issue of the normalizing constant. Numerical illustrations as well as a real dat…
A common concern with Bayesian methodology in scientific contexts is that inferences can be heavily influenced by subjective biases. As presented here, there are two types of bias for some quantity of interest: bias against and bias in favor. Based upon the principle of evidence, it is shown how to measure and control these biases for both hypothesis assessment and estimation problems. Optimality results are established for the principle of evidence as the basis of the approach to these problems. A close relationship is established between measuring bias in Bayesian inferences and frequentist …
As machine learning algorithms are increasingly deployed for high-impact automated decision making, ethical and increasingly also legal standards demand that they treat all individuals fairly, without discrimination based on their age, gender, race or other sensitive traits. In recent years much progress has been made on ensuring fairness and reducing bias in standard machine learning settings. Yet, for network embedding, with applications in vulnerable domains ranging from social network analysis to recommender systems, current options remain limited both in number and performance. We thus pr…
University of Maryland, Baltimore County（马里兰大学巴尔的摩县分校）
Intersectionality is a framework that analyzes how interlocking systems of power and oppression affect individuals along overlapping dimensions including race, gender, sexual orientation, class, and disability. Intersectionality theory therefore implies it is important that fairness in artificial intelligence systems be protected with regard to multi-dimensional protected attributes. However, the measurement of fairness becomes statistically challenging in the multi-dimensional setting due to data sparsity, which increases rapidly in the number of dimensions, and in the values per dimension. W…
Overcoming Selection Bias in Statistical Studies With Amortized Bayesian Inference
University of Bonn（波恩大学）；Institut Pasteur（巴斯德研究所）；University of Freiburg（弗赖堡大学）；LMU University Hospital（慕尼黑大学附属医院）；German Center for Infection Research（德国感染研究中心）；Fraunhofer Institute ITMP（弗劳恩霍夫ITMP研究所）；Max von Pettenkofer Institute（马克斯·冯·佩滕科弗研究所）；LMU Munich（慕尼黑大学）；German Research Center for Environmental Health（德国环境健康研究中心）；Institut Necker-Enfants Malades（内克-儿童疾病研究所）
Meta-analysis is a statistical method used in evidence synthesis for combining, analyzing and summarizing studies that have the same target endpoint and aims to derive a pooled quantitative estimate using fixed and random effects models or network models. Differences among included studies depend on variations in target populations (i.e. heterogeneity) and variations in study quality due to study design and execution (i.e. bias). The risk of bias is usually assessed qualitatively using critical appraisal, and quantitative bias analysis can be used to evaluate the influence of bias on the quant…
Online field experiments are the gold-standard way of evaluating changes to real-world interactive machine learning systems. Yet our ability to explore complex, multi-dimensional policy spaces - such as those found in recommendation and ranking problems - is often constrained by the limited number of experiments that can be run simultaneously. To alleviate these constraints, we augment online experiments with an offline simulator and apply multi-task Bayesian optimization to tune live machine learning systems. We describe practical issues that arise in these types of applications, including bi…
Bayesian generative models can flag performance loss, bias, and out-of-distribution image content
Instituto Universitario de Investigación en Tecnología Centrada en el Ser Humano, Universitat Politècnica de València（瓦伦西亚理工大学人类中心技术研究大学学院）；Technical University of Denmark（丹麦技术大学）
We design a double-or-quits game to compare the speed of learning one's specific ability with the speed of rising confidence as the task gets increasingly difficult. We find that people on average learn to be overconfident faster than they learn their true ability and we present an intuitive-Bayesian model of confidence which integrates confidence biases and learning. Uncertainty about one's true ability to perform a task in isolation can be responsible for large and stable confidence biases, namely limited discrimination, the hard--easy effect, the Dunning--Kruger effect, conservative learnin…
This paper introduces a novel parameter estimation method for the probability tables of Bayesian network classifiers (BNCs), using hierarchical Dirichlet processes (HDPs). The main result of this paper is to show that improved parameter estimation allows BNCs to outperform leading learning methods such as Random Forest for both 0-1 loss and RMSE, albeit just on categorical datasets. As data assets become larger, entering the hyped world of "big", efficient accurate classification requires three main elements: (1) classifiers with low-bias that can capture the fine-detail of large datasets (2) …
Overparameterised deep neural networks (DNNs) are highly expressive and so can, in principle, generate almost any function that fits a training dataset with zero error. The vast majority of these functions will perform poorly on unseen data, and yet in practice DNNs often generalise remarkably well. This success suggests that a trained DNN must have a strong inductive bias towards functions with low generalisation error. Here we empirically investigate this inductive bias by calculating, for a range of architectures and datasets, the probability $P_{SGD}(f\mid S)$ that an overparameterised DNN…
Evidence of bias in the Eurovision song contest: modelling the votes using Bayesian hierarchical models
欧洲歌唱大赛（Eurovision Song Contest）是欧洲广播联盟（European Broadcasting Union）自1956年以来举办的年度音乐比赛。该活动在欧洲各地进行电视直播。每个参赛国家都将献上一首歌曲，并通过电话投票和陪审团投票的方式进行投票。多年来，这导致了对战术投票的猜测，歧视了一些参与者，从而导致最终结果的偏见。在本文中，我们通过贝叶斯分层模型研究了参与国的地理邻近性，移民和文化特征的投票中存在的积极或消极偏见（这可能大致表明偏袒或歧视）。我们的分析没有发现负面偏见的证据，尽管轻微的正面偏见似乎系统地出现，将选民与表演者联系起来。
University of Amsterdam（阿姆斯特丹大学）；Leiden University（莱顿大学）；Statistics Netherlands（荷兰统计局）；Tinbergen Institute（丁伯根研究所）
In this paper, we study the accuracy of values aggregated over classes predicted by a classification algorithm. The problem is that the resulting aggregates (e.g., sums of a variable) are known to be biased. The bias can be large even for highly accurate classification algorithms, in particular when dealing with class-imbalanced data. To correct this bias, the algorithm's classification error rates have to be estimated. In this estimation, two issues arise when applying existing bias correction methods. First, inaccuracies in estimating classification error rates have to be taken into account.…
We present a robust Bayesian method to analyze forestry data when samples are selected with probability proportional to length from a finite population of unknown size. Specifically, we use Bayesian predictive inference to estimate the finite population mean of shrub widths in a limestone quarry dominated by re-growth of mountain mahogany. The data on shrub widths are collected using transect sampling and it is assumed that the probability that a shrub is selected is proportional to its width; this is length-biased sampling. In this type of sampling, the population size is also unknown and thi…
Institute of Mathematical Sciences (ICMAT-CSIC)（数学科学研究所（ICMAT-CSIC））；Escuela Politécnica Superior, Universidad Autónoma de Madrid（马德里自治大学高等理工学院）；Institute of Corpuscular Physics, University of Valencia and CSIC（瓦伦西亚大学与西班牙国家研究委员会粒子物理研究所）
Model selection in machine learning (ML) is a crucial part of the Bayesian learning procedure. Model choice may impose strong biases on the resulting predictions, which can hinder the performance of methods such as Bayesian neural networks and neural samplers. On the other hand, newly proposed approaches for Bayesian ML exploit features of approximate inference in function space with implicit stochastic processes (a generalization of Gaussian processes). The approach of Sparse Implicit Processes (SIP) is particularly successful in this regard, since it is fully trainable and achieves flexible …
The Receiver Operating Characteristic (ROC) surface is a generalization of ROC curve and is widely used for assessment of the accuracy of diagnostic tests on three categories. A complication called the verification bias, meaning that not all subjects have their true disease status verified often occur in real application of ROC analysis. This is a common problem since the gold standard test, which is used to generate true disease status, can be invasive and expensive. In this paper, we will propose a Bayesian approach for estimating the ROC surface based on continuous data under a semi-paramet…
Bayesian hybrid models fuse physics-based insights with machine learning constructs to correct for systematic bias. In this paper, we compare Bayesian hybrid models against physics-based glass-box and Gaussian process black-box surrogate models. We consider ballistic firing as an illustrative case study for a Bayesian decision-making workflow. First, Bayesian calibration is performed to estimate model parameters. We then use the posterior distribution from Bayesian analysis to compute optimal firing conditions to hit a target via a single-stage stochastic program. The case study demonstrates t…
In this paper we present a technique to couple non-traditional data with statistics based on survey data, in order to partially correct for the bias produced by non-random sample selections. All major social media platforms represent huge samples of the general population, generated by a self-selection process. This implies that they are not representative of the larger public, and there are problems in extrapolating conclusions drawn from these samples to the whole population. We present an algorithm to integrate these massive data with ones coming from traditional sources, with the propertie…
In epidemiological surveys, data missing not at random (MNAR) due to survey nonresponse may potentially lead to a bias in the risk factor estimates. We propose an approach based on Bayesian data augmentation and survival modelling to reduce the nonresponse bias. The approach requires additional information based on follow-up data. We present a case study of smoking prevalence using FINRISK data collected between 1972 and 2007 with a follow-up to the end of 2012 and compare it to other commonly applied missing at random (MAR) imputation approaches. A simulation experiment is carried out to stud…
J.P. Morgan AI Research（摩根大通人工智能研究院）；Carnegie Mellon University（卡内基梅隆大学）
We consider the problem of aggregating predictions or measurements from a set of human forecasters, models, sensors or other instruments which may be subject to bias or miscalibration and random heteroscedastic noise. We propose a Bayesian consensus estimator that adjusts for miscalibration and noise and show that this estimator is unbiased and asymptotically more efficient than naive alternatives. We further propose a Hierarchical Bayesian Model that leverages our proposed estimator and apply it to two real world forecasting challenges that require consensus estimates from error prone individ…
The validity of conclusions from meta-analysis is potentially threatened by publication bias. Most existing procedures for correcting publication bias assume normality of the study-specific effects that account for between-study heterogeneity. However, this assumption may not be valid, and the performance of these bias correction procedures can be highly sensitive to departures from normality. Further, there exist few measures to quantify the magnitude of publication bias based on selection models. In this paper, we address both of these issues. First, we explore the use of heavy-tailed distri…
We present a Bayesian reconstruction algorithm to generate unbiased samples of the underlying dark matter field from halo catalogues. Our new contribution consists of implementing a non-Poisson likelihood including a deterministic non-linear and scale-dependent bias. In particular we present the Hamiltonian equations of motions for the negative binomial (NB) probability distribution function. This permits us to efficiently sample the posterior distribution function of density fields given a sample of galaxies using the Hamiltonian Monte Carlo technique implemented in the Argo code. We have tes…
We investigate the problem of truth discovery based on opinions from multiple agents who may be unreliable or biased. We consider the case where agents' reliabilities or biases are correlated if they belong to the same community, which defines a group of agents with similar opinions regarding a particular event. An agent can belong to different communities for different events, and these communities are unknown a priori. We incorporate knowledge of the agents' social network in our truth discovery framework and develop Laplace variational inference methods to estimate agents' reliabilities, co…
Queen Mary University of London（伦敦玛丽女王大学）；Agena Ltd（阿格娜有限公司）；Koch Engineered Solutions（科赫工程解决方案公司）
Contradictory conclusions have been made about whether unarmed blacks are more likely to be shot by police than unarmed whites using the same data. The problem is that, by relying only on data of 'police encounters', there is the possibility that genuine bias can be hidden. We provide a causal Bayesian network model to explain this bias, which is called collider bias or Berkson's paradox, and show how the different conclusions arise from the same model and data. We also show that causal Bayesian networks provide the ideal formalism for considering alternative hypotheses and explanations of bia…
Consider the problem of modeling hysteresis for finite-state random walks using higher-order Markov chains. This Letter introduces a Bayesian framework to determine, from data, the number of prior states of recent history upon which a trajectory is statistically dependent. The general recommendation is to use leave-one-out cross validation, using an easily-computable formula that is provided in closed form. Importantly, Bayes factors using flat model priors are biased in favor of too-complex a model (more hysteresis) when a large amount of data is present and the Akaike information criterion (…
The sample mean is often used to aggregate different unbiased estimates of a parameter, producing a final estimate that is unbiased but possibly high-variance. This paper introduces the Bayesian median of means, an aggregation rule that roughly interpolates between the sample mean and median, resulting in estimates with much smaller variance at the expense of bias. While the procedure is non-parametric, its squared bias is asymptotically negligible relative to the variance, similar to maximum likelihood estimators. The Bayesian median of means is consistent, and concentration bounds for the es…
We propose an analysis for a novel experiment intended to locate the genetic locus Xce (X-chromosome controlling element), which biases the stochastic process of X-inactivation in the mouse. X-inactivation bias is a phenomenon where cells in the embryo randomly choose one parental chromosome to inactivate, but show an average bias towards one parental strain. Measurement of allele-specific gene-expression through pyrosequencing was conducted on mouse crosses of an uncharacterized parent with known carriers. Our Bayesian analysis is suitable for this adaptive experimental design, accounting for…
Multivariate meta-analysis (MMA) is a powerful tool for jointly estimating multiple outcomes' treatment effects. However, the validity of results from MMA is potentially compromised by outcome reporting bias (ORB), or the tendency for studies to selectively report outcomes. Until recently, ORB has been understudied. Since ORB can lead to biased conclusions, it is crucial to correct the estimates of effect sizes and quantify their uncertainty in the presence of ORB. With this goal, we develop a Bayesian selection model to adjust for ORB in MMA. We further propose a measure for quantifying the i…
Estimates of the under-five mortality rate (U5MR) are used to track progress in reducing child mortality and to evaluate countries' performance related to Millennium Development Goal 4. However, for the great majority of developing countries without well-functioning vital registration systems, estimating the U5MR is challenging due to limited data availability and data quality issues. We describe a Bayesian penalized B-spline regression model for assessing levels and trends in the U5MR for all countries in the world, whereby biases in data series are estimated through the inclusion of a multil…
Distributed lag models (DLMs) express the cumulative and delayed dependence between pairs of time-indexed response and explanatory variables. In practical application, users of DLMs examine the estimated influence of a series of lagged covariates to assess patterns of dependence. Much recent methodological work has sought to de- velop flexible parameterisations for smoothing the associated lag parameters that avoid overfitting. However, this paper finds that some widely-used DLMs introduce bias in the estimated lag influence, and are sensitive to the maximum lag which is typically chosen in ad…
Bias reduction in tail estimation has received considerable interest in extreme value analysis. Estimation methods that minimize the bias while keeping the mean squared error (MSE) under control, are especially useful when applying classical methods such as the Hill (1975) estimator. In Caeiro et al. (2005) minimum variance reduced bias estimators of the Pareto tail index were first proposed where the bias is reduced without increasing the variance with respect to the Hill estimator. This method is based on adequate external estimation of a pair of second-order parameters. Here we revisit this…
Inverse Probability Weighting in a Post-Bayesian World
Oslo Centre for Biostatistics and Epidemiology, Oslo University Hospital（奥斯陆生物统计与流行病学中心，奥斯陆大学医院）；Integreat, the Norwegian Centre for Knowledge-driven Machine Learning（Integreat，挪威知识驱动机器学习中心）；Oslo Centre for Biostatistics and Epidemiology, Department of Biostatistics, University of Oslo（奥斯陆生物统计与流行病学中心，奥斯陆大学生物统计系）
