The task of mixture proportion estimation (MPE) is to estimate the weight of a component distribution in a mixture, given observations from both the component and mixture. Previous work on MPE adopts the irreducibility assumption, which ensures identifiablity of the mixture proportion. In this paper, we propose a more general sufficient condition that accommodates several settings of interest where irreducibility does not hold. We further present a resampling-based meta-algorithm that takes any existing MPE algorithm designed to work under irreducibility and adapts it to work under our more general condition. Our approach empirically exhibits improved estimation performance relative to baseline methods and to a recently proposed regrouping-based algorithm.
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We suggest a simple Gaussian mixture model for data generation that complies with Feldman's long tail theory (2020). We demonstrate that a linear classifier cannot decrease the generalization error below a certain level in the proposed model, whereas a nonlinear classifier with a memorization capacity can. This confirms that for long-tailed distributions, rare training examples must be considered for optimal generalization to new data. Finally, we show that the performance gap between linear and nonlinear models can be lessened as the tail becomes shorter in the subpopulation frequency distribution, as confirmed by experiments on synthetic and real data.
Treatment effect estimates are often available from randomized controlled trials as a single average treatment effect for a certain patient population. Estimates of the conditional average treatment effect (CATE) are more useful for individualized treatment decision making, but randomized trials are often too small to estimate the CATE. Examples in medical literature make use of the relative treatment effect (e.g. an odds-ratio) reported by randomized trials to estimate the CATE using large observational datasets. One approach to estimating these CATE models is by using the relative treatment effect as an offset, while estimating the covariate-specific untreated risk. We observe that the odds-ratios reported in randomized controlled trials are not the odds-ratios that are needed in offset models because trials often report the marginal odds-ratio. We introduce a constraint or regularizer to better use marginal odds-ratios from randomized controlled trials and find that under the standard observational causal inference assumptions this approach provides a consistent estimate of the CATE. Next, we show that the offset approach is not valid for CATE estimation in the presence of unobserved confounding. We study if the offset assumption and the marginal constraint lead to better approximations of the CATE relative to the alternative of using the average treatment effect estimate from the randomized trial. We empirically show that when the underlying CATE has sufficient variation, the constraint and offset approaches lead to closer approximations to the CATE.
Hypertension is a highly prevalent chronic medical condition and a strong risk factor for cardiovascular disease (CVD), as it accounts for more than $45\%$ of CVD. The relation between blood pressure (BP) and its risk factors cannot be explored clearly by standard linear models. Although the fractional polynomials (FPs) can act as a concise and accurate formula for examining smooth relationships between response and predictors, modelling conditional mean functions observes the partial view of a distribution of response variable, as the distributions of many response variables such as BP measures are typically skew. Then modelling 'average' BP may link to CVD but extremely high BP could explore CVD insight deeply and precisely. So, existing mean-based FP approaches for modelling the relationship between factors and BP cannot answer key questions in need. Conditional quantile functions with FPs provide a comprehensive relationship between the response variable and its predictors, such as median and extremely high BP measures that may be often required in practical data analysis generally. To the best of our knowledge, this is new in the literature. Therefore, in this paper, we employ Bayesian variable selection with quantile-dependent prior for the FP model to propose a Bayesian variable selection with parametric nonlinear quantile regression model. The objective is to examine a nonlinear relationship between BP measures and their risk factors across median and upper quantile levels using data extracted from the 2007-2008 National Health and Nutrition Examination Survey (NHANES). The variable selection in the model analysis identified that the nonlinear terms of continuous variables (body mass index, age), and categorical variables (ethnicity, gender and marital status) were selected as important predictors in the model across all quantile levels.
This paper proposes a learning model of online ad auctions that allows for the following four key realistic characteristics of contemporary online auctions: (1) ad slots can have different values and click-through rates depending on users' search queries, (2) the number and identity of competing advertisers are unobserved and change with each auction, (3) advertisers only receive partial, aggregated feedback, and (4) payment rules are only partially specified. We model advertisers as agents governed by an adversarial bandit algorithm, independent of auction mechanism intricacies. Our objective is to simulate the behavior of advertisers for counterfactual analysis, prediction, and inference purposes. Our findings reveal that, in such richer environments, "soft floors" can enhance key performance metrics even when bidders are drawn from the same population. We further demonstrate how to infer advertiser value distributions from observed bids, thereby affirming the practical efficacy of our approach even in a more realistic auction setting.
In many modern statistical problems, the limited available data must be used both to develop the hypotheses to test, and to test these hypotheses-that is, both for exploratory and confirmatory data analysis. Reusing the same dataset for both exploration and testing can lead to massive selection bias, leading to many false discoveries. Selective inference is a framework that allows for performing valid inference even when the same data is reused for exploration and testing. In this work, we are interested in the problem of selective inference for data clustering, where a clustering procedure is used to hypothesize a separation of the data points into a collection of subgroups, and we then wish to test whether these data-dependent clusters in fact represent meaningful differences within the data. Recent work by Gao et al. [2022] provides a framework for doing selective inference for this setting, where a hierarchical clustering algorithm is used for producing the cluster assignments, which was then extended to k-means clustering by Chen and Witten [2022]. Both these works rely on assuming a known covariance structure for the data, but in practice, the noise level needs to be estimated-and this is particularly challenging when the true cluster structure is unknown. In our work, we extend this work to the setting of noise with unknown variance, and provide a selective inference method for this more general setting. Empirical results show that our new method is better able to maintain high power while controlling Type I error when the true noise level is unknown.
We study partially linear models in settings where observations are arranged in independent groups but may exhibit within-group dependence. Existing approaches estimate linear model parameters through weighted least squares, with optimal weights (given by the inverse covariance of the response, conditional on the covariates) typically estimated by maximising a (restricted) likelihood from random effects modelling or by using generalised estimating equations. We introduce a new 'sandwich loss' whose population minimiser coincides with the weights of these approaches when the parametric forms for the conditional covariance are well-specified, but can yield arbitrarily large improvements in linear parameter estimation accuracy when they are not. Under relatively mild conditions, our estimated coefficients are asymptotically Gaussian and enjoy minimal variance among estimators with weights restricted to a given class of functions, when user-chosen regression methods are used to estimate nuisance functions. We further expand the class of functional forms for the weights that may be fitted beyond parametric models by leveraging the flexibility of modern machine learning methods within a new gradient boosting scheme for minimising the sandwich loss. We demonstrate the effectiveness of both the sandwich loss and what we call 'sandwich boosting' in a variety of settings with simulated and real-world data.
Machine learning (ML) and deep learning models are extensively used for parameter optimization and regression problems. However, not all inverse problems in ML are ``identifiable,'' indicating that model parameters may not be uniquely determined from the available data and the data model's input-output relationship. In this study, we investigate the notion of model parameter identifiability through a case study focused on parameter estimation from motion sensor data. Utilizing a bipedal-spring mass human walk dynamics model, we generate synthetic data representing diverse gait patterns and conditions. Employing a deep neural network, we attempt to estimate subject-wise parameters, including mass, stiffness, and equilibrium leg length. The results show that while certain parameters can be identified from the observation data, others remain unidentifiable, highlighting that unidentifiability is an intrinsic limitation of the experimental setup, necessitating a change in data collection and experimental scenarios. Beyond this specific case study, the concept of identifiability has broader implications in ML and deep learning. Addressing unidentifiability requires proven identifiable models (with theoretical support), multimodal data fusion techniques, and advancements in model-based machine learning. Understanding and resolving unidentifiability challenges will lead to more reliable and accurate applications across diverse domains, transcending mere model convergence and enhancing the reliability of machine learning models.
In inverse problems, one attempts to infer spatially variable functions from indirect measurements of a system. To practitioners of inverse problems, the concept of "information" is familiar when discussing key questions such as which parts of the function can be inferred accurately and which cannot. For example, it is generally understood that we can identify system parameters accurately only close to detectors, or along ray paths between sources and detectors, because we have "the most information" for these places. Although referenced in many publications, the "information" that is invoked in such contexts is not a well understood and clearly defined quantity. Herein, we present a definition of information density that is based on the variance of coefficients as derived from a Bayesian reformulation of the inverse problem. We then discuss three areas in which this information density can be useful in practical algorithms for the solution of inverse problems, and illustrate the usefulness in one of these areas -- how to choose the discretization mesh for the function to be reconstructed -- using numerical experiments.
Many food products involve mixtures of ingredients, where the mixtures can be expressed as combinations of ingredient proportions. In many cases, the quality and the consumer preference may also depend on the way in which the mixtures are processed. The processing is generally defined by the settings of one or more process variables. Experimental designs studying the joint impact of the mixture ingredient proportions and the settings of the process variables are called mixture-process variable experiments. In this article, we show how to combine mixture-process variable experiments and discrete choice experiments, to quantify and model consumer preferences for food products that can be viewed as processed mixtures. First, we describe the modeling of data from such combined experiments. Next, we describe how to generate D- and I-optimal designs for choice experiments involving mixtures and process variables, and we compare the two kinds of designs using two examples.
Graph neural networks (GNNs) have been proven to be effective in various network-related tasks. Most existing GNNs usually exploit the low-frequency signals of node features, which gives rise to one fundamental question: is the low-frequency information all we need in the real world applications? In this paper, we first present an experimental investigation assessing the roles of low-frequency and high-frequency signals, where the results clearly show that exploring low-frequency signal only is distant from learning an effective node representation in different scenarios. How can we adaptively learn more information beyond low-frequency information in GNNs? A well-informed answer can help GNNs enhance the adaptability. We tackle this challenge and propose a novel Frequency Adaptation Graph Convolutional Networks (FAGCN) with a self-gating mechanism, which can adaptively integrate different signals in the process of message passing. For a deeper understanding, we theoretically analyze the roles of low-frequency signals and high-frequency signals on learning node representations, which further explains why FAGCN can perform well on different types of networks. Extensive experiments on six real-world networks validate that FAGCN not only alleviates the over-smoothing problem, but also has advantages over the state-of-the-arts.
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