The Regression Discontinuity (RD) design is a widely used non-experimental method for causal inference and program evaluation. While its canonical formulation only requires a score and an outcome variable, it is common in empirical work to encounter RD implementations where additional variables are used for adjustment. This practice has led to misconceptions about the role of covariate adjustment in RD analysis, from both methodological and empirical perspectives. In this chapter, we review the different roles of covariate adjustment in RD designs, and offer methodological guidance for its correct use in applications.
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Randomized response, as a basic building-block for differentially private mechanism, has given rise to great interest and found various potential applications in science communities. In this work, we are concerned with three-elements randomized response (RR$_{3}$) along with relevant applications to the analysis of weighted bipartite graph upon differentially private guarantee. We develop a principled framework for estimating statistics produced by RR$_{3}$-based mechanisms, and then prove the corresponding estimations to be unbiased. At the same time, we study in detail several fundamental and significant members in RR$_{3}$ family, and derive the closed-form solutions to unbiased estimations. Next, we show potential applications of several RR$_{3}$-based mechanisms into the estimation of average degree and average weighted value on weighted bipartite graph when requiring local differential privacy guarantee. In the meantime, we determine the lower bounds for choice of relevant parameters by minimizing variance of statistics in order to design optimal RR$_{3}$-based local differential private mechanisms, with which we optimize previous protocols in the literature and put forward a version that achieves the tight bound. Last but most importantly, we observe that in the analysis of relational data such as weighted bipartite graph, a portion of privacy budget in local differential private mechanism is sometimes "consumed" by mechanism itself accidentally, resulting to a more stronger privacy guarantee than we would get by simply sequential compositions.
Quasi-experimental research designs, such as regression discontinuity and interrupted time series, allow for causal inference in the absence of a randomized controlled trial, at the cost of additional assumptions. In this paper, we provide a framework for discontinuity-based designs using Bayesian model comparison and Gaussian process regression, which we refer to as 'Bayesian nonparametric discontinuity design', or BNDD for short. BNDD addresses the two major shortcomings in most implementations of such designs: overconfidence due to implicit conditioning on the alleged effect, and model misspecification due to reliance on overly simplistic regression models. With the appropriate Gaussian process covariance function, our approach can detect discontinuities of any order, and in spectral features. We demonstrate the usage of BNDD in simulations, and apply the framework to determine the effect of running for political positions on longevity, of the effect of an alleged historical phantom border in the Netherlands on Dutch voting behaviour, and of Kundalini Yoga meditation on heart rate.
Understanding the effects of social determinants of health on health outcomes requires data on characteristics of the neighborhoods in which subjects live. However, estimates of these characteristics are often aggregated over space and time in a fashion that diminishes their utility. Take, for example, estimates from the American Community Survey (ACS), a multi-year nationwide survey administered by the U.S. Census Bureau: estimates for small municipal areas are aggregated over 5-year periods, whereas 1-year estimates are only available for municipal areas with populations $>$65,000. Researchers may wish to use ACS estimates in studies of population health to characterize neighborhood-level exposures. However, 5-year estimates may not properly characterize temporal changes or align temporally with other data in the study, while the coarse spatial resolution of the 1-year estimates diminishes their utility in characterizing neighborhood exposure. To circumvent this issue, in this paper we propose a modeling framework to disaggregate estimates of proportions derived from sampling surveys which explicitly accounts for the survey design effect. We illustrate the utility of our model by applying it to the ACS data, generating estimates of poverty for the state of Michigan at fine spatio-temporal resolution.
In practice functional data are sampled on a discrete set of observation points and often susceptible to noise. We consider in this paper the setting where such data are used as explanatory variables in a regression problem. If the primary goal is prediction, we show that the gain by embedding the problem into a scalar-on-function regression is limited. Instead we impose a factor model on the predictors and suggest regressing the response on an appropriate number of factor scores. This approach is shown to be consistent under mild technical assumptions, numerically efficient and gives good practical performance in both simulations as well as real data settings.
Over the last few decades, various methods have been proposed for estimating prediction intervals in regression settings, including Bayesian methods, ensemble methods, direct interval estimation methods and conformal prediction methods. An important issue is the calibration of these methods: the generated prediction intervals should have a predefined coverage level, without being overly conservative. In this work, we review the above four classes of methods from a conceptual and experimental point of view. Results on benchmark data sets from various domains highlight large fluctuations in performance from one data set to another. These observations can be attributed to the violation of certain assumptions that are inherent to some classes of methods. We illustrate how conformal prediction can be used as a general calibration procedure for methods that deliver poor results without a calibration step.
We provide guarantees for approximate Gaussian Process (GP) regression resulting from two common low-rank kernel approximations: based on random Fourier features, and based on truncating the kernel's Mercer expansion. In particular, we bound the Kullback-Leibler divergence between an exact GP and one resulting from one of the afore-described low-rank approximations to its kernel, as well as between their corresponding predictive densities, and we also bound the error between predictive mean vectors and between predictive covariance matrices computed using the exact versus using the approximate GP. We provide experiments on both simulated data and standard benchmarks to evaluate the effectiveness of our theoretical bounds.
Fitting regression models with many multivariate responses and covariates can be challenging, but such responses and covariates sometimes have tensor-variate structure. We extend the classical multivariate regression model to exploit such structure in two ways: first, we impose four types of low-rank tensor formats on the regression coefficients. Second, we model the errors using the tensor-variate normal distribution that imposes a Kronecker separable format on the covariance matrix. We obtain maximum likelihood estimators via block-relaxation algorithms and derive their computational complexity and asymptotic distributions. Our regression framework enables us to formulate tensor-variate analysis of variance (TANOVA) methodology. This methodology, when applied in a one-way TANOVA layout, enables us to identify cerebral regions significantly associated with the interaction of suicide attempters or non-attemptor ideators and positive-, negative- or death-connoting words in a functional Magnetic Resonance Imaging study. Another application uses three-way TANOVA on the Labeled Faces in the Wild image dataset to distinguish facial characteristics related to ethnic origin, age group and gender.
The main objective of this work is to calculate the multivariate double truncated expectation (MDTE) and covariance (MDTCov) for elliptical distributions. We also consider double truncated expectation (DTE) and variance (DTV) for univariate elliptical distributions. The exact expressions of MDTE and MDTCov are derived for some special cases of the family, such as normal, student-$t$, logistic, Laplace and Pearson type VII distributions. As numerical illustration, the DTE, DTV, MDTE and MDTCov for normal distribution are computed in details. Finally, we discuss MDTE and MDTCov of three industry segments' (Banks, Insurance, Financial and Credit Service) stock return in London stock exchange.
Heatmap-based methods dominate in the field of human pose estimation by modelling the output distribution through likelihood heatmaps. In contrast, regression-based methods are more efficient but suffer from inferior performance. In this work, we explore maximum likelihood estimation (MLE) to develop an efficient and effective regression-based methods. From the perspective of MLE, adopting different regression losses is making different assumptions about the output density function. A density function closer to the true distribution leads to a better regression performance. In light of this, we propose a novel regression paradigm with Residual Log-likelihood Estimation (RLE) to capture the underlying output distribution. Concretely, RLE learns the change of the distribution instead of the unreferenced underlying distribution to facilitate the training process. With the proposed reparameterization design, our method is compatible with off-the-shelf flow models. The proposed method is effective, efficient and flexible. We show its potential in various human pose estimation tasks with comprehensive experiments. Compared to the conventional regression paradigm, regression with RLE bring 12.4 mAP improvement on MSCOCO without any test-time overhead. Moreover, for the first time, especially on multi-person pose estimation, our regression method is superior to the heatmap-based methods. Our code is available at //github.com/Jeff-sjtu/res-loglikelihood-regression
Discovering causal structure among a set of variables is a fundamental problem in many empirical sciences. Traditional score-based casual discovery methods rely on various local heuristics to search for a Directed Acyclic Graph (DAG) according to a predefined score function. While these methods, e.g., greedy equivalence search, may have attractive results with infinite samples and certain model assumptions, they are usually less satisfactory in practice due to finite data and possible violation of assumptions. Motivated by recent advances in neural combinatorial optimization, we propose to use Reinforcement Learning (RL) to search for the DAG with the best scoring. Our encoder-decoder model takes observable data as input and generates graph adjacency matrices that are used to compute rewards. The reward incorporates both the predefined score function and two penalty terms for enforcing acyclicity. In contrast with typical RL applications where the goal is to learn a policy, we use RL as a search strategy and our final output would be the graph, among all graphs generated during training, that achieves the best reward. We conduct experiments on both synthetic and real datasets, and show that the proposed approach not only has an improved search ability but also allows a flexible score function under the acyclicity constraint.
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