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Causal mediation analysis concerns the pathways through which a treatment affects an outcome. While most of the mediation literature focuses on settings with a single mediator, a flourishing line of research has examined settings involving multiple mediators, under which path-specific effects (PSEs) are often of interest. We consider estimation of PSEs when the treatment effect operates through K(\geq1) causally ordered, possibly multivariate mediators. In this setting, the PSEs for many causal paths are not nonparametrically identified, and we focus on a set of PSEs that are identified under Pearl's nonparametric structural equation model. These PSEs are defined as contrasts between the expectations of 2^{K+1} potential outcomes and identified via what we call the generalized mediation functional (GMF). We introduce an array of regression-imputation, weighting, and "hybrid" estimators, and, in particular, two K+2-robust and locally semiparametric efficient estimators for the GMF. The latter estimators are well suited to the use of data-adaptive methods for estimating their nuisance functions. We establish the rate conditions required of the nuisance functions for semiparametric efficiency. We also discuss how our framework applies to several estimands that may be of particular interest in empirical applications. The proposed estimators are illustrated with a simulation study and an empirical example.

We present deterministic algorithms for the Hidden Subgroup Problem. The first algorithm, for abelian groups, achieves the same asymptotic worst-case query complexity as the optimal randomized algorithm, namely O($\sqrt{ n}\,$), where $n$ is the order of the group. The analogous algorithm for non-abelian groups comes within a $\sqrt{ \log n}$ factor of the optimal randomized query complexity. The best known randomized algorithm for the Hidden Subgroup Problem has expected query complexity that is sensitive to the input, namely O($\sqrt{ n/m}\,$), where $m$ is the order of the hidden subgroup. In the first version of this article (arXiv:2104.14436v1 [cs.DS]), we asked if there is a deterministic algorithm whose query complexity has a similar dependence on the order of the hidden subgroup. Prompted by this question, Ye and Li (arXiv:2110.00827v1 [cs.DS]) present deterministic algorithms for abelian groups which solve the problem with O($\sqrt{ n/m }\,$ ) queries, and find the hidden subgroup with O($\sqrt{ n (\log m) / m} + \log m$) queries. Moreover, they exhibit instances which show that in general, the deterministic query complexity of the problem may be o($\sqrt{ n/m } \,$), and that of finding the entire subgroup may also be o($\sqrt{ n/m } \,$) or even $\omega(\sqrt{ n/m } \,)$. We present a different deterministic algorithm for the Hidden Subgroup Problem that also has query complexity O($\sqrt{ n/m }\,$) for abelian groups. The algorithm is arguably simpler. Moreover, it works for non-abelian groups, and has query complexity O($\sqrt{ (n/m) \log (n/m) }\,$) for a large class of instances, such as those over supersolvable groups. We build on this to design deterministic algorithms to find the hidden subgroup for all abelian and some non-abelian instances, at the cost of a $\log m$ multiplicative factor increase in the query complexity.

This paper introduces a $K$-function for assessing second-order properties of inhomogeneous random measures generated by marked point processes. The marks can be geometric objects like fibers or sets of positive volume, and the presented $K$-function takes into account geometric features of the marks, such as tangent directions of fibers. The $K$-function requires an estimate of the inhomogeneous density function of the random measure. We introduce parametric estimates for the density function based on parametric models that represent large scale features of the inhomogeneous random measure. The proposed methodology is applied to simulated fiber patterns as well as a three-dimensional data set of steel fibers in concrete.

We consider the stochastic score classification problem. There are several binary tests, where each test $i$ is associated with a probability $p_i$ of being positive and a cost $c_i$. The score of an outcome is a weighted sum of all positive tests, and the range of possible scores is partitioned into intervals corresponding to different classes. The goal is to perform tests sequentially (and possibly adaptively) so as to identify the class at the minimum expected cost. We provide the first constant-factor approximation algorithm for this problem, which improves over the previously-known logarithmic approximation ratio. Moreover, our algorithm is $non$ $adaptive$: it just involves performing tests in a $fixed$ order until the class is identified. Our approach also extends to the $d$-dimensional score classification problem and the "explainable" stochastic halfspace evaluation problem (where we want to evaluate some function on $d$ halfspaces). We obtain an $O(d^2\log d)$-approximation algorithm for both these extensions. Finally, we perform computational experiments that demonstrate the practical performance of our algorithm for score classification. We observe that, for most instances, the cost of our algorithm is within $50\%$ of an information-theoretic lower bound on the optimal value.

We observe $n$ pairs of independent random variables $X_{1}=(W_{1},Y_{1}),\ldots,X_{n}=(W_{n},Y_{n})$ and assume, although this might not be true, that for each $i\in\{1,\ldots,n\}$, the conditional distribution of $Y_{i}$ given $W_{i}$ belongs to a given exponential family with real parameter $\theta_{i}^{\star}=\boldsymbol{\theta}^{\star}(W_{i})$ the value of which is an unknown function $\boldsymbol{\theta}^{\star}$ of the covariate $W_{i}$. Given a model $\boldsymbol{\overline\Theta}$ for $\boldsymbol{\theta}^{\star}$, we propose an estimator $\boldsymbol{\widehat \theta}$ with values in $\boldsymbol{\overline\Theta}$ the construction of which is independent of the distribution of the $W_{i}$. We show that $\boldsymbol{\widehat \theta}$ possesses the properties of being robust to contamination, outliers and model misspecification. We establish non-asymptotic exponential inequalities for the upper deviations of a Hellinger-type distance between the true distribution of the data and the estimated one based on $\boldsymbol{\widehat \theta}$. We deduce a uniform risk bound for $\boldsymbol{\widehat \theta}$ over the class of H\"olderian functions and we prove the optimality of this bound up to a logarithmic factor. Finally, we provide an algorithm for calculating $\boldsymbol{\widehat \theta}$ when $\boldsymbol{\theta}^{\star}$ is assumed to belong to functional classes of low or medium dimensions (in a suitable sense) and, on a simulation study, we compare the performance of $\boldsymbol{\widehat \theta}$ to that of the MLE and median-based estimators. The proof of our main result relies on an upper bound, with explicit numerical constants, on the expectation of the supremum of an empirical process over a VC-subgraph class. This bound can be of independent interest.

Bayesian modelling enables us to accommodate complex forms of data and make a comprehensive inference, but the effect of partial misspecification of the model is a concern. One approach in this setting is to modularize the model, and prevent feedback from suspect modules, using a cut model. After observing data, this leads to the cut distribution which normally does not have a closed-form. Previous studies have proposed algorithms to sample from this distribution, but these algorithms have unclear theoretical convergence properties. To address this, we propose a new algorithm called the Stochastic Approximation Cut algorithm (SACut) as an alternative. The algorithm is divided into two parallel chains. The main chain targets an approximation to the cut distribution; the auxiliary chain is used to form an adaptive proposal distribution for the main chain. We prove convergence of the samples drawn by the proposed algorithm and present the exact limit. Although SACut is biased, since the main chain does not target the exact cut distribution, we prove this bias can be reduced geometrically by increasing a user-chosen tuning parameter. In addition, parallel computing can be easily adopted for SACut, which greatly reduces computation time.

The estimation of covariance matrices of multiple classes with limited training data is a difficult problem. The sample covariance matrix (SCM) is known to perform poorly when the number of variables is large compared to the available number of samples. In order to reduce the mean squared error (MSE) of the SCM, regularized (shrinkage) SCM estimators are often used. In this work, we consider regularized SCM (RSCM) estimators for multiclass problems that couple together two different target matrices for regularization: the pooled (average) SCM of the classes and the scaled identity matrix. Regularization toward the pooled SCM is beneficial when the population covariances are similar, whereas regularization toward the identity matrix guarantees that the estimators are positive definite. We derive the MSE optimal tuning parameters for the estimators as well as propose a method for their estimation under the assumption that the class populations follow (unspecified) elliptical distributions with finite fourth-order moments. The MSE performance of the proposed coupled RSCMs are evaluated with simulations and in a regularized discriminant analysis (RDA) classification set-up on real data. The results based on three different real data sets indicate comparable performance to cross-validation but with a significant speed-up in computation time.

Data on high-dimensional spheres arise frequently in many disciplines either naturally or as a consequence of preliminary processing and can have intricate dependence structure that needs to be understood. We develop exploratory factor analysis of the projected normal distribution to explain the variability in such data using a few easily interpreted latent factors. Our methodology provides maximum likelihood estimates through a novel fast alternating expectation profile conditional maximization algorithm. Results on simulation experiments on a wide range of settings are uniformly excellent. Our methodology provides interpretable and insightful results when applied to tweets with the $\#MeToo$ hashtag in early December 2018, to time-course functional Magnetic Resonance Images of the average pre-teen brain at rest, to characterize handwritten digits, and to gene expression data from cancerous cells in the Cancer Genome Atlas.

Purpose: The purpose of this study is to identify the important determinants responsible for the variation in women's attitude towards intimate partner violence (IPV). Methods: A nationally representative Bangladesh Demographic and Health Survey 2014 data of 17,863 women is used to address the research questions. In the study, two response variables are constructed from the five attitude questions, and a series of individual and community-level predictors are tested. The preliminary statistical methods employed in the study include univariate and bivariate distributions, while the adopted statistical models include binary logistic, ordinal logistic, mixed-effects multilevel logistic models for each response variable, and finally, the generalized ordinal logistic regression. Results: Statistical analyses reveal that among the individual-level independent variables age at first marriage, respondent's education, decision score, religion, NGO membership, access to information, husband's education, normalized wealth score, and division indicator have significant effects on the women's attitude towards IPV. Among the three community-level variables, only the mean decision score is found significant in lowering the likelihood. Conclusions: It is evident that other than religion, NGO membership, and division indicator, the higher the value of the variable, the lower the likelihood of justifying IPV. However, being a Muslim, NGO member, and resident of other divisions, women are found more tolerant of IPV from their respective counterparts. These findings suggest the government, policymakers, practitioners, academicians, and all other stakeholders to work on the significant determinants to divert women's wrong attitude towards IPV, and thus help to take away this deep-rooted problem from society.

We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.