Interacting particle systems undergoing repeated mutation and selection steps model genetic evolution, and also describe a broad class of sequential Monte Carlo methods. The genealogical tree embedded into the system is important in both applications. Under neutrality, when fitnesses of particles are independent from those of their parents, rescaled genealogies are known to converge to Kingman's coalescent. Recent work has established convergence under non-neutrality, but only for finite-dimensional distributions. We prove weak convergence of non-neutral genealogies on the space of c\`adl\`ag paths under standard assumptions, enabling analysis of the whole genealogical tree.
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In this paper, we consider recent progress in estimating the average treatment effect when extreme inverse probability weights are present and focus on methods that account for a possible violation of the positivity assumption. These methods aim at estimating the treatment effect on the subpopulation of patients for whom there is a clinical equipoise. We propose a systematic approach to determine their related causal estimands and develop new insights into the properties of the weights targeting such a subpopulation. Then, we examine the roles of overlap weights, matching weights, Shannon's entropy weights, and beta weights. This helps us characterize and compare their underlying estimators, analytically and via simulations, in terms of the accuracy, precision, and root mean squared error. Moreover, we study the asymptotic behaviors of their augmented estimators (that mimic doubly robust estimators), which lead to improved estimations when either the propensity or the regression models are correctly specified. Based on the analytical and simulation results, we conclude that overall overlap weights are preferable to matching weights, especially when there is moderate or extreme violations of the positivity assumption. Finally, we illustrate the methods using a real data example marked by extreme inverse probability weights.
The current best approximation algorithms for $k$-median rely on first obtaining a structured fractional solution known as a bi-point solution, and then rounding it to an integer solution. We improve this second step by unifying and refining previous approaches. We describe a hierarchy of increasingly-complex partitioning schemes for the facilities, along with corresponding sets of algorithms and factor-revealing non-linear programs. We prove that the third layer of this hierarchy is a $2.613$-approximation, improving upon the current best ratio of $2.675$, while no layer can be proved better than $2.588$ under the proposed analysis. On the negative side, we give a family of bi-point solutions which cannot be approximated better than the square root of the golden ratio, even if allowed to open $k+o(k)$ facilities. This gives a barrier to current approaches for obtaining an approximation better than $2 \sqrt{\phi} \approx 2.544$. Altogether we reduce the approximation gap of bi-point solutions by two thirds.
As humans, we have a remarkable capacity for reading the characteristics of objects only by observing how another person carries them. Indeed, how we perform our actions naturally embeds information on the item features. Collaborative robots can achieve the same ability by modulating the strategy used to transport objects with their end-effector. A contribution in this sense would promote spontaneous interactions by making an implicit yet effective communication channel available. This work investigates if humans correctly perceive the implicit information shared by a robotic manipulator through its movements during a dyadic collaboration task. Exploiting a generative approach, we designed robot actions to convey virtual properties of the transported objects, particularly to inform the partner if any caution is required to handle the carried item. We found that carefulness is correctly interpreted when observed through the robot movements. In the experiment, we used identical empty plastic cups; nevertheless, participants approached them differently depending on the attitude shown by the robot: humans change how they reach for the object, being more careful whenever the robot does the same. This emerging form of motor contagion is entirely spontaneous and happens even if the task does not require it.
In this paper, we present an abstract framework of many-valued modal logic with the interpretation of atomic propositions and modal operators as predicate lifting over coalgebras for an endofunctor on the category of sets. It generalizes Pattinson's stratification method for colagebraic modal logic to the many-valued setting. In contrast to standard techniques of canonical model construction and filtration, this method employs an induction principle to prove the soundness, completeness, and finite model property of the logics. As a consequence, we can lift a restriction on the previous approach~ \cite{Lin2022} that requires the underlying language must have the expressive power to internalize the meta-level truth valuation operations.
The discrete distribution of the length of longest increasing subsequences in random permutations of order $n$ is deeply related to random matrix theory. In a seminal work, Baik, Deift and Johansson provided an asymptotics in terms of the distribution of the largest level of the large matrix limit of GUE. As a numerical approximation, however, this asymptotics is inaccurate for small lengths and has a slow convergence rate, conjectured to be just of order $n^{-1/3}$. Here, we suggest a different type of approximation, based on Hayman's generalization of Stirling's formula. Such a formula gives already a couple of correct digits of the length distribution for $n$ as small as $20$ but allows numerical evaluations, with a uniform error of apparent order $n^{-3/4}$, for $n$ as large as $10^{12}$; thus closing the gap between a table of exact values (that has recently been compiled for up to $n=1000$) and the random matrix limit. Being much more efficient and accurate than Monte-Carlo simulations for larger $n$, the Stirling-type formula allows for a precise numerical understanding of the first few finite size correction terms to the random matrix limit, a study that has recently been initiated by Forrester and Mays, who visualized the form of the first such term. We display also the second one, of order $n^{-2/3}$, and derive (heuristically) expansions of expected value and variance of the length, exhibiting several more terms than previously known.
Understanding the impact of the most effective policies or treatments on a response variable of interest is desirable in many empirical works in economics, statistics and other disciplines. Due to the widespread winner's curse phenomenon, conventional statistical inference assuming that the top policies are chosen independent of the random sample may lead to overly optimistic evaluations of the best policies. In recent years, given the increased availability of large datasets, such an issue can be further complicated when researchers include many covariates to estimate the policy or treatment effects in an attempt to control for potential confounders. In this manuscript, to simultaneously address the above-mentioned issues, we propose a resampling-based procedure that not only lifts the winner's curse in evaluating the best policies observed in a random sample, but also is robust to the presence of many covariates. The proposed inference procedure yields accurate point estimates and valid frequentist confidence intervals that achieve the exact nominal level as the sample size goes to infinity for multiple best policy effect sizes. We illustrate the finite-sample performance of our approach through Monte Carlo experiments and two empirical studies, evaluating the most effective policies in charitable giving and the most beneficial group of workers in the National Supported Work program.
Motivated by a recently established result saying that within the class of bivariate Archimedean copulas standard pointwise convergence implies weak convergence of almost all conditional distributions this contribution studies the class $\mathcal{C}_{ar}^d$ of all $d$-dimensional Archimedean copulas with $d \geq 3$ and proves the afore-mentioned implication with respect to conditioning on the first $d-1$ coordinates. Several proper\-ties equivalent to pointwise convergence in $\mathcal{C}_{ar}^d$ are established and - as by-product of working with conditional distributions (Markov kernels) - alternative simple proofs for the well-known formulas for the level set masses $\mu_C(L_t)$ and the Kendall distribution function $F_K^d$ as well as a novel geometrical interpretation of the latter are provided. Viewing normalized generators $\psi$ of $d$-dimensional Archimedean copulas from the perspective of their so-called Williamson measures $\gamma$ on $(0,\infty)$ is then shown to allow not only to derive surprisingly simple expressions for $\mu_C(L_t)$ and $F_K^d$ in terms of $\gamma$ and to characterize pointwise convergence in $\mathcal{C}_{ar}^d$ by weak convergence of the Williamson measures but also to prove that regularity/singularity properties of $\gamma$ directly carry over to the corresponding copula $C_\gamma \in \mathcal{C}_{ar}^d$. These results are finally used to prove the fact that the family of all absolutely continuous and the family of all singular $d$-dimensional copulas is dense in $\mathcal{C}_{ar}^d$ and to underline that despite of their simple algebraic structure Archimedean copulas may exhibit surprisingly singular behavior in the sense of irregularity of their conditional distribution functions.
Recent findings have shown that highly over-parameterized Neural Networks generalize without pretraining or explicit regularization. It is achieved with zero training error, i.e., complete over-fitting by memorizing the training data. This is surprising, since it is completely against traditional machine learning wisdom. In our empirical study we fortify these findings in the domain of fine-grained image classification. We show that very large Convolutional Neural Networks with millions of weights do learn with only a handful of training samples and without image augmentation, explicit regularization or pretraining. We train the architectures ResNet018, ResNet101 and VGG19 on subsets of the difficult benchmark datasets Caltech101, CUB_200_2011, FGVCAircraft, Flowers102 and StanfordCars with 100 classes and more, perform a comprehensive comparative study and draw implications for the practical application of CNNs. Finally, we show that a randomly initialized VGG19 with 140 million weights learns to distinguish airplanes and motorbikes with up to 95% accuracy using only 20 training samples per class.
The essence of multivariate sequential learning is all about how to extract dependencies in data. These data sets, such as hourly medical records in intensive care units and multi-frequency phonetic time series, often time exhibit not only strong serial dependencies in the individual components (the "marginal" memory) but also non-negligible memories in the cross-sectional dependencies (the "joint" memory). Because of the multivariate complexity in the evolution of the joint distribution that underlies the data generating process, we take a data-driven approach and construct a novel recurrent network architecture, termed Memory-Gated Recurrent Networks (mGRN), with gates explicitly regulating two distinct types of memories: the marginal memory and the joint memory. Through a combination of comprehensive simulation studies and empirical experiments on a range of public datasets, we show that our proposed mGRN architecture consistently outperforms state-of-the-art architectures targeting multivariate time series.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
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