The use of propensity score (PS) methods has become ubiquitous in causal inference. At the heart of these methods is the positivity assumption. Violation of the positivity assumption leads to the presence of extreme PS weights when estimating average causal effects of interest, such as the average treatment effect (ATE) or the average treatment effect on the treated (ATT), which renders invalid related statistical inference. To circumvent this issue, trimming or truncating the extreme estimated PSs have been widely used. However, these methods require that we specify a priori a threshold and sometimes an additional smoothing parameter. While there are a number of methods dealing with the lack of positivity when estimating ATE, surprisingly there is no much effort in the same issue for ATT. In this paper, we first review widely used methods, such as trimming and truncation in ATT. We emphasize the underlying intuition behind these methods to better understand their applications and highlight their main limitations. Then, we argue that the current methods simply target estimands that are scaled ATT (and thus move the goalpost to a different target of interest), where we specify the scale and the target populations. We further propose a PS weight-based alternative for the average causal effect on the treated, called overlap weighted average treatment effect on the treated (OWATT). The appeal of our proposed method lies in its ability to obtain similar or even better results than trimming and truncation while relaxing the constraint to choose a priori a threshold (or even specify a smoothing parameter). The performance of the proposed method is illustrated via a series of Monte Carlo simulations and a data analysis on racial disparities in health care expenditures.
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A new mechanical model on noncircular shallow tunnelling considering initial stress field is proposed in this paper by constraining far-field ground surface to eliminate displacement singularity at infinity, and the originally unbalanced tunnel excavation problem in existing solutions is turned to an equilibrium one of mixed boundaries. By applying analytic continuation, the mixed boundaries are transformed to a homogenerous Riemann-Hilbert problem, which is subsequently solved via an efficient and accurate iterative method with boundary conditions of static equilibrium, displacement single-valuedness, and traction along tunnel periphery. The Lanczos filtering technique is used in the final stress and displacement solution to reduce the Gibbs phenomena caused by the constrained far-field ground surface for more accurte results. Several numerical cases are conducted to intensively verify the proposed solution by examining boundary conditions and comparing with existing solutions, and all the results are in good agreements. Then more numerical cases are conducted to investigate the stress and deformation distribution along ground surface and tunnel periphery, and several engineering advices are given. Further discussions on the defects of the proposed solution are also conducted for objectivity.
Indirect reciprocity is a mechanism that explains large-scale cooperation in human societies. In indirect reciprocity, an individual chooses whether or not to cooperate with another based on reputation information, and others evaluate the action as good or bad. Under what evaluation rule (called ``social norm'') cooperation evolves has long been of central interest in the literature. It has been reported that if individuals can share their evaluations (i.e., public reputation), social norms called ``leading eight'' can be evolutionarily stable. On the other hand, when they cannot share their evaluations (i.e., private assessment), the evolutionary stability of cooperation is still in question. To tackle this problem, we create a novel method to analyze the reputation structure in the population under private assessment. Specifically, we characterize each individual by two variables, ``goodness'' (what proportion of the population considers the individual as good) and ``self-reputation'' (whether an individual thinks of him/herself as good or bad), and analyze the stochastic process of how these two variables change over time. We discuss evolutionary stability of each of the leading eight social norms by studying the robustness against invasions of unconditional cooperators and defectors. We identify key pivots in those social norms for establishing a high level of cooperation or stable cooperation against mutants. Our finding gives an insight into how human cooperation is established in a real-world society.
Composite likelihood usually ignores dependencies among response components, while variational approximation to likelihood ignores dependencies among parameter components. We derive a Gaussian variational approximation to the composite log-likelihood function for Poisson and Gamma regression models with crossed random effects. We show consistency and asymptotic normality of the estimates derived from this approximation and support this theory with some simulation studies. The approach is computationally much faster than a Gaussian variational approximation to the full log-likelihood function.
The prevailing statistical approach to analyzing persistence diagrams is concerned with filtering out topological noise. In this paper, we adopt a different viewpoint and aim at estimating the actual distribution of a random persistence diagram, which captures both topological signal and noise. To that effect, Chazel and Divol (2019) proved that, under general conditions, the expected value of a random persistence diagram is a measure admitting a Lebesgue density, called the persistence intensity function. In this paper, we are concerned with estimating the persistence intensity function and a novel, normalized version of it -- called the persistence density function. We present a class of kernel-based estimators based on an i.i.d. sample of persistence diagrams and derive estimation rates in the supremum norm. As a direct corollary, we obtain uniform consistency rates for estimating linear representations of persistence diagrams, including Betti numbers and persistence surfaces. Interestingly, the persistence density function delivers stronger statistical guarantees.
The effects of continuous treatments are often characterized through the average dose response function, which is challenging to estimate from observational data due to confounding and positivity violations. Modified treatment policies (MTPs) are an alternative approach that aim to assess the effect of a modification to observed treatment values and work under relaxed assumptions. Estimators for MTPs generally focus on estimating the conditional density of treatment given covariates and using it to construct weights. However, weighting using conditional density models has well-documented challenges. Further, MTPs with larger treatment modifications have stronger confounding and no tools exist to help choose an appropriate modification magnitude. This paper investigates the role of weights for MTPs showing that to control confounding, weights should balance the weighted data to an unobserved hypothetical target population, that can be characterized with observed data. Leveraging this insight, we present a versatile set of tools to enhance estimation for MTPs. We introduce a distance that measures imbalance of covariate distributions under the MTP and use it to develop new weighting methods and tools to aid in the estimation of MTPs. We illustrate our methods through an example studying the effect of mechanical power of ventilation on in-hospital mortality.
Bidirectional typing is a discipline in which the typing judgment is decomposed explicitly into inference and checking modes, allowing to control the flow of type information in typing rules and to specify algorithmically how they should be used. Bidirectional typing has been fruitfully studied and bidirectional systems have been developed for many type theories. However, the formal development of bidirectional typing has until now been kept confined to specific theories, with general guidelines remaining informal. In this work, we give a generic account of bidirectional typing for a general class of dependent type theories. This is done by first giving a general definition of type theories (or equivalently, a logical framework), for which we define declarative and bidirectional type systems. We then show, in a theory-independent fashion, that the two systems are equivalent. This equivalence is then explored to establish the decidability of typing for weak normalizing theories, yielding a generic type-checking algorithm that has been implemented in a prototype and used in practice with many theories.
The majority of data assimilation (DA) methods in the geosciences are based on Gaussian assumptions. While these assumptions facilitate efficient algorithms, they cause analysis biases and subsequent forecast degradations. Non-parametric, particle-based DA algorithms have superior accuracy, but their application to high-dimensional models still poses operational challenges. Drawing inspiration from recent advances in the field of generative artificial intelligence (AI), this article introduces a new nonlinear estimation theory which attempts to bridge the existing gap in DA methodology. Specifically, a Conjugate Transform Filter (CTF) is derived and shown to generalize the celebrated Kalman filter to arbitrarily non-Gaussian distributions. The new filter has several desirable properties, such as its ability to preserve statistical relationships in the prior state and convergence to highly accurate observations. An ensemble approximation of the new theory (ECTF) is also presented and validated using idealized statistical experiments that feature bounded quantities with non-Gaussian distributions, a prevalent challenge in Earth system models. Results from these experiments indicate that the greatest benefits from ECTF occur when observation errors are small relative to the forecast uncertainty and when state variables exhibit strong nonlinear dependencies. Ultimately, the new filtering theory offers exciting avenues for improving conventional DA algorithms through their principled integration with AI techniques.
Over the last decade, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples $m$. Our work focuses on providing theoretical approximation guarantees for the class of $(\boldsymbol{b},\varepsilon)$-holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of $m$-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds.
The aim of this paper is to show the relationship that lies in the fact of a person being right or left handed, in their skateboarding stance. Starting from the null hypothesis that there is no relationship, the Pearson's X^2 with Yates correction tests, as well as its respective p-value will be used to test the hypothesis. It will also be calculated and analyzed the residuals, Cramer's V and the Risk and Odds Ratios, with their respective confidence intervals to know the intensity of the association.
The goal of explainable Artificial Intelligence (XAI) is to generate human-interpretable explanations, but there are no computationally precise theories of how humans interpret AI generated explanations. The lack of theory means that validation of XAI must be done empirically, on a case-by-case basis, which prevents systematic theory-building in XAI. We propose a psychological theory of how humans draw conclusions from saliency maps, the most common form of XAI explanation, which for the first time allows for precise prediction of explainee inference conditioned on explanation. Our theory posits that absent explanation humans expect the AI to make similar decisions to themselves, and that they interpret an explanation by comparison to the explanations they themselves would give. Comparison is formalized via Shepard's universal law of generalization in a similarity space, a classic theory from cognitive science. A pre-registered user study on AI image classifications with saliency map explanations demonstrate that our theory quantitatively matches participants' predictions of the AI.
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