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重新审视免带宽推断：样本分割方法AI 解读

Abstract The bandwidth-free tests for a multi-dimensional parameter have attracted considerable attention in econometrics and statistics literature. These tests can be conveniently implemented due to their tuning-parameter free nature and possess more accurate size as compared to the traditional heteroskedasticity and autocorrelation consistent-based approaches. However, when sample size is small/medium, these bandwidth-free tests exhibit large size distortion when both the…

Art Owen 对《通过投注估计有界随机变量均值》的讨论意见AI 解读

I congratulate the authors on some very interesting work making connections between likelihood, betting and martingales. What caught my eye was the connection to the empirical likelihood (EL) (Owen, 2001) and the dual likelihood (Mykland, 1995). The hindsight optimal λ solves 0=∑i=1t(xi−m)/(1+λT(xi−m))=0 corresponding to the observation weights wi∝1/(1+λT(xi−m)) used in EL. Two common alternatives use weights wi∝(1−λT(xi−m)) and wi∝exp(−λT(xi−m)) arising from L2 and entropy…

David Siegmund 对《通过下注估计有界随机变量均值》的讨论AI 解读

This contribution by Waudby-Smith and Ramdas provides a host of new insights combined with a concise, yet comprehensive, historical review of “any time valid inference”, which has blossomed in recent years after the hiatus that followed its beginnings in the mid-20th Century. Particularly, attractive is the authors’ systematic use of martingales of the form (29) to facilitate the betting interpretation and give unified derivations of a large diversity of inequalities. Given…

Thomas等对《通过下注估计有界随机变量均值》的讨论意见AI 解读

We enthusiastically agree about the importance of developing tighter confidence intervals and sequences for the mean—the topic of this paper. There are increasing concerns about the safety and fairness of machine learning systems, and we believe one powerful tool for ensuring that machine learning systems are safe, fair, and otherwise reliable is for them to provide guarantees in the form of their confidence that the models they produce will satisfy certain safety or…

Davison与Rodionov对《通过下注估计有界随机变量均值》的讨论AI 解读

We congratulate the authors on a wide-ranging contribution to a statistical hardy perennial. The restriction that the support be known and bounded appears highly restrictive, yet Proposition 3 suggests much wider potential for applications, one of which we briefly explore below. The Pickands–Balkema–de Haan theorem (Balkema & de Haan, 1974; Pickands, 1975) establishes that the distribution of a random variable X, conditioned on its exceeding a high threshold u can in rather…

Philip B. Stark 对《通过投注估计有界随机变量均值》的讨论AI 解读

This paper is a gem, and I predict it will have lasting impact on theoretical and applied statistics. Drawing on intellectual heritage spanning more than 80 years, illustrating and leveraging deep connections among results in probability, finance, information theory, computer science, and statistics, Waudby-Smith and Ramdas present mature, intuitive, flexible, computable, and powerful methods for two fundamental (indeed, canonical) nonparametric inference problems:…

Howard对《通过下注估计有界随机变量均值》的讨论AI 解读

I congratulate the authors on this creative and well-written contribution. It’s always a delight to see new insights into such a fundamental problem as inference for the mean of bounded observations, and certainly inspires excitement about the future of game-theoretic statistical inference. Besides the impressive technical contributions, I appreciate how the authors draw previously unappreciated connections between work in information theory, finance, and online learning,…

李佳怡、李远通与戴晓武对“通过下注估计有界随机变量均值”的讨论AI 解读

We congratulate Waudby-Smith and Ramdas for their interesting paper Waudby-Smith and Ramdas (2020b) in generating confidence intervals and time-uniform confidence sequences for mean estimation with bounded observations. Their methodology utilises composite non-negative martingales and establishes a connection to game-theoretic probability. Our comments will focus on numerical comparisons with alternative methods. The corresponding code is available at…

王若度对《通过下注估计有界随机变量均值》的讨论AI 解读

I congratulate Ian Waudby-Smith and Aaditya Ramdas for this excellent contribution to the theory of e-testing (or testing by betting). Over the past several years, this theory has been developed rapidly in multiple exciting directions. The contribution of Waudby-Smith and Ramdas is an important piece in this active stream of literature. I have a few comments on the approach in the current paper and its generalisations. The main approach taken in the paper (see its Theorem…

对Waudby-Smith与Ramdas的致谢动议附议及《以投注估计有界随机变量均值》的讨论AI 解读

The paper proposes an elegant new framework for deriving time-uniform confidence intervals for the mean of a sequence of random variables. The proposed technique is based on a game-theoretic view of statistical inference, rephrasing the problem of building confidence intervals as identifying a set of coin-betting games with a certain ‘plausibility’ property: if one can get ‘implausibly rich’ in a game by betting on the mean being lower (resp. higher) than a certain…

Angelopoulos 对《通过下注估计有界随机变量均值》的讨论AI 解读

I congratulate Waudby-Smith and Ramdas for their paper, which develops algorithms for confidence sequences motivated by betting. An impressive consequence of their work is the derivation of confidence intervals for a fixed sample size n, given by Theorem 3/Remark 3 and Theorem 4/Remark 4. These bounds, which my colleagues and I have referred to eponymously as the WSR bounds, prove quite useful for quantifying the uncertainty of artificial intelligence algorithms. I will give…

江荣与余克明对《通过下注估计有界随机变量均值》的讨论AI 解读

We want to congratulate the authors on estimating means of bounded random variables in both with- and without-replacement settings. The authors constructed confidence intervals and time-uniform confidence sequences for the mean of a bounded random variable using test supermartingale technique. It deepens our understanding of the confidence sequence. Confidence sequence is one particular tool in sequential design that facilitates anytime-valid inference. In particular,…

Ryan Martin对《通过下注估计有界随机变量均值》的讨论AI 解读

Congratulations to Waudby-Smith and Ramdas (WSR) for their excellent contribution to the rapidly growing literature on anytime-valid inference. Since some of my work involves imprecise probability, Professor Ramdas asked me privately what, if any, connections there are between e-values, etc. and imprecise probability. The answer to his question might be of general interest, so I’ll share it here. Following WSR, let (Xt:t≥1) be a [0,1]-valued process with distribution P∈Pμ…

GRASP：分类学习的拟合优度检验AI 解读

Abstract Performance of classifiers is often measured in terms of average accuracy on test data. Despite being a standard measure, average accuracy fails in characterising the fit of the model to the underlying conditional law of labels given the features vector (Y∣X), e.g. due to model misspecification, over fitting, and high-dimensionality. In this paper, we consider the fundamental problem of assessing the goodness-of-fit for a general binary classifier. Our framework…

估计函数的经验减偏调整AI 解读

Abstract We develop a novel, general framework for reduced-bias M-estimation from asymptotically unbiased estimating functions. The framework relies on an empirical approximation of the bias by a function of derivatives of estimating function contributions. Reduced-bias M-estimation operates either implicitly, solving empirically adjusted estimating equations, or explicitly, subtracting the estimated bias from the original M-estimates, and applies to partially or fully…

度修正及相关随机块模型的Monte Carlo拟合优度检验AI 解读

Abstract We construct Bayesian and frequentist finite-sample goodness-of-fit tests for three different variants of the stochastic blockmodel for network data. Since all of the stochastic blockmodel variants are log-linear in form when block assignments are known, the tests for the latent block model versions combine a block membership estimator with the algebraic statistics machinery for testing goodness-of-fit in log-linear models. We describe Markov bases and marginal…

用于贝叶斯模型诊断的留出预测检验AI 解读

Abstract Bayesian modelling helps applied researchers to articulate assumptions about their data and develop models tailored for specific applications. Thanks to good methods for approximate posterior inference, researchers can now easily build, use, and revise complicated Bayesian models for large and rich data. These capabilities, however, bring into focus the problem of model criticism. Researchers need tools to diagnose the fitness of their models, to understand where…

Abstract We derive confidence intervals (CIs) and confidence sequences (CSs) for the classical problem of estimating a bounded mean. Our approach generalizes and improves on the celebrated Chernoff method, yielding the best closed-form "empirical-Bernstein" CSs and CIs (converging exactly to the oracle Bernstein width) as well as non-closed-form "betting" CSs and CIs. Our method combines new composite nonnegative (super)martingales with Ville's maximal inequality, with…

鞅后验分布AI 解读

Abstract The prior distribution is the usual starting point for Bayesian uncertainty. In this paper, we present a different perspective that focuses on missing observations as the source of statistical uncertainty, with the parameter of interest being known precisely given the entire population. We argue that the foundation of Bayesian inference is to assign a distribution on missing observations conditional on what has been observed. In the i.i.d. setting with an observed…

Huk、Pacchiardi、Dutta 与 Steel 对《鞅后验分布》的讨论AI 解读

We congratulate the authors for this thought-provoking paper. The recursive definition of the predictive distributions through a bivariate copula (Section 4) depends on a hyperparameter ρ, which the authors tune by minimising the prequential log-score −∑i=1nlogpi−1(yi) (Section 4.5.2). The prequential framework nicely connects with the predictive resampling approach used later. However, other strictly proper scoring rules (Gneiting & Raftery, 2007) could be used in place of…

用于稀疏贝叶斯推断的快速异步 MCMC 采样器AI 解读

Abstract We propose a very fast approximate Markov chain Monte Carlo sampling framework that is applicable to a large class of sparse Bayesian inference problems. The computational cost per iteration in several regression models is of order O(n(s+J)), where n is the sample size, s is the underlying sparsity of the model, and J is the size of a randomly selected subset of regressors. This cost can be further reduced by data sub-sampling when stochastic gradient Langevin…

Draper 与 Guo 对《鞅后验分布》的讨论AI 解读

We have two comments motivated by this interesting paper. The idea, introduced early in the paper, that ‘the object of interest is fully defined once all the observations have been viewed’ is almost exactly 100 years old: it was a cornerstone of the remarkable paper by Fisher (1922) and has been referred to for many decades as Fisher consistency. We are surprised that the authors did not make this connection. Theorem (Draper & Guo, 2023) Under the conditions detailed above,…

对《鞅后验分布》的致谢动议附议与讨论AI 解读

Let me first congratulate the authors for an interesting, thought provoking and inspiring article, with many fine examples. However, I wonder whether the specific setup has sufficient generality to cover interesting cases, and I should have liked to see the ideas in this paper confronted with some different situations. Firstly, let me remind everyone of the standard Bayesian setup, where we have observablesX,Y⁠, a parameter Θ of interest that in principle should be added to…

David Rossell对《鞅后验分布》的讨论意见AI 解读

Congratulations on a thought-provoking piece. Building Bayesian inference from a (likelihood, predictive) pair, rather than a (likelihood, prior), enriches the paradigm and provides new ways to think about, formulate, and solve problems. A few respectful remarks. First, although the authors never claim this, it is worth emphasising that the framework is not prior-free. There is a posterior and a likelihood, hence the prior is proportional to their ratio. The key is that said…