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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…

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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…

网络上的平稳不可分离时空协方差函数AI 解读

Abstract The advent of data science has provided an increasing number of challenges with high data complexity. This paper addresses the challenge of space-time data where the spatial domain is not a planar surface, a sphere, or a linear network, but a generalised network (termed a graph with Euclidean edges). Additionally, data are repeatedly measured over different temporal instants. We provide new classes of stationary nonseparable space-time covariance functions where…

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…

Kolyan Ray 与 Botond Szabo 对“鞅后验分布”的讨论AI 解读

We congratulate the authors for their thought-provoking article, which includes proposing a copula-based update for the predictive density and establishing its frequentist consistency under relatively mild assumptions. In this contribution, we further explore the frequentist properties of (Bayesian) predictive densities and illustrate through a simple example that, similarly to the posterior distribution, the predictive distribution can also be inconsistent. One therefore…

Isadora Antoniano Villalobos 对《鞅后验分布》的讨论AI 解读

I thank the authors for their refreshing ideas, providing many insights and intuitions on the process of statistical learning. For years, Bayesians have considered two levels of uncertainty, the likelihood and the prior. By shifting the focus to the predictive distribution as a single expression of uncertainty about a future observation (given data), the sequential nature of statistical learning is made explicit, together with some of its limitations. If each observation…

Mena Chavez 对《鞅后验分布》的讨论AI 解读

I congratulate the authors for introducing a novel approach for Bayesian inference, which avoids directly using the posterior distribution, often not available or computationally expensive. In addition to opening several lines of thinking about the foundations of Bayesian statistics, their proposal unveils many interesting directions to be explored and applied within the world of statistics. With an infinite population and the epistemic view of uncertainty (see, e.g.…

Wijayatunga对《鞅后验分布》的讨论意见AI 解读

Statistical uncertainty in (an estimate of) a parameter of a probability distribution is due to missing (unseen) observations (when it is estimated), as authors have noted. We can think that an estimate of the parameter has the maximal uncertainty when no observation is used for it, and no uncertainty when all possible observations are used for it. For example, in Bayesian sense, for a Bernoulli parameter, Beta distribution with parameter values α=1 and β=1 represents the…

布莱克·莫亚对《鞅后验分布》的讨论意见AI 解读

I would like to congratulate the authors on an expository and intuitive representation of statistical uncertainty. To assist in the further investigation of predictive resampling techniques, I have developed a software package for R (R Core Team, 2022) which implements some of the algorithms presented here as well as from subsequent work (Moya & Walker, 2023). The CopRe package (Moya, 2022), named for the copula resampling technique described in Section 4, can be installed…

贾琦顾与郭胜尹对《鞅后验分布》的讨论AI 解读

We congratulate the authors for their thought-provoking paper, especially in the three aspects as follows. Instead of imposing a prior π(θ) on the parameter θ of a likelihood function fθ(y)⁠, where both π and fθ are elicited subjectively, the authors proposed the framework of martingale posterior to directly model the predictive p(yn+1:∞∣y1:n) for which only the prior predictive p(y) needs to be specified. As a result, Bayesian inference can be conducted on any parameter (or…

Catalano、Fasano 与 Rebaudo 对《鞅后验分布》的讨论AI 解读

We congratulate the authors for an insightful foundational contribution to the statistical literature. This work builds on well-established methodologies, such as predictive inference, Doob’s theorem, and conditional independent sequences, providing a unifying framework and a novel understanding of Bayesian uncertainty. A central role is played by the predictive characterisation of the random parameter in terms of observable (yet unobserved) quantities, which are regarded as…