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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…

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…

Pietro Rigo 对《鞅后验分布》的讨论AI 解读

This paper is really interesting and offers a number of intriguing hints. Here, I just make a few isolated remarks without any claim of being exhaustive. I would give more emphasis to the Ionescu-Tulcea theorem (ITT). This theorem is quoted only in passing, but it is the cornerstone of this paper. By ITT, the distribution of Y1:∞=(Y1,Y2,…) is completely determined by the assignment of {Pn:n≥0}⁠, where P0(⋅)=P(Y1∈⋅) and Pn(⋅)=P(Yn+1∈⋅∣Y1:n)⁠. Exploiting ITT has at least two…

Bertrand Clarke 对《鞅后验分布》的讨论AI 解读

To see the effect of the paper on statistical practice, consider the following hypothetical dialogue between a Quantitative Scientist (QS) and a Predictive Statistician (PS). QS: I’ve been thinking about the data generating mechanism and what we can safely assume pre-experimentally. PS: Actually, I’m more interested in what you think your n+1 data point would be. If you were going to predict your n+1 measurement, how would you do it? QS: I’d guess what the true model is and…

Ascolani、Lijoi与Prünster对《鞅后验分布》的讨论AI 解读

We would like to congratulate the authors on their fine and insightful contribution, which provides an original perspective on Bayesian inference and opens up new exciting research directions. The key point is a novel interpretation of the role of prediction: in a Bayesian framework all the uncertainty lies in the unobserved data Yn+1,∞ and, once they are imputed through the predictive distributions, inference is straightforward. Therefore, prediction rules, besides allowing…

对《鞅后验分布》的致谢提议与讨论AI 解读

Philip Dawid; Proposal of Vote of Thanks for “Martingale Posterior Distributions” by Edwin Fong, Chris Holmes and Stephen G. Walker, Journal of the Royal Statis

Ben Swallow对《鞅后验分布》的讨论意见AI 解读

Fong et al. (2023) present an interesting and novel approach to conducting Bayesian posterior estimation utilising a joint predictive distribution over missing potential observations and justify how this approach of generating one-step-ahead predictive distributions naturally aligns with the Bayesian likelihood-prior paradigm. The authors mention time series and hierarchical data as areas of potential future development. However, one of the major topics in Bayesian…

基于 l₁ 球先验的贝叶斯推断：用精确零值求解组合问题AI 解读

Abstract The l1-regularisation is very popular in high-dimensional statistics—it changes a combinatorial problem of choosing which subset of the parameter is zero, into a simple continuous optimisation. Using a continuous prior concentrated near zero, the Bayesian counterparts are successful in quantifying the uncertainty in the variable selection problems; nevertheless, the lack of exact zeros makes it difficult for broader problems such as change-point detection and rank…

自适应变分贝叶斯：最优性、计算与应用AI 解读

In this paper, we explore adaptive inference based on variational Bayes. Although several studies have been conducted to analyze the contraction properties of variational posteriors, there is still a lack of a general and computationally tractable variational Bayes method that performs adaptive inference. To fill this gap, we propose a novel adaptive variational Bayes framework, which can operate on a collection of models. The proposed framework first computes a variational…

学习企业的消费风险暴露AI 解读

We structurally estimate an investment-based asset pricing model, in which firms' exposure to macroeconomic risk is unknown. Bayesian beliefs about this parameter are updated from firms' and industry peers' comovement between their productivity and consumption growth. The model implies that discount rates rise endogenously with the perceived risk exposure of firms, thereby depressing investment and valuation ratios. We test these predictions in the data and find strong…

识别材料基因的算子诱导结构变量选择AI 解读

In the emerging field of materials informatics, a fundamental task is to identify physicochemically meaningful descriptors, or materials genes, which are engineered from primary features and a set of elementary algebraic operators through compositions. Standard practice directly analyzes the high-dimensional candidate predictor space in a linear model; statistical analyses are then substantially hampered by the daunting challenge posed by the astronomically large number of…

采用结构化尖峰厚尾先验的贝叶斯病灶估计AI 解读

Neural demyelination and brain damage accumulated in white matter appear as hyperintense areas on T2-weighted MRI scans in the form of lesions. Modeling binary images at the population level, where each voxel represents the existence of a lesion, plays an important role in understanding aging and inflammatory diseases. We propose a scalable hierarchical Bayesian spatial model, called BLESS, capable of handling binary responses by placing continuous spike-and-slab mixture…

估计孕期空气污染个体化效应的异质分布滞后模型AI 解读

Children’s health studies support an association between maternal environmental exposures and children’s birth outcomes. A common goal is to identify critical windows of susceptibility—periods during gestation with increased association between maternal exposures and a future outcome. The timing of the critical windows and magnitude of the associations are likely heterogeneous across different levels of individual, family, and neighborhood characteristics. Using an…

贝叶斯推断中自适应求积的随机收敛速率及应用AI 解读

We provide the first stochastic convergence rates for a family of adaptive quadrature rules used to normalize the posterior distribution in Bayesian models. Our results apply to the uniform relative error in the approximate posterior density, the coverage probabilities of approximate credible sets, and approximate moments and quantiles, therefore, guaranteeing fast asymptotic convergence of approximate summary statistics used in practice. The family of quadrature rules…