arXiv subjects

Chuanhai Liu

Publications and source records attributed to Chuanhai Liu.

At least 19 recordsLinked to original sources

arXiv2026

Inferential Models: The Power of Auxiliary Variables for Reasoning with Scientific Uncertainty

A central challenge in scientific inference is to produce uncertainty assessments that are both situation-specific and frequency-calibrated. This article examines inferential models (IMs) as a framework for prior-free probabilistic reasoning with scientific uncertainty. The central IM idea is to view the auxiliary variables in a sampling model as the source of model-based uncertainty. R. A. Fisher's fiducial inference transfers auxiliary randomness to the parameter space before applying probability calculus; IMs instead predict the unobserved auxiliary value with calibrated predictive random sets (PRSs) and transfer the resulting plausibility statements only afterward. This change in order yields valid uncertainty assessments and clarifies the relations among Fisherian fiducial reasoning, Neymanian confidence theory, Dempster-Shafer belief functions, generalized fiducial inference, and IMs. By comparing IMs with objective-prior Bayesian inference, the article argues that E. T. Jaynes' logic-of-science ambition can be continued without forcing all scientific uncertainty into a precise prior distribution because calibrated imprecision is often essential. Finally, the article suggests that a differential-geometric theory of IMs may be within reach, offering a possible route to foundational questions traditionally framed in terms of the likelihood principle.

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arXiv2026

Revisiting the Behrens-Fisher Problem: Validity-First Optimality

The Behrens--Fisher problem concerns inference on the difference of two normal means when both variances are unknown and unequal. It is a classical example in which nuisance parameters prevent ordinary exact fixed-sample inference, and it has long served as a benchmark for the foundations of inference. We revisit it through the inferential model (IM) framework of Martin and Liu. After conditioning and regular marginalization, the exact association is two-dimensional, with one coordinate for the standardized mean contrast and one for the variance ratio. Their one-dimensional generalized marginal IM is then best understood as a cylindrical two-dimensional predictive random set: sharp in its mean-contrast projection, by Hsu's stochastic domination, and vacuous in the variance ratio. Our main result is a precise validity-first optimality: among prior-free procedures that retain exact, uniform, finite-sample validity, the IM interval is the shortest. We prove minimaxity and admissibility in the cylindrical class and, by a projection argument, extend this to rectangular and general two-dimensional predictive random sets. A companion tradeoff principle shows that any adaptive procedure can only redistribute interval width across variance-ratio regimes, never shorten it uniformly. A Monte Carlo study bears this out: Welch and the bootstrap under-cover, whereas the conservative fiducial does not dominate the IM interval, being shorter only where the latter over-covers and longer where validity binds.

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arXiv2025

The typicality principle and its implications for statistics and data science

A central focus of data science is the transformation of empirical evidence into knowledge. As such, the key insights and scientific attitudes of deep thinkers like Fisher, Popper, and Tukey are expected to inspire exciting new advances in machine learning and artificial intelligence in years to come. Along these lines, the present paper advances a novel {\em typicality principle} which states, roughly, that if the observed data is sufficiently ``atypical'' in a certain sense relative to a posited theory, then that theory is unwarranted. This emphasis on typicality brings familiar but often overlooked background notions like model-checking to the inferential foreground. One instantiation of the typicality principle is in the context of parameter estimation, where we propose a new typicality-based regularization strategy that leans heavily on goodness-of-fit testing. The effectiveness of this new regularization strategy is illustrated in three non-trivial examples where ordinary maximum likelihood estimation fails miserably. We also demonstrate how the typicality principle fits within a bigger picture of reliable and efficient uncertainty quantification.

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arXiv2024

Towards Strong AI: Transformational Beliefs and Scientific Creativity

Strong artificial intelligence (AI) is envisioned to possess general cognitive abilities and scientific creativity comparable to human intelligence, encompassing both knowledge acquisition and problem-solving. While remarkable progress has been made in weak AI, the realization of strong AI remains a topic of intense debate and critical examination. In this paper, we explore pivotal innovations in the history of astronomy and physics, focusing on the discovery of Neptune and the concept of scientific revolutions as perceived by philosophers of science. Building on these insights, we introduce a simple theoretical and statistical framework of weak beliefs, termed the Transformational Belief (TB) framework, designed as a foundation for modeling scientific creativity. Through selected illustrative examples in statistical science, we demonstrate the TB framework's potential as a promising foundation for understanding, analyzing, and even fostering creativity -- paving the way toward the development of strong AI. We conclude with reflections on future research directions and potential advancements.

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arXiv2024

Estimation of Over-parameterized Models from an Auto-Modeling Perspective

From a model-building perspective, we propose a paradigm shift for fitting over-parameterized models. Philosophically, the mindset is to fit models to future observations rather than to the observed sample. Technically, given an imputation method to generate future observations, we fit over-parameterized models to these future observations by optimizing an approximation of the desired expected loss function based on its sample counterpart and an adaptive $\textit{duality function}$. The required imputation method is also developed using the same estimation technique with an adaptive $m$-out-of-$n$ bootstrap approach. We illustrate its applications with the many-normal-means problem, $n < p$ linear regression, and neural network-based image classification of MNIST digits. The numerical results demonstrate its superior performance across these diverse applications. While primarily expository, the paper conducts an in-depth investigation into the theoretical aspects of the topic. It concludes with remarks on some open problems.

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arXiv2024

Finite Sample Valid Inference via Calibrated Bootstrap

While widely used as a general method for uncertainty quantification, the bootstrap method encounters difficulties that raise concerns about its validity in practical applications. This paper introduces a new resampling-based method, termed $\textit{calibrated bootstrap}$, designed to generate finite sample-valid parametric inference from a sample of size $n$. The central idea is to calibrate an $m$-out-of-$n$ resampling scheme, where the calibration parameter $m$ is determined against inferential pivotal quantities derived from the cumulative distribution functions of loss functions in parameter estimation. The method comprises two algorithms. The first, named $\textit{resampling approximation}$ (RA), employs a $\textit{stochastic approximation}$ algorithm to find the value of the calibration parameter $m=m_α$ for a given $α$ in a manner that ensures the resulting $m$-out-of-$n$ bootstrapped $1-α$ confidence set is valid. The second algorithm, termed $\textit{distributional resampling}$ (DR), is developed to further select samples of bootstrapped estimates from the RA step when constructing $1-α$ confidence sets for a range of $α$ values is of interest. The proposed method is illustrated and compared to existing methods using linear regression with and without $L_1$ penalty, within the context of a high-dimensional setting and a real-world data application. The paper concludes with remarks on a few open problems worthy of consideration.

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arXiv2023

Circularly Symmetric Tests of Goodness-of-Fit

It is realized that existing powerful tests of goodness-of-fit are all based on sorted uniforms and, consequently, can suffer from the confounded effect of different locations and various signal frequencies in the deviations of the distributions under the alternative hypothesis from those under the null. This paper proposes circularly symmetric tests that are obtained by circularizing reweighted Anderson-Darling tests, with the focus on the circularized versions of Anderson-Darling and Zhang test statistics. Two specific types of circularization are considered, one is obtained by taking the average of the corresponding so-called scan test statistics and the other by using the maximum. To a certain extent, this circularization technique effectively eliminates the location effect and allows the weights to focus on the various signal frequencies. A limited but arguably convincing simulation study on finite-sample performance demonstrates that the circularized Zhang method outperforms the circularized Anderson-Darling and that the circularized tests outperform their parent methods. Large-sample theoretical results are also obtained for the average type of circularization. The results show that both the circularized Anderson-Darling and circularized Zhang have asymptotic distributions that are a weighted sum of an infinite number of independent squared standard normal random variables. In addition, the kernel matrices and functions are circulant. As a result, asymptotic approximations are computationally efficient via the fast Fourier transform.

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arXiv2023

On Existence Theorems for Conditional Inferential Models

The framework of Inferential Models (IMs) has recently been developed in search of what is referred to as the holy grail of statistical theory, that is, prior-free probabilistic inference. Its method of Conditional IMs (CIMs) is a critical component in that it serves as a desirable extension of the Bayes theorem for combining information when no prior distribution is available. The general form of CIMs is defined by a system of first-order homogeneous linear partial differential equations (PDEs). When admitting simple solutions, they are referred to as regular, whereas when no regular CIMs exist, they are used as the so-called local CIMs. This paper provides conditions for regular CIMs, which are shown to be equivalent to the existence of a group-theoretical representation of the underlying statistical model. It also establishes existence theorems for CIMs, which state that under mild conditions, local CIMs always exist. Finally, the paper concludes with a simple example and a few remarks on future developments of CIMs for applications to popular but inferentially nontrivial statistical models.

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arXiv2023

Elucidating Inferential Models with the Cauchy Distribution

Statistical inference as a formal scientific method to covert experience to knowledge has proven to be elusively difficult. While frequentist and Bayesian methodologies have been accepted in the contemporary era as two dominant schools of thought, it has been a good part of the last hundred years to see growing interests in development of more sound methods, both philosophically, in terms of scientific meaning of inference, and mathematically, in terms of exactness and efficiency. These include Fisher's fiducial argument, the Dempster-Shafe theory of belief functions, generalized fiducial, Confidence Distributions, and the most recently proposed inferential framework, called Inferential Models. Since it is notoriously challenging to make exact and efficient inference about the Cauchy distribution, this article takes it as an example to elucidate different schools of thought on statistical inference. It is shown that the standard approach of Inferential Models produces exact and efficient prior-free probabilistic inference on the location and scale parameters of the Cauchy distribution, whereas all other existing methods suffer from various difficulties.

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arXiv2023

Partial Conditioning for Inference of Many-Normal-Means with H\"older Constraints

Inferential models have been proposed for valid and efficient prior-free probabilistic inference. As it gradually gained popularity, this theory is subject to further developments for practically challenging problems. This paper considers the many-normal-means problem with the means constrained to be in the neighborhood of each other, formally represented by a H\"older space. A new method, called partial conditioning, is proposed to generate valid and efficient marginal inference about the individual means. It is shown that the method outperforms both a fiducial-counterpart in terms of validity and a conservative-counterpart in terms of efficiency. We conclude the paper by remarking that a general theory of partial conditioning for inferential models deserves future development.

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arXiv2022

Reweighted and Circularized Anderson-Darling Tests of Goodness-of-Fit

This paper takes a look at omnibus tests of goodness of fit in the context of reweighted Anderson-Darling tests and makes threefold contributions. The first contribution is to provide a geometric understanding. It is argued that the test statistic with minimum variance for exchangeable distributional deviations can serve as a good general-purpose test. The second contribution is to propose better omnibus tests, called circularly symmetric tests and obtained by circularizing reweighted Anderson-Darling test statistics or, more generally, test statistics based on the observed order statistics. The resulting tests are called circularized tests. A limited but arguably convincing simulation study on finite-sample performance demonstrates that circularized tests have good performance, as they typically outperform their parent methods in the simulation study. The third contribution is to establish new large-sample results.

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arXiv2022

Estimation of High-Dimensional Normal Means through Inferential Models

The classical multivariate normal means problem remains conceptually unresolved. While shrinkage and empirical Bayes methods improve risk by imposing external geometric or hierarchical structure, they fail to explain how information is shared across independent coordinates for a fixed, unstructured mean vector. We address this gap using the prior-free Inferential Models framework. By formulating a generalized probability integral transform (GPIT) for independent, non-i.i.d~observations combined with a reweighted Anderson-Darling predictive random set, we leverage the global shape of ordered observations for valid, efficient inference. Crucially, the auxiliary structure of this formulation provides a novel explanation for Stein's paradox, demonstrating that the maximum likelihood estimator becomes structurally implausible for $n\geq 3$. To ensure scalability, we introduce an i.i.d. sampling-with-replacement surrogate that connects our exact fixed-mean formulation to overparameterized $g$-modeling. Furthermore, we develop a maximin criterion for combining plausibility contours. Under squared error loss, our estimators are competitive with state-of-the-art auto-modeling methods and outperform classical shrinkage and empirical Bayes methods.

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arXiv2021

Inferential models and possibility measures

The inferential model (IM) framework produces data-dependent, non-additive degrees of belief about the unknown parameter that are provably valid. The validity property guarantees, among other things, that inference procedures derived from the IM control frequentist error rates at the nominal level. A technical complication is that IMs are built on a relatively unfamiliar theory of random sets. Here we develop an alternative -- and practically equivalent -- formulation, based on a theory of possibility measures, which is simpler in many respects. This new perspective also sheds light on the relationship between IMs and Fisher's fiducial inference, as well as on the construction of optimal IMs.

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arXiv2018

An Asynchronous Distributed Expectation Maximization Algorithm For Massive Data: The DEM Algorithm

The family of Expectation-Maximization (EM) algorithms provides a general approach to fitting flexible models for large and complex data. The expectation (E) step of EM-type algorithms is time-consuming in massive data applications because it requires multiple passes through the full data. We address this problem by proposing an asynchronous and distributed generalization of the EM called the Distributed EM (DEM). Using DEM, existing EM-type algorithms are easily extended to massive data settings by exploiting the divide-and-conquer technique and widely available computing power, such as grid computing. The DEM algorithm reserves two groups of computing processes called \emph{workers} and \emph{managers} for performing the E step and the maximization step (M step), respectively. The samples are randomly partitioned into a large number of disjoint subsets and are stored on the worker processes. The E step of DEM algorithm is performed in parallel on all the workers, and every worker communicates its results to the managers at the end of local E step. The managers perform the M step after they have received results from a $γ$-fraction of the workers, where $γ$ is a fixed constant in $(0, 1]$. The sequence of parameter estimates generated by the DEM algorithm retains the attractive properties of EM: convergence of the sequence of parameter estimates to a local mode and linear global rate of convergence. Across diverse simulations focused on linear mixed-effects models, the DEM algorithm is significantly faster than competing EM-type algorithms while having a similar accuracy. The DEM algorithm maintains its superior empirical performance on a movie ratings database consisting of 10 million ratings.

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arXiv2018

Exact and efficient inference for Partial Bayes problems

Bayesian methods are useful for statistical inference. However, real-world problems can be challenging using Bayesian methods when the data analyst has only limited prior knowledge. In this paper we consider a class of problems, called Partial Bayes problems, in which the prior information is only partially available. Taking the recently proposed Inferential Model approach, we develop a general inference framework for Partial Bayes problems, and derive both exact and efficient solutions. In addition to the theoretical investigation, numerical results and real applications are used to demonstrate the superior performance of the proposed method.

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arXiv2016

Valid uncertainty quantification about the model in a linear regression setting

In scientific applications, there often are several competing models that could be fit to the observed data, so quantification of the model uncertainty is of fundamental importance. In this paper, we develop an inferential model (IM) approach for simultaneously valid probabilistic inference over a collection of assertions of interest without requiring any prior input. Our construction guarantees that the approach is optimal in the sense that it is the most efficient among those which are valid. Connections between the IM's simultaneous validity and post-selection inference are also made. We apply the general results to obtain valid uncertainty quantification about the set of predictor variables to be included in a linear regression model.

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arXiv2015

Simulating from a gamma distribution with small shape parameter

Simulating from a gamma distribution with small shape parameter is a challenging problem. Towards an efficient method, we obtain a limiting distribution for a suitably normalized gamma distribution when the shape parameter tends to zero. Then this limiting distribution provides insight to the construction of a new, simple, and highly efficient acceptance--rejection algorithm. Comparisons based on acceptance rates show that the proposed procedure is more efficient than existing acceptance--rejection methods.

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arXiv2014

Discussion: Foundations of Statistical Inference, Revisited

This is an invited contribution to the discussion on Professor Deborah Mayo's paper, "On the Birnbaum argument for the strong likelihood principle," to appear in Statistical Science. Mayo clearly demonstrates that statistical methods violating the likelihood principle need not violate either the sufficiency or conditionality principle, thus refuting Birnbaum's claim. With the constraints of Birnbaum's theorem lifted, we revisit the foundations of statistical inference, focusing on some new foundational principles, the inferential model framework, and connections with sufficiency and conditioning. [arXiv:1302.7021]

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