arXiv subjects

Chuanhai Liu

Publications and source records attributed to Chuanhai Liu.

28 records · Page 2Linked to original sources

arXiv2014

Marginal inferential models: prior-free probabilistic inference on interest parameters

The inferential models (IM) framework provides prior-free, frequency-calibrated, posterior probabilistic inference. The key is the use of random sets to predict unobservable auxiliary variables connected to the observable data and unknown parameters. When nuisance parameters are present, a marginalization step can reduce the dimension of the auxiliary variable which, in turn, leads to more efficient inference. For regular problems, exact marginalization can be achieved, and we give conditions for marginal IM validity. We show that our approach provides exact and efficient marginal inference in several challenging problems, including a many-normal-means problem. In non-regular problems, we propose a generalized marginalization technique and prove its validity. Details are given for two benchmark examples, namely, the Behrens--Fisher and gamma mean problems.

math.ST↗

arXiv2014

Frameworks for prior-free posterior probabilistic inference

The development of statistical methods for valid and efficient probabilistic inference without prior distributions has a long history. Fisher's fiducial inference is perhaps the most famous of these attempts. We argue that, despite its seemingly prior-free formulation, fiducial and its various extensions are not prior-free and, therefore, do not meet the requirements for prior-free probabilistic inference. In contrast, the inferential model (IM) framework is genuinely prior-free and is shown to be a promising new method for generating both valid and efficient probabilistic inference. With a brief introduction to the two fundamental principles, namely, the validity and efficiency principles, the three-step construction of the basic IM framework is discussed in the context of the validity principle. Efficient IM methods, based on conditioning and marginalization are illustrated with two benchmark examples, namely, the bivariate normal with unknown correlation coefficient and the Behrens--Fisher problem.

math.ST↗

arXiv2014

A note on p-values interpreted as plausibilities

P-values are a mainstay in statistics but are often misinterpreted. We propose a new interpretation of p-value as a meaningful plausibility, where this is to be interpreted formally within the inferential model framework. We show that, for most practical hypothesis testing problems, there exists an inferential model such that the corresponding plausibility function, evaluated at the null hypothesis, is exactly the p-value. The advantages of this representation are that the notion of plausibility is consistent with the way practitioners use and interpret p-values, and the plausibility calculation avoids the troublesome conditioning on the truthfulness of the null. This connection with plausibilities also reveals a shortcoming of standard p-values in problems with non-trivial parameter constraints.

math.ST↗

arXiv2014

Conditional inferential models: combining information for prior-free probabilistic inference

The inferential model (IM) framework provides valid prior-free probabilistic inference by focusing on predicting unobserved auxiliary variables. But, efficient IM-based inference can be challenging when the auxiliary variable is of higher dimension than the parameter. Here we show that features of the auxiliary variable are often fully observed and, in such cases, a simultaneous dimension reduction and information aggregation can be achieved by conditioning. This proposed conditioning strategy leads to efficient IM inference, and casts new light on Fisher's notions of sufficiency, conditioning, and also Bayesian inference. A differential equation-driven selection of a conditional association is developed, and validity of the conditional IM is proved under some conditions. For problems that do not admit a valid conditional IM of the standard form, we propose a more flexible class of conditional IMs based on localization. Examples of local conditional IMs in a bivariate normal model and a normal variance components model are also given.

math.ST↗

arXiv2013

Inferential models: A framework for prior-free posterior probabilistic inference

Posterior probabilistic statistical inference without priors is an important but so far elusive goal. Fisher's fiducial inference, Dempster-Shafer theory of belief functions, and Bayesian inference with default priors are attempts to achieve this goal but, to date, none has given a completely satisfactory picture. This paper presents a new framework for probabilistic inference, based on inferential models (IMs), which not only provides data-dependent probabilistic measures of uncertainty about the unknown parameter, but does so with an automatic long-run frequency calibration property. The key to this new approach is the identification of an unobservable auxiliary variable associated with observable data and unknown parameter, and the prediction of this auxiliary variable with a random set before conditioning on data. Here we present a three-step IM construction, and prove a frequency-calibration property of the IM's belief function under mild conditions. A corresponding optimality theory is developed, which helps to resolve the non-uniqueness issue. Several examples are presented to illustrate this new approach.

math.ST↗

arXiv2012

Optimal inferential models for a Poisson mean

Statistical inference on the mean of a Poisson distribution is a fundamentally important problem with modern applications in, e.g., particle physics. The discreteness of the Poisson distribution makes this problem surprisingly challenging, even in the large-sample case. Here we propose a new approach, based on the recently developed framework of inferential models (IMs). Specifically, we construct optimal, or at least approximately optimal, IMs for two important classes of assertions/hypotheses about the Poisson mean. For point assertions, we develop a novel recursive sorting algorithm to construct this optimal IM. Numerical comparisons of the proposed method to existing methods are given, for both the mean and the more challenging mean-plus-background problem.

stat.ME↗

arXiv2011

Parameter Expansion and Efficient Inference

This EM review article focuses on parameter expansion, a simple technique introduced in the PX-EM algorithm to make EM converge faster while maintaining its simplicity and stability. The primary objective concerns the connection between parameter expansion and efficient inference. It reviews the statistical interpretation of the PX-EM algorithm, in terms of efficient inference via bias reduction, and further unfolds the PX-EM mystery by looking at PX-EM from different perspectives. In addition, it briefly discusses potential applications of parameter expansion to statistical inference and the broader impact of statistical thinking on understanding and developing other iterative optimization algorithms.

stat.ME↗

arXiv2010

Dempster--Shafer Theory and Statistical Inference with Weak Beliefs

The Dempster--Shafer (DS) theory is a powerful tool for probabilistic reasoning based on a formal calculus for combining evidence. DS theory has been widely used in computer science and engineering applications, but has yet to reach the statistical mainstream, perhaps because the DS belief functions do not satisfy long-run frequency properties. Recently, two of the authors proposed an extension of DS, called the weak belief (WB) approach, that can incorporate desirable frequency properties into the DS framework by systematically enlarging the focal elements. The present paper reviews and extends this WB approach. We present a general description of WB in the context of inferential models, its interplay with the DS calculus, and the maximal belief solution. New applications of the WB method in two high-dimensional hypothesis testing problems are given. Simulations show that the WB procedures, suitably calibrated, perform well compared to popular classical methods. Most importantly, the WB approach combines the probabilistic reasoning of DS with the desirable frequency properties of classical statistics.

stat.ME↗

arXiv2010

The Dynamic ECME Algorithm

The ECME algorithm has proven to be an effective way of accelerating the EM algorithm for many problems. Recognising the limitation of using prefixed acceleration subspace in ECME, we propose the new Dynamic ECME (DECME) algorithm which allows the acceleration subspace to be chosen dynamically. Our investigation of an inefficient special case of DECME, the classical Successive Overrelaxation (SOR) method, leads to an efficient, simple, and widely applicable DECME implementation, called DECME_v1. The fast convergence of DECME_v1 is established by the theoretical result that, in a small neighbourhood of the maximum likelihood estimate (MLE), DECME_v1 is equivalent to a conjugate direction method. Numerical results show that DECME_v1 and its two variants are very stable and often converge faster than EM by a factor of one hundred in terms of number of iterations and a factor of thirty in terms of CPU time when EM is very slow.

stat.CO↗

arXiv2009

Estimating limits from Poisson counting data using Dempster--Shafer analysis

We present a Dempster--Shafer (DS) approach to estimating limits from Poisson counting data with nuisance parameters. Dempster--Shafer is a statistical framework that generalizes Bayesian statistics. DS calculus augments traditional probability by allowing mass to be distributed over power sets of the event space. This eliminates the Bayesian dependence on prior distributions while allowing the incorporation of prior information when it is available. We use the Poisson Dempster--Shafer model (DSM) to derive a posterior DSM for the ``Banff upper limits challenge'' three-Poisson model. The results compare favorably with other approaches, demonstrating the utility of the approach. We argue that the reduced dependence on priors afforded by the Dempster--Shafer framework is both practically and theoretically desirable.

stat.AP↗