PhD candidate/TA, Rutgers University
A new statistical paradigm and a new computational technique for complicated inferential problems.
There are several competing paradigms within the field of statistics. Recently, researchers have been interested in attempting to unify these different paradigms both for philosophical and practical purposes. Confidence distribution theory -or CD theory for short- is one such area of research.
In our work, we are inspired by a popular Bayesian computational technique, called ABC. This technique originated in population genetics, a field in which statistical inference can difficult as a result of studying biological procedures with complex mathematical representations. We develop a new method, Approximate CD Computing (or ACC for short), within the umbrella of CD theory. The result is a highly flexible procedure that contains ABC as a special case. Our new algorithm affords the user better opportunities for increased computational efficiency. Furthermore, CD theory lends support that helps establish statistical guarantees on the performance of the ACC method.
Abstract: Approximate Bayesian computation (ABC) is becoming an accepted tool for statistical analysis in models with intractable likelihoods. With the initial focus being primarily on the practical import of ABC, exploration of its formal statistical properties has begun to attract more attention. In this paper we consider the asymptotic behavior of the posterior obtained from ABC and the ensuing posterior mean. We give general results on: (i) the rate of concentration of the ABC posterior on sets containing the true parameter (vector); (ii) the limiting shape of the posterior; and\ (iii) the asymptotic distribution of the ABC posterior mean. These results hold under given rates for the tolerance used within ABC, mild regularity conditions on the summary statistics, and a condition linked to identification of the true parameters. Using simple illustrative examples that have featured in the literature, we demonstrate that the required identification condition is far from guaranteed. The implications of the theoretical results for practitioners of ABC are also highlighted.
Pub.: 23 Jul '16, Pinned: 27 Jun '17
Abstract: We present an informal review of recent work on the asymptotics of Approximate Bayesian Computation (ABC). In particular we focus on how does the ABC posterior, or point estimates obtained by ABC, behave in the limit as we have more data? The results we review show that ABC can perform well in terms of point estimation, but standard implementations will over-estimate the uncertainty about the parameters. If we use the regression correction of Beaumont et al. then ABC can also accurately quantify this uncertainty. The theoretical results also have practical implications for how to implement ABC.
Pub.: 23 Jun '17, Pinned: 27 Jun '17
Abstract: The notion of confidence distribution (CD), an entirely frequentist concept, is in essence a Neymanian interpretation of Fisher's Fiducial distribution. It contains information related to every kind of frequentist inference. In this article, a CD is viewed as a distribution estimator of a parameter. This leads naturally to consideration of the information contained in CD, comparison of CDs and optimal CDs, and connection of the CD concept to the (profile) likelihood function. A formal development of a multiparameter CD is also presented.
Pub.: 07 Aug '07, Pinned: 27 Jun '17
Abstract: Approximate Bayesian computing (ABC) is a likelihood-free method that has grown increasingly popular since early applications in population genetics. However, the theoretical justification for Bayesian inference (e.g. construction of credible intervals) based on this method has not yet been fully addressed when using non-sufficient summary statistics. We introduce a more general computational technique, approximate confidence distribution computing (ACC), to overcome a few issues associated with the ABC method, for instance, the lack of theory supporting for constructing credible (or confidence) intervals when the ACC method uses non-sufficient summary statistics, the long computing time, and the necessity of a prior assumption. Specifically, we establish frequentist coverage properties for the outcome of the ACC method by using the theory of confidence distributions, and thus inference based on ACC is justified, even if reliant upon a non-sufficient summary statistic. Furthermore, the ACC method is very broadly applicable; in fact, the ABC algorithm can be viewed as a special case of an ACC method without damaging the integrity of ACC based inference. We supplement the theory with simulation studies and an epidemiological application to illustrate the benefits of the ACC method. It is demonstrated that a well-tended ACC algorithm can greatly increase its computing efficiency over a typical ABC algorithm.
Pub.: 29 May '17, Pinned: 27 Jun '17