## GENERAL THEORY OF INFERENTIAL MODELS II. MARGINAL INFERENCE

Citations: | 1 - 0 self |

### BibTeX

@MISC{Martin_generaltheory,

author = {Ryan Martin and Jing-shiang Hwang and Chuanhai Liu and Academia Sinica},

title = {GENERAL THEORY OF INFERENTIAL MODELS II. MARGINAL INFERENCE},

year = {}

}

### OpenURL

### Abstract

This paper is a continuation of the authors ’ theoretical investigation of inferential model (IMs); see Martin, Hwang and Liu (2010). The fundamental idea is that prior-free posterior probability-like inference with desirable long-run frequency properties can be achieved through a system based on predicting unobserved auxiliary variables. In Part I, an intermediate conditioning step was proposed to reduce the dimension of the auxiliary variable to be predicted, making the construction of efficient IMs more manageable. Here we consider the problem of inference in the presence of nuisance parameters, and we show that such problems admit a further auxiliary variable reduction via marginalization. Unlike classical procedures that use optimization or integration, the proposed framework eliminates nuisance parameters via a set union operation. Sufficient conditions are given for when this marginalization operation can be performed without loss of information, and in such cases we prove that an appropriately constructed IM is calibrated, in a frequentist sense, for marginal inference. In problems where these sufficient conditions are not met, we propose a marginalization technique based on parameter expansion that leads to conservative marginal inference. The marginal IM approach is illustrated on a number of examples, including Stein’s problem and the Behrens-Fisher problem.

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Citation Context ...n detail in Zhang and Liu (2010), Martin, Zhang and Liu (2010), and Martin, Hwang and Liu (2010). Here we give a quick review for completeness. 3.1. Belief functions. Belief functions (Dempster 1967; =-=Shafer 1976-=-) are similar to, but more general than, probability measures. The DempsterShafer (DS) theory for statistical inference constructs a belief function on Θ as follows. For observed data X = x, define th... |

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Citation Context ...also prove a “conditional version” of the fundamental Theorem 3.1 of Zhang and Liu (2010), and draw parallels between this conditional IM framework and Fisher’s notion of sufficiency and ancillarity (=-=Fisher 1925-=-, 1934, 1935) in group transformation models. This conditioning step, reviewed in Section 3, will often reduce the dimension of the auxiliary variable to that of the parameter. But in marginal inferen... |

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