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\footnotesize \noindent Statistica Sinica {\bf 21} (2011), 475-494

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\centerline{\large\bf DEMPSTER-SHAFER INFERENCE WITH  WEAK BELIEFS}
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\author{Jianchun Zhang and Chuanhai Liu}
\vspace{.4cm} \centerline{\it Purdue University} \vspace{.55cm}
\begin{quotation}
\noindent {\it Abstract:} The work of A. P. Dempster in 1960s
extending Fisher's fiducial inference for parametric inference using
multivalued mapping, and that of G. Shafer in 1970s on the
assessment and combination of evidence led to what is now known as
the Dempster-Shafer (DS) theory of belief functions. However,
application of DS for parametric inference has been limited due,
perhaps, to its computational difficulty, non-uniqueness, and lack
of frequency properties. In this paper, we return to Dempster's
original approach to constructing belief functions for parametric
inference, called basic DS models (BDSMs), which are usual
probability models on the space of the so-called focal elements. We
propose to modify BDSMs by enlarging focal elements to obtain belief
functions that have desired frequency properties. We call our method
Weak Belief (WB).
 When it enlarges the focal elements
no more than necessary, the method of WB is called Maximal Belief
(MB). The MB method is illustrated with two examples: $(i)$
inference about a binomial proportion,
 and $(ii)$ inference about the number of outliers $(\mu_i\neq 0)$
based on the observed data $X_1,\ldots,X_n$ with the model
$X_i\stackrel{ind}{\sim}N(\mu_i,1)$.\par \vspace{9pt} \noindent {\it
Key words and phrases:} Belief functions, fiducial inference,
frequentist evaluation, hypothesis testing, maximal belief,
predictive random sets. \par
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