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DempsterShafer theory and statistical inference with weak beliefs
, 2008
"... DempsterShafer (DS) theory is a powerful tool for probabilistic reasoning and decisionmaking based on a formal calculus for combining statistical and nonstatistical evidence, as represented by a system of belief functions. DS theory has been widely used in computer science and engineering applica ..."
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DempsterShafer (DS) theory is a powerful tool for probabilistic reasoning and decisionmaking based on a formal calculus for combining statistical and nonstatistical evidence, as represented by a system of belief functions. 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 the familiar longrun frequency properties statisticians are used to. Recently, two of the authors proposed an extension of DS, called the weak belief (WB) approach, which has the ability to incorporate desirable frequency properties into the DS framework. The present paper reviews and extends this WB approach. We present a general description of the WB approach, its interplay with the DS calculus, and the resulting maximal belief solution, some simple illustrative examples, and some new perspectives, namely, frequency properties/interpretations and the potential of WB for situationspecific inference. New applications of the WB method in two interesting statistical problems—largescale simultaneous testing and nonparametrics—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, making it possible to solve challenging inference problems in a situationspecific way.
DEMPSTERSHAFER INFERENCE WITH WEAK BELIEFS
"... Beliefs specified for predicting an unobserved realization of pivotal variables in the context of the fiducial and DempsterShafer (DS) inference can be weakened for credible inference. We consider predictive random sets for predicting an unobserved random sample from a known distribution, e.g., t ..."
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Cited by 15 (11 self)
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Beliefs specified for predicting an unobserved realization of pivotal variables in the context of the fiducial and DempsterShafer (DS) inference can be weakened for credible inference. We consider predictive random sets for predicting an unobserved random sample from a known distribution, e.g., the uniform distribution U(0, 1). More specifically, we choose our beliefs for inference in two steps: (i) define a class of weak beliefs in terms of DS models for predicting an unobserved sample, and (ii) seek a belief within that class to balance the tradeoff between credibility and efficiency of the resulting DS inference. We call this approach the Maximal Belief (MB) method. The MB method is illustrated with two examples: (1) inference about µ based on a sample n from the Gaussian model N(µ,1), and (2) inference about the number of outliers (µi ̸ = 0) based on the observed data ind X1,..., Xn with the model Xi ∼ N(µi,1). The first example shows that MBDS analysis does a type of conditional inference. The second example demonstrates that MB posterior probabilities are easy to interpret for hypothesis testing.
Inference about constrained parameters using the elastic belief method,” Internat
 J. Approx. Reason
, 2011
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On the Behavior of Dempster’s Rule of Combination and the Foundations of DempsterShafer Theory,” (Best paper award
 in Proc. of IEEE IS’2012
"... Abstract—On the base of simple emblematic example we analyze and explain the inconsistent and inadequate behavior of DempsterShafer’s rule of combination as a valid method to combine sources of evidences. We identify the cause and the effect of the dictatorial power behavior of this rule and of its ..."
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Cited by 8 (6 self)
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Abstract—On the base of simple emblematic example we analyze and explain the inconsistent and inadequate behavior of DempsterShafer’s rule of combination as a valid method to combine sources of evidences. We identify the cause and the effect of the dictatorial power behavior of this rule and of its impossibility to manage the conflicts between the sources. For a comparison purpose, we present the respective solution obtained by the more efficient PCR5 fusion rule proposed originally in DezertSmarandache Theory framework. Finally, we identify and prove the inherent contradiction of DempsterShafer Theory foundations. Keywords—Belief functions; DempsterShafer Theory; DSmT; PCR5; contradiction.
A betting interpretation for probabilities and Dempster– Shafer degrees of belief
 Journal International Journal of Approximate Reasoning
, 2011
"... There are at least two ways to interpret numerical degrees of belief in terms of betting: 1. You can offer to bet at the odds defined by the degrees of belief. 2. You can make the judgement that a strategy for taking advantage of such betting offers will not multiply the capital it risks by a large ..."
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Cited by 6 (0 self)
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There are at least two ways to interpret numerical degrees of belief in terms of betting: 1. You can offer to bet at the odds defined by the degrees of belief. 2. You can make the judgement that a strategy for taking advantage of such betting offers will not multiply the capital it risks by a large factor. Both interpretations can be applied to ordinary additive probabilities and used to justify updating by conditioning. Only the second can be applied to DempsterShafer degrees of belief and used to justify Dempster’s rule
SITUATIONSPECIFIC INFERENCE USING THE DEMPSTERSHAFER THEORY
"... R.A. Fisher questioned the samplingbased approach to statistical inference on the grounds that it often cannot really answer the scientific question of interest. Fisher’s fiducial argument and the DempsterShafer (DS) theory are inferential methods that strive towards answering these situationspec ..."
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R.A. Fisher questioned the samplingbased approach to statistical inference on the grounds that it often cannot really answer the scientific question of interest. Fisher’s fiducial argument and the DempsterShafer (DS) theory are inferential methods that strive towards answering these situationspecific questions. For some important problems, such as testing of a sharp null hypothesis, these alternative theories suffer from the same drawbacks as their samplingbased counterparts. The Weak Belief (WB) extension of DS is applied in such cases to achieve the best of both worlds: the desirable personal probabilitybased inference of DS with the additional flexibility of WB. We formulate a general framework for situation specific inference, which we call the WBDS method. Applications of the WBDS method are illustrated in two important statistical problems, namely largescale simultaneous hypothesis testing and nonparametrics. We show in simulations that the WBDS procedures, suitably calibrated,
A note on pvalues interpreted as plausibilities
 Statist. Sinica
, 2014
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Constructing and Reasoning about Alternative Frames of Discernment
 In: Proceedings of the Workshop on the Theory of Belief Functions, paper 24
, 2010
"... Abstract⎯We construct alternative frames of discernment from input belief functions. We assume that the core of each belief function is a subset of a so far unconstructed frame of discernment. The alternative frames are constructed as different cross products of unions of different cores. With the f ..."
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Cited by 2 (2 self)
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Abstract⎯We construct alternative frames of discernment from input belief functions. We assume that the core of each belief function is a subset of a so far unconstructed frame of discernment. The alternative frames are constructed as different cross products of unions of different cores. With the frames constructed the belief functions are combined for each alternative frame. The appropriateness of each frame is evaluated in two ways: (i) we measure the aggregated uncertainty (an entropy measure) of the combined belief functions for that frame to find if the belief functions are interacting in interesting ways, (ii) we measure the conflict in Dempster’s rule when combining the belief functions to make sure they do not exhibit too much internal conflict. A small frame typically yields a small aggregated uncertainty but a large conflict, and vice versa. The most appropriate frame of discernment is that which minimizes a probabilistic sum of the conflict and a normalized aggregated uncertainty of all combined belief functions for that frame of discernment. Keywords: DempsterShafer theory, belief function, representation, frame of discernment, induction. I.