Statistical Reasoning with Imprecise Probabilities

Walley, Peter

doi:10.1007/978-1-4899-3472-7

Cited by 2,081 publications

(2,617 citation statements)

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“…The model obtained in this way is a special version of the imprecise Dirichlet model [40] and is coherent in the strong sense of Walley [39,Section 7.8]. In this framework s is interpreted as a degree of caution that Walley suggests choosing in the interval [1,2].…”

Section: The Naive Credal Classifiermentioning

confidence: 96%

“…Overall, sets of probability distributions appear as a well-founded framework suited to model ignorance. Sets of probability distributions belong to the wider theory of imprecise probabilities [39] (see http://www.sipta.org for up-to-date information).…”

Section: Imprecise Probabilities and Credal Classificationmentioning

confidence: 99%

“…2.3). The NCC generalizes the wellknown discrete Naive Bayes Classifier (or NBC [4]) to imprecise probabilities [39], a well-founded generalized framework for uncertain reasoning. The NCC models both prior ignorance and ignorance about the likelihood originated by missing data, by sets of probability distributions (see Sect.…”

Section: Introductionmentioning

confidence: 99%

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Reliable diagnoses of dementia by the naive credal classifier inferred from incomplete cognitive data

Zaffalon

Wesnes²,

Petrini³

2003

Artificial Intelligence in Medicine

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Dementia is a serious personal, medical and social problem. Recent research indicates early and accurate diagnoses as the key to effectively cope with it. No definitive cure is available but in some cases when the impairment is still mild the disease can be contained. This paper describes a diagnostic tool that jointly uses the naive credal classifier and the most widely used computerized system of cognitive tests in dementia research, the Cognitive Drug Research system. The naive credal classifier extends the discrete naive Bayes classifier to imprecise probabilities. The naive credal classifier models both prior ignorance and ignorance about the likelihood by sets of probability distributions. This is a new way to deal with small or incomplete data sets that departs significantly from most established classification methods. In the empirical study presented here the naive credal classifier provides reliability and unmatched predictive performance. It delivers up to 95% correct predictions while being very robust with respect to the partial ignorance due to the largely incomplete data. The diagnostic tool also proves to be very effective in discriminating between Alzheimer's disease and Dementia with Lewy Bodies.

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Section: The Naive Credal Classifiermentioning

confidence: 96%

Section: Imprecise Probabilities and Credal Classificationmentioning

confidence: 99%

See 1 more Smart Citation

Reliable diagnoses of dementia by the naive credal classifier inferred from incomplete cognitive data

Zaffalon

Wesnes²,

Petrini³

2003

Artificial Intelligence in Medicine

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“…The problem is that moving from a single distribution to a convex set of distributions introduces a plurality of independence concepts, and this splintering of probabilistic independence has ramifications for rational choice (Levi 1980, Seidenfeld and Wasserman 1993, Kyburg and Pittarelli 1996, statistical inference (Walley 1991), and probabilistic logic (Haenni et al 2011), mainly because some inferences from independence conditions which are perfectly sound in the context of a single probability distribution are fallacious in the context of a set of distributions.…”

Section: Probabilitymentioning

confidence: 99%

Objective Bayesian Calibration and the Problem of Non-convex Evidence

Wheeler

2012

The British Journal for the Philosophy of Science

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Jon Williamson's Objective Bayesian Epistemology relies upon a calibration norm to constrain credal probability by both quantitative and qualitative evidence. One role of the calibration norm is to ensure that evidence works to constrain a convex set of probability functions. This essay brings into focus a problem for Williamson's theory when qualitative evidence specifies non-convex constraints.Jon Williamson (2010) provides a spirited defense of a version of Objective Bayesianism which relies upon a calibration norm to constrain credal probability by evidence. According to this norm, an agent's degrees of belief should be constrained by two types of evidence: quantitative evidence, which directly constrains admissible values of chance functions, and qualitative evidence, such as logical or causal constraints on chance variables, which may indirectly constrain chance functions. Once those constraints are in place, the theory maintains that the agent's degrees of belief should be maximally equivocal between the basic outcomes.I have discussed the calibration norm with sympathy (Wheeler and Williamson 2011), although in that setting our aim was to reconcile the reference class reasoning machinery of Evidential Probability (Kyburg and Teng 2001) with Williamson's Objective Bayesian Epistemology (OBE), so we naturally played down the differences between the two theories. Nevertheless, I have reservations about viewing OBE as a general theory for rational belief. So, in response to Williamson's call to go whole hog, I would like to bring into focus a problem for OBE that arises when qualitative evidence prescribes non-convex constraints on a set of chance functions.The mechanics of OBE depend on a convex set of probability functions, and one role the calibration norm plays is to enforce this convexity condition. This can be a reasonable approach when the endpoints of an interval designate upper and lower bounds on admissible degrees of belief and the question to answer is what point within this range represents the most cautious position for an agent to take. That is the question that OBE is set up to answer, and it does so by advising the agent to pick the most equivocal point within the convex hull of admissible options

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“…[0,1], which would indicate, in this case, an underlying continuous value anywhere between impossibility and certainty. The range of the upper and lower bounds would express the degree of second order uncertainty (Walley, 1991). For example, the eventual number of cases of CJD resulting from the UK BSE epidemic in the 1980s might range from the low hundreds to several million.…”

Section: Expectations Where Relative Frequencies Have Not Been Firmlymentioning

confidence: 99%