Decision And Reasoning In Incompleteness Or Uncertainty Conditions

2016·0.88 MB·

by  Gerardo Iovane

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/ACCESS.2020.3003726, IEEE Access
DigitalObjectIdentifier
Decision and Reasoning in
Incompleteness or Uncertainty
conditions
GERARDOIOVANE1,PATRIZIADIGIRONIMO2,MARTACHINNICI3,ANTONIORAPUANO1
1DipartimentodiInformatica,UniversityofSalerno,Italy(e-mail:[email protected],[email protected])
2DipartimentodiMatematica,UniversityofSalerno,Italy(e-mail:[email protected])
3EnergyTechnologiesDepartment,ICTDivision(DTE-ICT),ENEA,Casaccia,Rome,Italy(e-mail:[email protected])
Correspondingauthor:MartaChinnici(e-mail:[email protected]).
ABSTRACT Whenwetrytosolveneworknownproblemstowhichwewanttogivenewsolutions,we
createnewknowledgeandrealizenewdiscoveries.Todate,thescientificmethodshaveusedtheprobability
in order to analyze problems, make inference and build forecasts. However, everyone agreed that most
problemsdonotfollowstandardprobabilisticrules.Inthisstudywewillbuildanuncertaintylogicbyusing
the concept of probability, with those of plausibility, credibility and possibility. We will provide several
models which treats uncertainty information and allow to perform more reliable forecasts. After that, we
willprovethemodelsreliabilitythroughafinalsimulationontheBiometricsandSportfieldsusingoneof
themodels;thesesimulationarefullyreplicabileforeachfieldandforeachoftheprovidedmodels.
INDEXTERMS Biometrics,Credibility,Incompleteness,Plausibility,Possibility,Probability,Sportsodds,
Uncertainty.
I. INTRODUCTION information not only uncertain, but also paradoxical1. This
reasoning has been fully studied: Cuzzolin in [1] defines
THE plausibility is a concept used in many inferential
thepropertiesofplausibility.Friedman,HalpernandBarnett
contexts and processes. A plausible inference theory
calculatedandmeasuredtheplausibility[2][3].Thisconcept
goesthroughapreliminarydefinitionofwhatis"uncertain"
has been used in real world applications like [4] as well.
and how it affects the inferential process. A plausible inter-
While Shved, Wanga and Wen provide a way to measure
ferencecanbeinvalidated,correctedorsimplyupdatedatthe
uncertainty[5][6][7].Inthispaper,wearegoingtodefinea
entryofnewdataorinformation.Inthissense,plausibilityis
newperspectiveontheplausiblereasoning.
notonlynon-monotonous,butoperatesinanopenconceptual
space. The data, the available information and inferential
rules are of a temporary nature and reviewable so, they are II. CONCEPTUALOVERVIEW
constantlychanging. This section in inspired by the work of author in [8]. The
Polya (1954) defines the plausible reasoning as opposed to formulationofatheoryofplausibilitywhichhasbeenknown
thedemonstrativereasoning.Theplausibleinferencehelpsto in antiquity is that of Sextus Empiricus. Truthly, he at-
acquirenewknowledge,usingtheinductionsandtheanalogy tributesthistheorytoCarneadeofCyrene,inhiscomposition
which are not certain and do not transmit the truth. The "AgainsttheLogicians".Plausibleheremeans"topersuade".
plausibleinferencehasafluid,temporaryandriskybehavior. Thecriteriaare:
Rescher(1976),conceivestheplausiblereasoningasaform
of deduction starting from uncertain premises, aimed at • somethingisplausibleifitseemstobetrue,
the treatment of "cognitive dissonances". In this theory the • is even more plausible if it seems to be true and is
compatible with other things that seem to be true (i.e.
premises and conclusions are provisional, controversial and
itisstable),
fluid.
For Dezert (2002), the plausible reasoning is an extension • isstableandistested.
of the evidence theory of Dempster-Shafer, and therefore
of the Bayesian probability. His theory allows to deal with 1admittingcontradictoryinformationthankstothe"hyper"set-power"
VOLUME4,2016 1
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So it is not necessary that something be true or believed, Pr{E/H} and therefore "have a numerical value, equal to
to be plausible. It simply has to satisfy some requirements, the probability that an event of the type predicted by E will
so as to allow something to be assessed as in accordance or happen,computedonthebasisofthestatisticalhypothesisH"
consistentwithrelatedfacts. although"credibilityandprobabilityaredifferentlydefined"
Uncertaintycanbeoftwotypes: [10].WhenwedealwiththeplausibilityofA,wearedealing
• Aleatory (also known as stochastic), when it concerns withthereliabilityofAandthestrengthofevidenceinfavor
therandomnessoftheobjectofknowledgesystem; ofA,inotherwords,ourtrustinA.Credibilityisexpressed
• Epistemic(alsoknownassubjective),i.e.whenitrefers in probabilistic terms, readjusting the Bayes’ theorem with
tothestateofknowledgeofthesubjectoverthesystem. the substitution of probability (Pr) with credibility (Cr). In
this view, plausibility and probability are indistinguishable,
Theplausibilityuncertaintybelongstothesecondtype;it
astheformercanbereducedtothesecond;howeveritisnot
isanexpressionofasubjectiverelationship.
always possible to compare numerically the plausibility of
AccordingtoWalton[9],itisacontroversialquestionifthe
twodifferentevents.TheprobabilisticconceptionofPolyais
plausibilityisdifferentfromtheprobability,anditisdifficult
affectedbyallthelimitationsoftheBayesianconception.
to rule out that plausibility can be resolved in some special
caseofprobability.
In order to model the epistemic uncertainty with the proba- B. DEMPSTER-SHAFER
bility,theCox’stheoremisused.Itestablishestheexistence To overcome the limits of the bayesian conception and the
of a precise relationship between the notion of plausibility, relativevisionofplausibleinference,theevidencetheoryof
plandthetheoryofprobabilitycalculation,pr.Thistheorem Dempster-Shaferhasbeendefined,hereafterdenotedasDS.
proves that any system of plausible inferences, which is not Also for this theory the bayesian conception is the basis for
isomorphictoprobabilitytheory,mustviolateatleastoneof theplausibilitynotion.Inparticular,it"includesthebayesain
thefollowingconditions: theoryasaspecialcaseandthereforepreservesatleastsome
of the attractions of that theory" [12] [13]. The DS theory
renounces the one-dimensional representation of belief: in
• pl(A|X)=0↔AfalseinX
otherwords,itrenouncestoassigntheremainingportionof
• pl(A|X)=1↔AtrueinX
belieftothenegationoftheproposition.TheDStheorystarts
• 0≤pl(A|X)≤1
from what is called a discernment or universe of discourse
• pl(A∧B|X)=pl(A|X)pl(B|X)
u, which is a series of mutually exclusive alternatives. The
• pl(¬A|X)=1−pl(A|X),
DS theory assigns a basic probability, a belief function and
where pl (A|X) indicates the plausibility of A over X. If it
a plausibility function to the u subsets which are called
doesnotviolateanyoftheconstraintsitispossibletoreplace
propositions.Wedefinethefollowingassignmenttoabasic
pr a pl and use the probability calculation without losing
probability:
information.
m:2u →[0,1]suchthatm()=0,(cid:80) m(A)=1
A⊆u
A. POLYA
which represents and expresses numerically the force of
AccordingtoPolya,theprototypesofplausibleinterference
evidence,theexactbeliefthatanagenthasinA.
aretheanalogyandtheinduction;infact"theanalogyandthe
The function m: 2u → [0,1] is called belief function Bel,
particular cases are the most abundant sources of plausible
whenitsatisfiestheconditions:
arguments"[10][11].Inparticular"theinferencebyanalogy
seems the most common and essential form of inference. It • Bel()=0,
produces more or less plausible conjectures which may or • Bel(u)=1.
may not be confirmed by experience or by more rigorous Plausibilitycannowbedefined.Weobservethat"aproposi-
reasoning". The plausible argument, as defined by Polya, is tion is plausible in the light of evidence to the point where
notdefinitive:itproducesprovisionalconclusions.Theplau- the evidence does not support its opposite" [12]. Then we
sibilisticreasoningisnotsubjective:themodelsofreasonings introduce the dubious function, Dub, placing it as equal to
areimpersonal,whileitistheforceoftheconclusionswhich Cr(¬A) where ¬A is the complement of A in u and we
is of a subjective nature and therefore not representable by define the function of plausibility, Pl, as Pl (A) = 1-Dub
means of quantity. As for the fundamental inductive model, (A) = 1 (Cr¬A). The plausibility function expresses what
they have a correspondence in the demonstrative logic. In shouldbebelievedinpropositionAifallthenotknownfacts
[10], the impersonal rules which establish some kind of at present state support A; it also expresses the maximum
evidencearedescribed.Whileitispersonalandsubjectiveto probabilistic value that can be allocated to a proposition
establish"whetheratesthasasufficientweightornot".For A: in particular "it measures the total mass of belief that
Polya the probability is the core in the plausibility notion. can be moved to A" [14]. The lower limit corresponds to
The Bayes theorem is the foundation of Polya’s concept of the belief function, and therefore the relation Cr (A) ≤ Pl
plausibility. The credibility of B is the confidence of A if B (A) is valid. The difference between belief and plausibility
is true. In fact, credibility can be a conditional probability is also reflected at the functional level: indeed the belief
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function"isoftenzeroforallatomicpropositionsincomplex propositionsare"candidates"tothetruthornot.Anargument
domains, unless a large number of trials are available". The plausibility can therefore be defined as the maximum value
otherbigdifferencebetweenbayesiantheoryandDStheory amongtheminimumplausibilisticvaluesoftheentemematic
is the evidence combination, that is the belief updating rule integrations which allows a deductive conclusion derivated
whenthereisanewevidence.Infact,intheDS’theorythe fromthepremises.
combinationrulesubstitutesthebayesianrule.Thecombina-
tionruleisalsocalledorthogonalsum.TheDScombination E. NOTMONOTONICITYANDPLAUSIBILITY
rulesuffersfromsomewell-knownstructurallimitations:in Anothernon-probabilisticapproachaimedattreatingplausi-
fairlysimplesituationsitcouldgenerateunexpectedandnon- bleinferenceisnon-monotonouslogics’one.Thisapproach
intuitiveresults,whichstronglylimiteffectiveness[15]. is characterized by the violation of one of the strongest
conditionsofclassicallogic,knownas"monotonicity".Ifthe
C. DEZERT-SMARANDACHE conclusion ϕ is a consequence of a premises set Γ, then it
TheDezert-Smarandache’stheory[16],hereafterdenotedas remains a consequence of any premises set ∆ that contains
DSm, has born as an explicit attempt to overcome some Γasitssubset.Accordingtothiscondition,aconclusioncan
difficultiesoftheDStheory.Plausibilityisanupperlimitof not be invalidated by the addition of new information, it re-
the probabilistic value which a proposition can assume. Ds’ mainstrueonceandforall,regardlessofthepremisesthatwe
plausibility theory conceives plausible reasoning as a con- canaddtothepremisessetΓ.Thevalidpropositionsnumber
ceptualmodelforguidingdecision-makingunderuncertainty increasesmonotonouslytothepremisesaddedtoΓincrease.
conditions,whichisbroadlydefinedrespectingtheDS’the- Ifthepropositionscanbeinvalidatedbytheadditionofnew
ory.Inparticular,theuncertaintyreferredbyDezertextends premises,thevalidassertionsnumbermaynotonlyincrease,
the one conceived in probability theory and DS’ theory. A but may also decrease. The non-monotonous logic is based
situationoraninformationstateis"rational"whenthebasic on a reduced vision of uncertainty and it simply treats in-
assignmentmisasumoneandtheclosureoftheoperators∩ ferences with exceptions. Inference forms modeled by non-
and∪ontheuniverseelementsis0;itisstrictly"uncertain" monotonic logics are uncertain as they produce provisional
when it is a sum one, and the closing of the operator ∪ conclusionswhichcanbeinvalidatedorreviewedinthelight
can be different from zero; it is "paradoxical" when it is of new information available, and this is a big limitation.
sum one and the closing of the operator ∩ can be different Furthermore, the non-monotonous approach conceives the
fromzero;itis"uncertainandparadoxical"whentheclosure plausiblereasoningasessentiallyaddressedtotheinferences
of both operators ∩ and ∪ can be different from zero. The treatmentpropertoeverydayexperience.Thus,itneglectsthe
assumptions’modificationproduceseffectsonthedeveloped wholepartoftheplausibleinferencedirectlyconnectedtothe
conceptual model, but does not allow to exceed its limits. scientific disciplines, including Mathematics. As theorized
AlthoughDezerttendstoamoreextensivediscussionofthe byPolyaandCellucci[18],theplausibleinferenceisfullof
plausible reasoning, his approach contributes to leaving it experimentalreasoningthattheclassicallogicdoesnotallow
anchoredtoastronglyreductionistview. tohandle.
D. THERESCHER’SDEDUCTIVISTCONCEPTION F. THECOLLINS-MICHALSKI’SCOGNITIVEAPPROACH
Among the non-probabilistic approaches to plausibility the- The cognitive approach to plausibility aims to show how
ory,Rescher’sdeductivistapproachstandsout.Accordingto factors of a subjective and psychological nature can play
Nicholas Rescher, the "plausibility theory aims to provide a decisive role in the process that leads to a choice. The
a rational instrument to treat cognitive dissonances" [17]. cognitiveapproachtoplausibilityismodeledintheCollins-
Rescherprovidesapreliminarydistinctionbetweenplausibil- Michalscki’s plausibility theory hereafter denoted as C-M.
ityandprobability:thefirst"classifiespropositionsaccording It intends to formalize the plausible inferences which fre-
tothestatusoftheevidentiarysourcesorthevalidatingprin- quently recur in people’s responses to questions for which
ciples that guarantee them", while "the probability weights theydonothaveareadyanswer[19].
variousalternativesandevaluatesthem".Insteadofadopting Information is defined as common knowledge to a group of
theprobabilitytheoryasabasisformodelingtheplausibility people if: all the group members know it; they know that
notion, he bases his approach on the traditional Theophras- othersknowit;theyknowthatothersknowthattheyknowit;
tus’ principle, pointing to a return to the historical roots of andsoon.Itisdifferentfromthesimplemutualinformation,
the plausibility theory. It never develops calculations which "which implies only that specific possession of information
merge quantities into new results, but proceeds only with andnoteventheawarenessoftheothers’knowledge".That
comparisons between different degrees of data. A further mutual knowledge may become a common knowledge, in
aspectofthedeductivisticapproachtotheplausibilitynotion presence of an "independent arbitrariness" to the various
is the link with the entymatic argument where the common agents who make inferences and share information. These
knowledge makes it possible to integrate the missing steps, reasoningformscreateplausibleinferencemodelswhich,al-
that is the non-explicit assumptions within an inferences thoughdependentoncognitiveandpsychologicalfactors,are
sequence.Theplausibilitytheoryallowsustoestabilishifthe notincludedortraceabletothetheoryofCollins-Michalsky.
VOLUME4,2016 3
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Thistheoryleavesoutsomefactswhichtheplausiblereason-
ingexplicitlyintendedtotreatandmodel.
G. OPENSYSTEMSANDNETWORKS
Thankstoitsrecentdevelopments,thenetworks’theorycan
be viewed as an analysis source and plausible inference
investigation. The knowledge representation in graph form
is capable of detecting essential aspects about the plausible
inference like nature and dynamics. We can consider an
hypothesis as ’plausible’ when it is connected to existing
knowledge and has a capability of self-adjustment and ro-
bustness (resistance to error): the greater the degree of con-
nection, the more likely the hypotesis to be plausible. Con-
FIGURE1:Howaproblemisdefined
nectivitywillmeasureitsintegrationandcompatibilitywith
existing knowledge degree and robustness. The success of
thisapproachdependsessentiallyonanadequateknowledge
dependscouldbehencedescribedeitherbydynamicdefined
representation in a reticular form, and in particular on the
laws or by some dynamics which are out of the analyst’s
chosengranularity.Plausibleinferencescanhaveverycom-
control.
plexforms,composedofmanystepsresultingfromdifferent
rules application (deductions, inductions, analogies), which
This indetermination could concern both the single vari-
create equally complex graphs. It is possible to define plau-
ableandthewayitaffectstheoutput.Inthefirstcasewehave
sibilityintermsoflocalorglobalproperties.Inparticular,a
adirectindeterminacy(causeindeterminacy)thatconcernsa
nodecanbeplausibleifithasnumerousincidentedgesorif
problemvariable.Inthesecondcasewehaveaderivativeor
therearedifferentpathsleadingtoit.
effectindeterminacy.
ImagineanindividualthathastomovefromAtoBbybus.
III. DECISIONINUNCERTAINTYDOMAINS
The first indetermination concerne when the bus will arrive
A. SCENARIO
(temporalindeterminacy),soontheinputvariable"time".It
In the knowledge and collective intelligence era, like ever,
is obvious that reaching the B position could depend on the
wearecalledtomakedecisions.Generally,weuseproblem
trafficwhichrepresentsaneffectindeterminacygivenbythe
solvingtoachivegoals(decisionmaking).Itisobviousthat
constraint"traffic"orroadfillingcoefficient.Havingknowl-
atargetcouldbereachedbyjoining"elementary"decisions.
edgeofmultiplepathsandtheirroadstatus("traffic"orroad
This highlights an important issue: in the modern decision-
fillingcoefficient)itisanobviousadvantagetominimizethe
making support systems, it is important to focus both on
Breachingtime,alsopayingaroutecost,correspondingtoa
the objectives and on the decisionaly path. The objective is
longerjourneyintermsofspace.
reachedbyjoiningsub-objectivesthroughanoptimalpath.
Thisexampleallowstounderstandthesimplicityofdealing
If we were in Physics, with a material system that has to
with non-deterministic problems when they can be reduced
transitfromAtoBinnfinitestepsandthroughmdecision-
todeterministiconeswithaseriesofchoices(constraints).
making paths, it would be easy to estimate the problem
If we think about how to choose which path to take, the
complexityandreduceitwithprinciplesofminimization,i.e.
answer leads to another dualism: infocompletness and in-
the problems of transaction energy, the transaction disorder
foincompletness. With the previous example it is simple to
ortimeminimization.
understand how the knowledge of a free path allows the
In the case of a logistic transport problem, in which the
driver to easily choose which road to follow to reach point
objectiveismovingmerchandiseorpeoplefromAtoB,the
B(goalofthetrip)intheshortesttime.Withthismethodwe
optimal solution may be achieved by maximizing merchan-
cantransformanondeterministicprobleminadeterministic
disetransportedinagiventimeinterval.
onethankstoachoicebasedonthescenarioknowledge.
Whentherearemoresolutionsoratargetsolutionreachable
In this way, we have introduced complex systems and
throughdifferentcompetitivestrategies,theseexamplesand
decision theory characterization: the info completeness-
theirgeneralizationsgiveustheopportunitytoformalizethe
incompleteness.
problemin(fig.1):
From the following three characterizations the stochasticity
• initialconditionsdefinition, isdefined2:
• constraintsfixing,
1) variablesdeterminism,variablesindeterminism;
• targetindividuation,
2) functional link between variables and their impact on
• solutionstrategydefinition,
thetargetdeterminism,targetindeterminism;
• solutionoptimization.
Afirstproblemclassificationistoestablishwhetheritisde-
2Thatisthenon-determinismofmanyproblemsinmostofthecognitive
terministicorstochastic.Thevariablesonwhichtheproblem areaswhereitisnecessarytomakedecisions
4 VOLUME4,2016
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3) completeness and incompleteness of the starting sce- methodologies to solve and to describe the uncertainty in a
nario(inputvarables),thearrivalscenario(outputvari- rigorous framework. An example is the price distribution of
ables), and the constraints (functional models which afinancialinstrumentaroundasamplepriceinwhichanot-
linkinputstooutputs). normaldistributionisobserved,withheavytails(largerthan
We give an example in the financial context where the Gaussiantails).
targetisthereachingofapricetarget: This means that some improbable events can still happen,
therefore it is important to estimate whether the event is
1) firsttypeindetermination:tickinstant;
plausible,credibleorpossibleandestimatereliability.
2) secondtypeindetermination:tickssuccession;
Info-incompleteness or the input indecision generate emo-
3) third type indetermination, info incompleteness: ex-
tionsandbeliefsthatmakedecision-makingprocesssubjec-
changedvolumesandpricevariation.
tive.
It is important to specify that the word stochastic could
So we understand why the rational operator becomes
not mean full random, because it is possible to build mod-
emotional-affected and takes sentiment and subjective
els which manipulate stochasticity and make predictions. In
choices.
otherwords,takingthepreviousexample,wewillbeableto
The purpose of decision-making support systems must be
give an estimation based on known constraints or analyst’s
to build simulated scenarios, like those of game theory. In
needs, so we will be able to make predictions on the price
gametheory,differentsolutions/targetscooperateorcompete
in a time frame. Stochastic dynamic systems solve many
tothebestchoice,wherethebestwordisbasedondifferent
problemseveniftheydonotprovideaperfectlydeterministic
parametersofdecision-makerinterest(targetconstraints).
answer.
Thanks to this methodology, related to technological help
It is important to define the right position of the problem.
andthesupportofsimulatedscenarios,anemotional-affected
This problem concerns the certainty of the target. About
operatorwillbebroughttorationally(inducedrationaloper-
the previous examples, we must ask if the target price is
ator)decidebeforetheresultachievement(decisionforward
reachable, or if the considered bus line stops at point A etc.
intime).
Sothequestionisifthetargetcanbepossiblyachievedorit
Informationincompletenessandthevariables/information,or
isimpossible.
functional bonds indetermination, generate emotions in the
Therefore, a set of linguistic nuances emerged between the
decisionmakerwhichinducebeliefs.
certaineventandtheimpossibleone.
Thankstothisworkwebringbacktheoperatortoberational
In fact, other terms have emerged linguistically: possible,
on a larger cognitive basis (expanded knowledge); we use
plausible and credible. These terms need to be formally
other approaches like sentiment analysis or emotional anal-
defined. Shortly, our study highlights that decision-making
ysistocontrolthisbeliefs.
problemsincomplexitycontextsaregeneratedbydualisms:
Thenewrealizationmomentwillleadtothehyperconscious-
• determinism,indeterminism; nessstatuswheretheoperatorisconsciousoftheaugmented
• completeness,incompleteness; scenariocalledhyperscenario.
whichgenerateanotherdualism:certainty,uncertaintyofthe
target which have two extremes of solution. This target is
B. CONTINUOUSOCCURENCEOFUNLIKELY
betweencertainevent,impossibleeventanduncertainevent,
POSSIBLE
whichisrealityanditisinthemiddle.
Howmanytimeshavewewitnessedtheoccurrenceofhighly
Uncertaindualismisfacedbyquantificationof:
improbableevents?
• plausible,implausible; How many times was a football match conclusion plausible
• credible,incredible; andthingshavegonedifferently?
• probable,improbable; Moreover,howmanytimesthepricedynamicsofafinancial
• possible,impossible; instrumenthavemadeustobelievethatwehaveunderstood
The goal of our work is to understand how to use, quantify thedominantdirection,butassoonasweenteredthemarket
andgeneralizethesedichotomiesinordertobuilddecision- ithadanoppositebehavior?
making support systems which are not only probability- These are some examples of how complex it is to make
based. previsions under uncertainty conditions (input, models and
Someone could be confused and wonder why to use so resultsuncertainty)orinformationincompleteness.
many different concepts and not rely on probability theory Overcome determinism and entered the stochastic or un-
orBayesianprocesses. certainty phenomena context we find the terms: probable,
There is a simple reason, some phenomena which consider plausible,credible,possible.
complexity or complex systems are not Gaussian (or other In literature there are different research lines and thousands
known distributions) distributed, but they have heavy tails works which describe, distinguish and model these terms
whicharenotGaussian. sometimesinaconflictualway.
When someone uses the term heavy tails, he is putting Inthisstudy,wepresentagradedorganicvisionoftheterms,
a patch on a problem, because he does not have suitable starting from the best modeled concept, rather probability,
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using that to model probable event, but that extend it to With this assumption it is obvious that in uncertainty or in-
describeotherconceptswherepossible. completenessinformationconditions,theimprobableevents
In other words, while György Pólya3 or Arthur Dempster occurrence opens the space to emotions which significantly
- Glenn Shafer4 have tried to extract,in probabilistic terms demarcatethepassagefromstrictuncertaintymethods(prob-
thebelieffunctionfromthedescribablecomponent.Herewe ability) to the uncertain events treatment under emotional
acceptthe differenceamong theterms; wehierarchizethem effectsaction.
instrenghttermsandwedescribethemwithamethodthatis In this scenario the closer concept to probability is plausi-
moregeneralandnotprobabilityexclusive. bility, even if plausibility and credibility both concern an
Inourvisiontheconcepts: estimation theory under emotional coditioning. For what
concerns affectivity it tends to provide a subjective answer
probable(P ), that aims to being objective. So we assign a probability to
r
an event based on the several people opinion, in the belief
plausible(P ), case this comparison is not required, there is no subjective
l
evaluationobjectification.Beliefbringsoutthepersonaland
credible(C ), subjective evaluation to be shared, surely possible, but not
r
necessarilysharable.
possible(P ), In other words, for the belief it is sufficient that the propo-
o
sition to which a subjective probability is to be assigned is
haveadecreasingstrength,eveniftheyalldescribeuncertain possible.
events.Soa: From these considerations it emerges that as the concept of
probability is more stringent than plausibility one, in the
• possibleeventisnotnecessarilycredible;
first we assume that even if there is uncertainty, there is
• credibleeventissurelypossible;
noemotionallydrivenassessment;insteadintheplausibility
• credibleeventisnotnecessarilyplausible,butitissurely
definition, we associate an emotional component to uncer-
possible;
tainty,butitrepresentsthemostpeopleopinion.Inthebelief,
• plausible event is not necessarily probable, but it is
emotionalitymakebringstheegoout,thatisapersonaland
surelycredibleandpossible;
subjective vision that does not require comparison with the
• probableeventissurelyplausible,credibleandpossible.
community,butonlyneedsthepossibility.
Fromnowon,itwillbenecessarytoindicateacharacteristic
Wesummarizethisintheconceptualprovocation:
multiple function for a process or phenomenon that is a
"when the improbable is highly possible we should not be
combinationofmorefunctions(i.e.distribution):
surprisedifitisrealized".
Formulated as the same way as a Edward Murphy6 law it
F1probability
shouldbeascientifictruthtoall.
In other words the current price p(t) is a certain form of
F2plausibility
uniformity, while the trend p(t-1)-p(t) is a more or less
reliable,butnotsuresource.
F3credibility
Using the Dempster-Shafer’s (denoted as DS) theory rather
thanbayesianmodelallowstoexceedthemultidimensional-
F4possibility
itylimittosumone.
In other words, price growth possibility with a subjective
whereF aremultiscalesingle-modefunctionsto1sum5.
i probability estimated, does not imply that the decrescence
We conduct an analysis that starts from a different perspec-
hasthecomplementto1,soadecreasingtrendprobabilityis
tivewhichnaturallyemergesfromartificialintelligencestud-
equaltooneminusuptrendprobability.
ieswhenitisdesideredtoextenddecision-makingcapability
Thanks to the DS theory it is possible to give a probability
ofanartificialagentwhichoperatesinuncertainconditions,
toanEeventandanEnegation(¬E),thatcouldbeanother
with increasing cognitive capacities which try to tend to
probabilitydifferentfrom1-P(E)withP(E)+P(¬E)≤1.
humanones.
InDStheory,C (E)≤P (E)meansthatcredibleislessstrong
r l
than plausible, because the plausibility is the probabilistic
3Pólya’stheory
4Dempster-Shafer’stheorymanagesthedistinctionbetweenuncertainty maximum value that can be allocated to E proposition,
and ignorance and calculates a proposition evidence, in their work it is while belief is the probabilistic minimum assignable value
calculatedthebelieffunctionBel(X),thatistheprobabilitythataproposition to E, above all because for plausible we mean something
issupportedbyevidence.
believableformorepeople.
5WhenanEeventishighlyimprobablei.e.P(E)≤1%,P(E)≤0,1%,
P(E) ≤ 0,01% we are moving in the plausibility of improbable events.
Then it is not important how the event is improbable, but how much the
improbableeventisplausible.Issimilarforcredibilityandpossibility. 6Murphylawisapseudo-scientificparadoxessetwithanironiccharacter
realisedbyArthurBlochwithdidacticphrasesinstatistical-mathematical
form
6 VOLUME4,2016
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Another difference element between our theory and DS We suppose that on this dynamic system we want to make
theoryis:whiletheDStheoryprovidesamethodtochange a position and speed (or momentum) measurement. Make
previousopinionsbecauseofnewbeliefs,equallytreatingthe a measurement means to send one or more photons (i.e. a
newandpreviousevidence,ourdynamicsystemsexperience beam)inordertoknowineachinstantwheretheelectronis
leadsustogivemoreweighttothenewevidencethanold,to andtheitsspeed.
refiningthecognitiveprocessandagradualapproachtothe Feynman’s observation is that by making this measure we
solution. are changing the state of motion of the electron system; we
In financial markets context, this means: if we do a price are perturbing it, because, to provide information about the
average for assuming a future price, then an average that electronspeedandtheelectronposition,thephoton,bumping
weighs recent prices more than older ones will be more itwillchangeitsspeedanditsmotion.
appropriate than an average that equally weighs new prices So, the effect of the measure represents a perturbation. The
andolderones. more we are sure of the position, having sent more photons
tomeasuretheposition,thelesswecansayaboutthespeed
NicholasRescher’snon-probabilistictheoryofplausibility because doing the different measures we have perturbed
can be useful when there are equal reliability conflicting (if not distorted) the electron motion and vice versa. The
information(newsorindicators). Heisenberg principle says that in the combined measure of
Asanexample,thinkthatasetofnewsorindicatorstellsus position and momentum it is impossible to be more precise
thatthepriceofafinancialinstrumentisdestinedtoriseina than a certain threshold value given by the reduced Planck
timeframeandtofallinanotherone. constant9.
Thanks to the Rescher’s theory, which is not sum 1, we can In the macroscopic scale reality, we understand that what
associate a value of bullish belief, for example 0.9, and a explained by Quantum Mechanics does not respond to our
bearerofthesamevalue,0.9too,sincewehavenotbounded commonsense.Letthemotionofacar,andweconsiderthe
it by sum one total like it happens to sum one total theories measurement process link to a beam of photons that collide
astheprobabilityone. with the vehicle to know the position and the speed. It is
Consequently,theresultisthatthepricewillnotcertainlybe evident that the photons can not disturb the vehicle motion
stationary,soafinancialinstrumentwithalateralbehavioris state.
tobeexcluded. In this case the observer or measurer is unrelated to the
vehiclemotionandanyuncertaintyisgenerated.
C. INFERENCEMETHODSFORFINANCIAL ForthisreasonthemacroscopicrealityisdescribedinClas-
INSTRUMENTSDOMINATEDBYUNCERTAINTY: sicalMechanics,startingfromtheIsaacNewton’sprinciples
INTERACTIONBETWEENFINANCIALOPERATORAND ofdynamics(inertiaprinciple,actionandreactionprinciple).
MARKET Manymacroscopicdynamicalsystemshaveasimilarbehav-
iortoquantumsystems.
When a financial operator joins the market, he performs an
Considering the price dynamics of a financial instrument, it
analysis (or someone does it for him) in order to decide
isclearthatwhatFeynmanexplainedisincompleteanalogy.
whethertoenteronafinancialinstrument,andwhatkindof
If we replace the electron position with the price level of a
positiontotake,intermsofsizeandofupwardordownward
financial instrument, we formulate an uncertainty financial
directions.
markets principle similar to Heisenberg’s principle. In this
Thefinancialoperatordoesnotremainanobserver,interacts
wayitisimpossibletocertaintypredictthepriceandvolatil-
withthefinancialmarketanddisturbsit.
ityofafinancialinstrumentinafuturemoment.
Inotherwords,theanalysisbecomesaninteractionwiththe
This because of the decision taken by the financial operator
market,oncetheoperatordecidestoenter.
This reasoning recalls that of Richard Feynman7, where he and therefore the entry to the market automatically could
imply the market transition from one state to another one.
explainsthequantumsystemsmeasurementprocessandthe
consequentWernerKarlHeisenberg’suncertaintyprinciple8. A state could be different from the previous one depending
ontheoperatorsize,(hisliquidityintroducedtothemarket)
About this Feynman explains that the fluctuations or uncer-
givingadirectiontoit.
tainties that introduce a description in terms of expectation
Think of the effects caused by market mover (such as the
and probability values to the state of a dynamic system,
effects of the decisions carried out by FED, ECB, BOE etc
proposes the following example. Suppose that the dynamic
...).
systemunderstudyisanelectron,oratleastaparticleofthe
What has been described explains the need to place the
microscopic world described through Quantum Mechanics.
financial markets in the complexity theory and to describe
7Feynman’sreasoningsaysthatonlyaquantumsystem,wherethereis them with increasingly accurate tools which can look at
uncertaintyorevenworseambiguity,cansimulatethebehaviorofanother
quantumsystem. 9Planck’s constant, also called action quantum and denoted by h, is a
8The Heisenberg principle establishes the limits in the knowledge of physicalconstantthatrepresentstheelementaryactionquantity,determining
valuesthatphysicalquantitiesassociatedwithoperatorswhodonotinteract that fundamental physical quantities do not evolve continuously, but are
witheachothertakeonaphysicalsystematthesametime. quantized,thatis,theycanonlyassumevaluesmultiplesofthisconstant.
VOLUME4,2016 7
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uncertaintyfromdifferentperspectivesandnotonlyfromthe this model, as in the previous one, all the Pi have the same
probabilistic,plausibilistic,credibilisticorpossibility. importance.Letusmakeanotherweatherforecastexample:
Becauseallthesevisionsinaninstant,indifferentquantities,
• P1=militaryairforcerainforecasts;
contributetotheformationoftheprice.
• P2=sentimentforthewind;
We intend to provide a further enrichment level for the
• P3=experts’forecastforthetemperatures;
MRQF theory [20] as a complementary work. For this aim,
• P4=users’forecastsforthehumidity.
weworkonthemodelingofuncertainty,andonhowthevar-
Thismodelcanbeusedforcalculatetheexpectationfunction
iousprobable,plausible,credible,possiblemodelscontribute
fortheevent"itwillrain,thewindwillblowat31km/h,there
totheconstructionofamarketstatefunctionwhichabsorbs
willbe18C°andthehumiditywillbe80%".Itisimportant
theindividualprobability,plausibility,credibility,possibility
togiveadifferentimportancetotheProbability,Plausibility,
functions.Afterthatharmonizesthemtoallowtheanalystto
CredibilityandPossibility;so,wedefineweightsforeachof
make inferences about future market conditions and to help
theseconcepts.
him to make decisions about entering the market, in what
dimensions,andwhethertobullorbear.
3) Thirdmodel:Weightedaveragemethod
ToextendthepreviousmodelsweassumethatnotallP have
i
D. UNCERTAINTYTREATMENTBYINFORMATION thesameimportanceintheexpectationfunctionconstruction.
FUSIONANDEXPECTATIONFUNCTIONMODELING So,webuildthefollowingweightedaveragetypemodel,
WtoafienPedd,ePfifrno,emCtha,nePae,pxsppurecochptartihtaiaotetnafuf(un∅nc)ct=itoion0nalaanc:dom2(cid:80)Ap2os→itiao(nA[0i,n)1=t]ero1mb.s- a3 = ((cid:80)(cid:80)4i=4i=11ααiPii),
r l r o i=1 i whereαiareweightsrelativetothedifferentPi.
Anexamplecouldbetogivea50%ofweighttoP thatis:
1
1) Firstmodel:Averagemodel α = 0,5; 25% to P , so α = 0,25; 15% to P , so α =
1 2 2 3 3
We indicate with P1 = Pr, P2 = Pl, P3 = Cr, P4 = Po. 0,15,and10%toP ,α =0,1;
4 4
Theexpectationfunctionsimplestmodelwecanbuildisthe inthiswaytheexpectationfunctionis
followingaveragemodel
a1 =a1(P1,P2,P3,P4)= 41(cid:80)4i=1Pi. a3 = (cid:80)(cid:80)4i=4i=11ααiPii =0,5P1+0,25P2+0,15P3+0,1P4.
Evidentlythismodelassignsthesameimportancetothedif-
4) Fourthmodel:Weightedproductmodel
ferentdistributionsoftheprobability,plausibility,credibility
In analogy, to extend the product model for the expectation
andpossibility,soitisbothanadvantageandadisadvantage
function,weconsiderthefollowingweigthedproductmodel
dependingonhowmuchexperttheanalystisandhowmuch
wthee pwroanbtabtiolitymathkeeoreyxpiserutsethdefosrycstaelmcu.laTtehimsomreoadletle,rnaastivine a4 = (cid:81)(cid:80)4i=4i=11ααiPii.
eventsexpectation.Letusmakeaweatherforecastexample: Also in this way, we will have the possibility to weigh the
different contribution of the probability, plausibility, credi-
• P1=placeAforecastsofthemilitaryairforce;
bilityandpossibilitytotheexpectationfunction.Inboththe
• P2=placeBforecastsbythesentiment;
weightedmodelswehavetheconstraintof
• P3=experts’forecastforplaceC;
• P4=users’forecastsforplaceD; (cid:88)4
a =1.
Thismodelcanbeusedforcalculatetheexpectationfunction i
fortheevent"itwillrainintheplaceAorintheplaceBor i=1
intheplaceCorintheplaceD". Thisisbecausewhenweuseanexpectationfunctionmodel,
we are trying to rebuild a probability which had uncertain
2) Secondmodel:Productmodel inputs.
Another way of estimating the event expectation is through
thePiproduct,formally 5) Overlapmodelwithshiftbasedonprobability
a =a (P ,P ,P ,P )=(cid:81)4 P Firstandthirdmodelsrepresentoverlapmodelsandweighted
2 1 1 2 3 4 i=1 i overlap of the different P contributions to the expectation
i
whereP :2A →[0,1]. function.
i
Ingeneral,weobservethat A further possible generalization could include a P hierar-
i
(cid:81)4 P ≤(cid:80)4 P . chicaluse.
i=1 i i=1 i Let P be the probability of the event, In detail we can
r
Tobeabletosaythata ≤ a ,itisrequiredamoredetailed see probability, plausibility, credibility and possibility as
2 1
studythatincludestheP functionalmodels,consideringthat conceptstobeusedindescendingorderofimportance.
i
the function a has the normalization factor equal to 1. In Forexamplewecanconsider
1 4
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mass (m) function, the belief(bel) function and the Demp-
 ster’splausibility(wecallitDpl)foreachP:
P if1%<P ≤100%
P1 if0,1%<rP ≤1% • m(P1),relativebel(P1)andDpl(P1);
a5 = 2 r • m(P2),relativebel(P2)andDpl(P2);
PP34 iiff0P,r0≤1%0,<01P%r.≤0,1% •• mm((PP34)),,rreellaattiivveebbeell((PP34))aannddDDppll((PP34));;
Thiswouldmeanthatuntil1%oftheprobabilitydistribu- And finally compose it with the Dempster’s composition
tion we will use only the classical probability(P1); beyond rules.
that probability we will define the improbable event, with
a plausibility (P2) coefficient between 0% and 100% in the E. RELIABILITY
intervalinwhich1%≤P ≤0,1%;similarly,however,0.1% Inclassicalreasoning,themethod,theelementorthesystem
r
< P ≤ 0.01% we define the improbable and implausible reliability,istheprobabilityofnotbreakingforacertainpe-
r
event, with credibility (P3) between 0% and 100%; finally, riodt,underdefinedoperatingandenvironmentalconditions;
for even lower values of probability we will have unlikely i.e.reliabilitydefinitionisinfrequentisticterms:
andimplausibleevents,withacoefficientofpossibility(P4)
R= nf,
between0%and100%. n
At the ends of these evaluations there is the certain event withnf numberoftimesthatthemethod,elementorsystem
P1 =100%andtheimpossibleoneP1 =0%: hasworked,andntestsnumberandsamplesusedforthetest.
By using our approach, reliability is obtained by changing
P :2A →[0,1].
1 the probability P with the expectation function a. Similarly
wecandefineunreliabilityasFault
6) OverlapmodelwithshiftbasedonPihierarchical
F=1-R.
As an alternative to the previous model, without making to
the probability play a pivotal role, but generalizing the Pi Soweareina1-dimensionalone-sumlogicandthereforeR
andevaluating,withownimportance,alsotheemotionaland andFaretimefunctions.
cognitiveelements,leadstotheformulationofanexpectation Thereliabilityislinkedtothenumberofsuccessesorfailures
functionwhichcanbedefinedas obtainedafterattime.
FormallythereliabilityRisthefollowingtimefunction:
 R:T →[0,1].
P ifP −3σ ≤p ≤P +3σ
P12 iafnPdPPp12t−>3PσPPP121 +≤3pσttP≤1 PPP12 +3σPP12 Wmqueeannatt,listsyoysdteefimn,eoarvmereaagne toimpeerattoingfaitliumree o(mfttfh)ethmeetfhoolldo,weilneg-
a = P ifP −3σ ≤p ≤P +3σ
6 3 P3 P3 t P3 P3
P4 iaafnnPddPpp4t−>>3PPσPP42 ++≤33pσσtP≤2.PP4 +3σP4 wInitohuλrsntuudmybwereocfofnasuidltesrm/edftaFRilu==reRs1/a(Tλt)t,;tiimnge.eneralR=R(T,C,
t P3 P3
A)wheretisthetime,Cisthemethodforjudgingsuccesses
Where P is the average price on the considered set of data andfailuresandAareenvironmentalconditions.
and σ is the standard deviation; the subscript Pi indicates Thisreliabilityconceptcanbeappliedasanestimatortothe
withwhichdistributionP andσarecalculated. probability,plausibility,credibilityandpossibilityfunctions,
From the provided models, we understand how general the inwhichbacktestingcanbeseenhowthedifferentinferences
issueofdecisionsinuncertaintyconditionsisandhoweasy worked.
it is to build models using the expectation function which Whenwedonotrefertothemethodbuttotheresult,weneed
describe the inference processes in info-incompleteness and tointroducethetrustworthinessconcept.
inputuncertaintyterms.
The proposed reasoning theory is a meta model where
F. TRUSTWORTHINESS
defining expectation function models makes it possible to
In some contexts, simple reliability and trustworthiness, as
constructnewandmoreopportuneoperativemodelscontex-
well as probability, plausibility, credibility and possibility,
tualisedtoacognitiveandinformativedomain.
can be considered synonyms, but this is not the case of
domainsandeventswherecomplexitytheoryisrequired.
7) ModelbasedonDempster’scompositionrules Aftersettingthemeasurementconditions,thetrustworthiness
Anothersignificantwaytoconstructtheexpectationfunction indicatesthecostancyofasetofresults.
istousetherulethatDempsterhasdefinedfortheplausibil- With this definition we understand how in an uncertainty
ity, extending it to the application of Pi case calculating the context,intermsofprobabilitiesanddistributionswhichare
VOLUME4,2016 9
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Fingerprint Face Iris Retina
dependent on emotion (such as plausibility and credibility),
Accuracy 90 76 93 94
thereliabilitycanpreciselyexpresshowthesamedynamics
aredifferentlyconsidered. Vascul. Ear Voice Signature
For example, a trader with the same trend at different times Accuracy 96 74 66 67
mayfeeltheneedtoact/reactdifferently. Hand Palm BodyMot. DNA
At this point the result trustworthiness acquires a different, Accuracy 67 88 78 98
butequallyimportantrolefromthepreviousconcepts.
Trustworthiness is not reliability, accuracy or validity of a TABLE1:Biometricsaccuracy
data synonymous; but we will say that an estimate is trust-
Fingerprint Iris Face
worthyiftheresultsremainconstantovertime.
Accuracy 90 93 76
Ifthisdoesnothappen,wewillsaythattheestimationorthe
resultisuntrustworthiness. TABLE2:Selectedbiometrics
There are several methods to estimate trustworthiness; we
will use the Pearson10 correlation index which is the basis
ofthevariousreliabilityestimates. • probability:thebiometricsstatisticalreliabilitythatisin
ThePearsoncorrelationindexbetweentwovariablesxandy the[0,1]interval,
is • plausibility:thebiometricssentimentthatisinthe[0,1]
r = σxy with -1≤r<1 interval,
σxσy • credibility:theawarenessandreliabilityoftherecogni-
whereσ xydenotethecovarianceandσ xσ ythestandard tionalgorithmthatisintheinterval[0,1],
deviationsofxandyrespectively. • possibility:thesystemfaultstatistic.
Wedistinguishthefollowingcases:
InSportodds,allthefeaturesareinthe[0,1]interval:
r <0:relatedvariables
• probability:thebookmakerquotesaverage,
r =0:nonrelatedvariables
• plausibility:theexpertsopinions,
r >0:anticorrelatedvariables
• credibility:thesentimentanalysis,
0<r <0.3:weaklycorrelatedvariables
• possibility:theusersforecasts.
0.3<r <0.7:moderatelycorrelatedvariables
Thesimulationhasbeendevelopedwithrstudio;wecreated
r >0.7:highlycorrelatedvariables
thetwosimulateddatasets11 andtheyhavebeendisturbedin
andsimilarlyforanti-correlationcases.
ordertosimulateuncertainty.
Generallyinthecaseofmorethantwovariables,thePearson
indexwillbecome
A. BIOMETRICSSIMULATION
r =r .
ij Whenwefusebiometricsinamultibiometricsystem,weare
We have a correlation matrix of r rows and r symmetric dealing with a decision-making problem. The decisions can
column r = r with coefficients on the diagonal equal to be joined by several methods: voting method, where, each
ij ji
1,asr =σ /σ2. classifier votes for a class, then the pattern will be assigned
ii ii i
Forexample,inthefinancialcasewecanestimatetherelia- to the most voted class, we focused on this method; Gupta,
bilityoftwoormoreperiodsofresults. Singh Walia and Sharma [22] propose a score based fusion
method;Divyakantetal.[21]describeotherfusionmethods.
Weinitializeeachbiometricwiththerelatedaccuracy:
IV. SIMULATION
In this section we will describe a simulation based on the Delacetal.[23]andTripathi[24]calculatedthebiometrics
describedtheory.Thissimulationhasbeendevelopedintwo reliability;wehaveextractedthebiometricsaccuraciesfrom
fields: these works (table 1). Suppose to build a multibiometrics
systemthatusethemostplausiblebiometricsintermsofac-
• Biometricfusionchoiches,
curacyanduser-acceptability(useracceptabilityisdescribed
• Sportodds.
byTripathi[24]andRui[25]).Weselect:
For each field, we suppose to have different features to be
appliedatprobability,plausibility,credibilityandpossibility. • Fingerprint,
Ouraimistocalculatehowmuchthesefeaturesweightinthe • Iris,
formula: • Face.
Inthetable2,wecanseetheselectedbiometricsaccuracy.
a3 = (cid:80)(cid:80)4i=4i=11ααiPii.
After that, we initialized three algorithms for each bio-
Inparticularinthebiometricsfieldwehave:
metricsandrelativesuccessrate:randomhigh,mediumand
10ThePearsoncorrelationindexbetweentwostatisticalvariablesisde- lowerrate.Thesuccessratecanbefoundintable3.
finedastheircovariancedividedbythestandarddeviationsproductofthe
twovariablesandexpressesapossiblelinearityrelationshipbetweenthem. 11onefield,onedataset
10 VOLUME4,2016
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