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Composition Operator for Credal SetsReconsidered
Jiiina Vejnarovri
Abstract This paper is the second altempr to introduce the composition operarotalready known from probability, possibiliiy, *la"r"" ""i'r"fuarion_based sysremstheories' also for credal sers. we try to avoid the discon,inurl *r,i"r, *u, present in theoriginar definition, bur simulhneously to keep all the f."p"ii., ""uurirg us to designcompositional models in a way analogous tothose rn thJ auove-mendoned theories.These compositionar models are aimid to be un arte-atiu" to Graphical MarkovModels. Theoretical resulrs achieved ln *ris paper are iiirrr.,"o by an example.I IntroductionIn the second half of 1990s amensionar probab, iry aistrru;,l"x lxx."iff :",T":f;1"J,;:ffi ililHl#i::.;Graphical. Markov Modeling. This approach i, Uur"a on u Ji.ple idea: a multidi_mensional distribution is conryosed fiom a system of row-dimensional distributionsby repetitive apprication of a speciar composition op".u,of*i,"r, is also the reasonwhy such models are called compositional models.Later, these compositional models were introduced also in possibility theory [7,8l (here the models are parameterized by a continuou;r_;;; and almosr ten yearsago also in evidence rheory [3, 4]. In ali these frameworkitre o.iginat idea is kept,but rhere exist some slight differences among these frameworks." f1 [9J we inroauced a composition_o-p..atJr fo. .."dJr"tJ, ir, au. to ,t e problemofdiscontinuity ir needed a revision. After a thorougr, .."""r"ia"".u,io, we decided topresent a new proposal avoiding this discontinuity. The goal of rhis paper is to showthat the revised composition operator.keeps th" nuri" p.o!"ni", ofirs counterparts inother frameworks, and therefore it will enable us ,o ln,.oir"" *.positional modelsfor m ultidimensional credal sers.
J. Vejnarov6 (E)Institute oflnformation Theorv and Automation of the Czech Academy of Sciences,Pod voddrcnskou vEil4, prague, czecrr nepuuiic -'- .--wu rrlaeeu'|J (e-mail: vejnar@utia.cas.cz@ Springer International publish.ing Swirzerland 2017M.B. Ferraro er al. leds.). Sofr Methods for Data Science, Advancesin Inrerligent svstems and ctmpuring 450, oor ro.roouisi8 -i'-iis-qzslz-q sn
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