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1791:. The open-world assumption is that the coin can be stolen in mid-air, disappear, break apart or otherwise fall sideways so that neither Head nor Tail occurs, so that the power set of {Head,Tail} is considered and there is a decomposition of the overall probability (i.e. 1) of the following form:
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Haenni, R. (2006). "Uncover
Dempster's Rule Where It Is Hidden" in: Proceedings of the 9th International Conference on Information Fusion (FUSION 2006), Florence, Italy, 2006.
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Haenni, R. (2006). "Uncover
Dempster's Rule Where It Is Hidden" in: Proceedings of the 9th International Conference on Information Fusion (FUSION 2006), Florence, Italy, 2006.
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to stress the fact that those probability functions are based on incomplete data, whose only purpose is a forced decision, e.g. to place a bet. This is in contrast to the
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that relaxes the assumption that all possible outcomes are known. Under the open world assumption
Dempster's rule of combination is adapted such that there is no
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Smets Ph. (1993) "An axiomatic justification for the use of belief function to quantify beliefs", IJCAI'93 (Inter. Joint Conf. on AI), Chambery, 598–603
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2201:"Forward-Backward-Viterbi procedures in the Transferable Belief Model for state sequence analysis using belief functions"
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Bayesian updating the first opinion with the second (or the other way round) implies certainty that the cause is
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181:. Doctor 1 says that the patient's illness is very likely to be caused by A (very likely, meaning probability
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is the set of unknown outcomes. Consequently, the combination rule underlying the TBM corresponds to
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Zadeh, A., L., (1984) "Review of shafer's a mathematical theory of evidence". AI Magazine, 5(3).
1873:{\displaystyle \Pr(\emptyset )+\Pr({\text{Head}})+\Pr({\text{Tail}})+\Pr({\text{Head,Tail}})=1.}
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Shafer, Glenn (1976). "A Mathematical Theory of
Evidence", Princeton University Press,
951:{\displaystyle m_{1,2}(A)=(m_{1}\otimes m_{2})(A)=\sum _{B\cap C=A}m_{1}(B)m_{2}(C)\,\!}
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is true from other propositions that are assigned probabilities. It was developed by
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holds true. The underlying idea is that the frame of discernment is not necessarily
130:. This adaptation violates the probabilistic character of the original DST and also
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Ph, Smets (1990). "The combination of evidence in the transferable belief model".
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Ph, Smets (1990). "The combination of evidence in the transferable belief model".
1346:{\displaystyle P_{\text{Bet}}(x)=\sum _{x\in A\subseteq X}{\frac {m(A)}{|A|}}\,\!}
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problem. A patient has an illness that can be caused by three different factors
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are equally distributed among the atoms of A. This strategy corresponds to the
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When tossing a coin one usually assumes that Head or Tail will occur, so that
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is not likely. (For this reason the standard
Bayesian approach is to adopt
2123:, ed. Smets Ph., Mamdani A, Dubois D. and Prade H. Academic Press, London
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Uncertainty
Management in Information Systems: from Needs to solutions
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provides insufficient context for those unfamiliar with the subject
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Smets, Ph.; Kennes, R. (1994). "The transferable belief model".
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Smets, Ph.; Kennes, R. (1994). "The transferable belief model".
2213:"Application of transferable belief model to navigation system"
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IEEE Transactions on
Pattern Analysis and Machine Intelligence
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IEEE Transactions on
Pattern Analysis and Machine Intelligence
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224:, since although the two doctors point at different causes,
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described above, whose purpose is representing the actual
1115:{\displaystyle \operatorname {bel} :2^{X}\rightarrow \,\!}
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209:= 0.05). How is one to make one's own opinion from this?
102:. In contrast to the original DST the TBM propagates the
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The transferable belief model for belief representation
1784:{\displaystyle \Pr({\text{Head}})+\Pr({\text{Tail}})=1}
1231:{\displaystyle \operatorname {bel} (\emptyset )=0.\,\!}
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Model for reasoning with uncertain beliefs and evidence
134:. Therefore, the authors substituted notation such as
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2238:"The Dempster–Shafer calculus for statisticians"
2017:"The Dempster–Shafer calculus for statisticians"
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1704:{\displaystyle P_{\text{Bet}}(\emptyset )=0\,\!}
1591:{\displaystyle P_{\text{Bet}}:X\rightarrow \,\!}
540:is not required to be zero, and hence generally
240:and avoid the use of 0 or 1 as probabilities.)
2242:International Journal of Approximate Reasoning
2022:International Journal of Approximate Reasoning
2211:Touil, K.; Zribi, M.; Benjelloun, M. (2007).
2199:Ramasso, E., Rombaut, M., Pellerin D. (2007)
585:, and thus belief allocated to a proposition
220:lead to the same result. This can be seen as
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2121:Non Standard Logics for Automated Reasoning
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2119:Smets Ph. (1988) "Belief function". In:
1908:This article includes a list of general
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2217:Integrated Computer-Aided Engineering
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189:is also possible but not likely (
2189:" In: Smets and Motro A. (eds.)
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1538:. Such a function satisfies the
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2271:The Transferable Belief Model
2175:10.1016/0004-3702(94)90026-4
2078:10.1016/0004-3702(94)90026-4
1491:principle of maximum entropy
1456:. Hence, probability masses
1511:are described by functions
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2285:Software for TBM in Matlab
2255:10.1016/j.ijar.2007.03.004
2036:10.1016/j.ijar.2007.03.004
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2236:Dempster, A.P. (2007).
2163:Artificial Intelligence
2066:Artificial Intelligence
2015:Dempster, A.P. (2007).
1929:more precise citations.
1252:pignistic probabilities
800:{\displaystyle m_{1,2}}
232:, they both agree that
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2229:10.3233/ICA-2007-14108
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1021:
1020:
1007:
1004:
1001:
996:
992:
988:
985:
982:
979:
976:
973:
959:
958:
945:
942:
939:
934:
930:
926:
923:
920:
915:
911:
905:
902:
899:
896:
893:
889:
885:
882:
879:
876:
873:
868:
864:
860:
855:
851:
847:
844:
841:
838:
835:
830:
827:
824:
820:
794:
791:
788:
784:
761:
757:
734:
730:
709:
706:
703:
700:
697:
694:
669:
647:
643:
638:
634:
630:
627:
605:
601:
597:
594:
570:
567:
564:
561:
558:
555:
552:
549:
529:
506:
486:
461:
457:
442:
441:
428:
425:
422:
419:
416:
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406:
402:
398:
395:
391:
376:
375:
362:
359:
356:
353:
350:
347:
342:
338:
334:
331:
308:
293:
290:
289:
288:
269:
245:
242:
159:
156:
92:Philippe Smets
70:
69:
31:
29:
22:
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13:
10:
9:
6:
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3:
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2309:
2307:
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2287:
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2282:
2280:
2279:
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2268:
2267:
2263:
2256:
2251:
2247:
2243:
2239:
2234:
2230:
2226:
2223:(1): 93–105.
2222:
2218:
2214:
2209:
2206:
2202:
2198:
2195:
2192:
2188:
2184:
2180:
2176:
2172:
2168:
2164:
2159:
2156:
2152:
2148:
2143:
2138:
2134:
2130:
2125:
2122:
2118:
2117:
2113:
2106:
2105:0-608-02508-9
2102:
2096:
2093:
2087:
2084:
2079:
2075:
2071:
2067:
2060:
2058:
2056:
2052:
2046:
2043:
2037:
2032:
2028:
2024:
2023:
2018:
2011:
2008:
2003:
1999:
1994:
1989:
1985:
1981:
1974:
1972:
1970:
1968:
1964:
1957:
1949:
1946:
1938:
1928:
1924:
1918:
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1911:
1906:
1897:
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1891:
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1883:
1867:
1864:
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1830:
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1775:
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1693:
1675:
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1666:
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1612:
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1577:
1574:
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1492:
1488:
1469:
1463:
1443:
1423:
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1385:
1369:
1366:
1363:
1330:
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1181:
1175:
1170:
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1164:
1161:
1154:
1150:
1144:
1138:
1135:
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1127:
1126:
1104:
1101:
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1087:
1083:
1079:
1076:
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1067:
1050:
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980:
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963:
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909:
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900:
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728:
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625:
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253:
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243:
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200:
196:
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188:
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184:
180:
176:
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168:
165:describes an
164:
157:
155:
153:
149:
145:
141:
137:
133:
129:
125:
121:
117:
113:
109:
108:normalization
105:
101:
97:
93:
89:
86:that a given
85:
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66:
63:
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35:
32:This article
30:
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20:
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2277:
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2245:
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2220:
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2190:
2166:
2162:
2132:
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2095:
2086:
2069:
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2045:
2026:
2020:
2010:
1983:
1979:
1941:
1932:
1913:
1736:
1726:
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1718:
1715:Philip Smets
1713:
1600:
1508:
1506:
1498:distribution
1494:
1355:
1247:
1245:
1124:
1024:
1022:
960:
443:
377:
295:
292:Credal level
276:
272:
261:
258:credal level
257:
247:
233:
229:
225:
213:
211:
206:
202:
198:
194:
190:
186:
182:
178:
174:
170:
161:
151:
147:
143:
139:
135:
126:outside the
75:
73:
58:
49:
38:Please help
33:
1927:introducing
1507:In the TBM
222:paradoxical
163:Lotfi Zadeh
88:proposition
84:probability
2295:Categories
2114:References
1910:references
1384:singletons
583:exhaustive
444:where the
124:hypothesis
2137:CiteSeerX
1988:CiteSeerX
1935:June 2010
1858:Head,Tail
1808:∅
1719:pignistic
1688:∅
1620:∈
1613:∑
1569:→
1367:∈
1301:⊆
1295:∈
1288:∑
1215:∅
1209:
1168:⊆
1162:≠
1159:∅
1155:∑
1139:
1093:→
1054:∅
1051:≠
1038:∈
1003:∅
1000:≠
987:∈
895:∩
888:∑
859:⊗
699:∅
642:∪
629:∈
596:∈
566:≤
560:∅
551:≤
528:∅
519:empty set
446:power set
397:∈
390:∑
346:→
281:decisions
116:empty set
2185:(1995) "
2183:Kruse R.
1884:See also
148:transfer
98:against
52:May 2016
1923:improve
1495:unknown
277:beliefs
262:beliefs
250:beliefs
2139:
2103:
1990:
1912:, but
1727:belief
1386:) and
1356:where
961:where
659:where
275:where
260:where
120:belief
2301:Logic
1958:Notes
1601:with
1125:with
378:with
122:in a
2101:ISBN
1841:Tail
1824:Head
1769:Tail
1752:Head
1254:by:
807:by:
747:and
228:and
146:and
138:and
74:The
2250:doi
2225:doi
2171:doi
2147:doi
2074:doi
2031:doi
1998:doi
1680:Bet
1633:Bet
1558:Bet
1524:Bet
1270:Bet
1206:bel
1136:bel
1077:bel
569:1.0
177:or
42:by
2297::
2246:48
2244:.
2240:.
2221:14
2219:.
2215:.
2203:,
2167:66
2165:.
2145:.
2133:12
2131:.
2070:66
2068:.
2054:^
2027:48
2025:.
2019:.
1996:.
1984:12
1982:.
1966:^
1868:1.
1851:Pr
1834:Pr
1817:Pr
1802:Pr
1762:Pr
1745:Pr
1729:.
1542::
1504:.
1224:0.
271:a
256:a
216:.
173:,
154:.
2258:.
2252::
2231:.
2227::
2207:.
2177:.
2173::
2153:.
2149::
2080:.
2076::
2039:.
2033::
2004:.
2000::
1948:)
1942:(
1937:)
1933:(
1919:.
1865:=
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1854:(
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1837:(
1831:+
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1820:(
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1805:(
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1765:(
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1748:(
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1685:(
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1520:P
1473:)
1470:A
1467:(
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1370:X
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1331:A
1327:|
1321:)
1318:A
1315:(
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1292:x
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1278:x
1275:(
1266:P
1221:=
1218:)
1212:(
1185:)
1182:A
1179:(
1176:m
1171:H
1165:A
1151:=
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1145:H
1142:(
1108:]
1105:1
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1099:0
1096:[
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1084:2
1080::
1046:X
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981:,
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892:B
884:=
881:)
878:A
875:(
872:)
867:2
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846:(
843:=
840:)
837:A
834:(
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823:1
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793:2
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760:2
756:m
733:1
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708:0
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702:)
696:(
693:m
668:e
646:e
637:X
633:2
626:A
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600:2
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563:)
557:(
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460:X
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418:A
415:(
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361:]
358:1
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349:[
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333::
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307:m
287:.
268:,
234:B
230:C
226:A
214:B
207:p
203:B
199:p
197:(
195:C
191:p
187:B
183:p
179:C
175:B
171:A
65:)
59:(
54:)
50:(
46:.
36:.
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