The transferable belief model (TBM) is an elaboration on the Dempster–Shafer theory (DST), which is a mathematical model used to evaluate the probability that a given proposition is true from other propositions that are assigned probabilities. It was developed by Philippe Smets who proposed his approach as a response to Zadeh’s example against Dempster's rule of combination. In contrast to the original DST the TBM propagates the open-world assumption 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 normalization. The underlying idea is that the probability mass pertaining to the empty set is taken to indicate an unexpected outcome, e.g. the belief in a hypothesis outside the frame of discernment. This adaptation violates the probabilistic character of the original DST and also Bayesian inference. Therefore, the authors substituted notation such as probability masses and probability update with terms such as degrees of belief and transfer giving rise to the name of the method: The transferable belief model.
Zadeh’s example in TBM context Lotfi Zadeh describes an information fusion problem. A patient has an illness that can be caused by three different factors A, B or C. Doctor 1 says that the patient's illness is very likely to be caused by A (very likely, meaning probability p = 0.95), but B is also possible but not likely (p = 0.05). Doctor 2 says that the cause is very likely C (p = 0.95), but B is also possible but not likely (p = 0.05). How is one to make one's own opinion from this? Bayesian updating the first opinion with the second (or the other way round) implies certainty that the cause is B. Dempster's rule of combination lead to the same result. This can be seen as paradoxical, since although the two doctors point at different causes, A and C, they both agree that B is not likely. (For this reason the standard Bayesian approach is to adopt Cromwell's rule and avoid the use of 0 or 1 as probabilities.)
Formal definition The TBM describes beliefs at two levels:
a credal level where beliefs are entertained and quantified by belief functions, a pignistic level where beliefs can be used to make decisions and are quantified by probability functions.
Credal level According to the DST, a probability mass function m {\displaystyle m} is defined such that:
m : 2 X → [ 0 , 1 ] {\displaystyle m:2^{X}\rightarrow [0,1]\,\!}
with
∑ A ∈ 2 X m ( A ) = 1 {\displaystyle \sum _{A\in 2^{X}}m(A)=1\,\!}
where the power set 2 X {\displaystyle 2^{X}} contains all possible subsets of the frame of discernment X {\displaystyle X} . In contrast to the DST the mass m {\displaystyle m} allocated to the empty set ∅ {\displaystyle \emptyset } is not required to be zero, and hence generally 0 ≤ m ( ∅ ) ≤ 1.0 {\displaystyle 0\leq m(\emptyset )\leq 1.0} holds true. The underlying idea is that the frame of discernment is not necessarily exhaustive, and thus belief allocated to a proposition A ∈ 2 X {\displaystyle A\in 2^{X}} , is in fact allocated to A ∈ 2 X ∪ e {\displaystyle A\in 2^{X}\cup {e}} where e {\displaystyle {e}} is the set of unknown outcomes. Consequently, the combination rule underlying the TBM corresponds to Dempster's rule of combination, except the normalization that grants m ( ∅ ) = 0 {\displaystyle m(\emptyset )=0} . Hence, in the TBM any two independent functions m 1 {\displaystyle m_{1}} and m 2 {\displaystyle m_{2}} are combined to a single function m 1 , 2 {\displaystyle m_{1,2}} by:
m 1 , 2 ( A ) = ( m 1 ⊗ m 2 ) ( A ) = ∑ B ∩ C = A m 1 ( B ) m 2 ( C ) {\displaystyle m_{1,2}(A)=(m_{1}\otimes m_{2})(A)=\sum _{B\cap C=A}m_{1}(B)m_{2}(C)\,\!}
where
A , B , C ∈ 2 X ≠ ∅ . {\displaystyle A,B,C\in 2^{X}\neq \emptyset .\,\!}
In the TBM the degree of belief in a hypothesis H ∈ 2 X ≠ ∅ {\displaystyle H\in 2^{X}\neq \emptyset } is defined by a function:
… excerpt ends here. Continue reading the full article.
