The transferable belief model (TBM) is an elaboration on the Dempster–Shafer theory (DST) of evidence 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 tiny adaptation encounters the probabilistic character of the original DST and also Bayesian inference. Therefore, the corresponding authors avoided probabilistic notations, e.g. probability masses or update in the sense of probabilistic calculus, and replaced them by technical terms such as degrees of beliefs and transfer. These technical terms deploy the name of the method: The transferable belief model.
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Zadeh’s example in TBM context
Lofti Zadeh describes an information fusion problem. A patient has an illness that can be caused by three different factors A, B and 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
with
where the power set
where
In the TBM the degree of belief in a hypothesis
with
Pignistic level
When a decision must be made the credal beliefs are transferred to pignistic probabilities by:
where
In the TBM pignistic probability functions are described by functions
with
Philip Smets introduced them as pignistic 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 credal beliefs described above, whose purpose is representing the actual belief.
Open world example
When tossing a coin one usually assumes that Head or Tail will occur, so that