Transferable belief model


The transferable belief model is an elaboration on the Dempster–Shafer theory 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 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

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, but B is also possible but not likely. Doctor 2 says that the cause is very likely C, but B is also possible but not likely. How is one to make one's own opinion from this?
Bayesian updating the first opinion with the second 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.

Formal definition

The TBM describes beliefs at two levels:
  1. a credal level where beliefs are entertained and quantified by belief functions,
  2. 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 is defined such that:
with
where the power set contains all possible subsets of the frame of discernment. In contrast to the DST the mass allocated to the empty set is not required to be zero, and hence generally holds true. The underlying idea is that the frame of discernment is not necessarily exhaustive, and thus belief allocated to a proposition, is in fact allocated to where 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. Hence, in the TBM any two independent functions and are combined to a single function by:
where
In the TBM the degree of belief in a hypothesis is defined by a function:
with

Pignistic level

When a decision must be made the credal beliefs are transferred to pignistic probabilities by:
where denote the atoms and the number of atoms that appear in. Hence, probability masses are equally distributed among the atoms of A.
This strategy corresponds to the principle of insufficient reason according to which an unknown distribution most probably corresponds to a uniform distribution.
In the TBM pignistic probability functions are described by functions. Such a function satisfies the probability axioms:
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. 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 is considered and there is a decomposition of the overall probability of the following form: