Transferable belief model

The transferable belief model (TBM) is an elaboration on the Dempster–Shafer theory of evidence.

Context

Consider the following classical problem of information fusion. 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.)

The transferable belief model (TBM) is an elaboration on the Dempster–Shafer theory of evidence developed by Philippe Smets, based on the intuition that in the situation above, the result should allocate most of the belief weight to the empty set (i.e. neither A, B, nor C). Technically, this would be done by using the TBM conjunction rule for non-interactive sources of information, which is the same as Dempster's rule of combination without renormalization.

While most other theories adhere to the axiom the probability (or belief mass) of the empty set is always zero, there is another intuitive reason to drop this axiom: the open-world assumption. It applies when the frame of reference is not exhaustive, so there are reason to believe that an event not described in this frame of reference will occur. For example, when tossing a coin one usually assumes that Head or Tail will occur, so that $\scriptstyle \Pr(\mathrm{Head}) + \Pr(\mathrm{Tail}) = 1$. The open-world assumption is that the coin can be stolen in mid-air, disappear, break apart or otherwise fall sideway so that neither Head nor Tail occurs, so that the power set of {Head,Tail} is considered and there is a decompostion of the overall probability (i.e. 1) of the following form:

$\Pr(\emptyset) + \Pr(\mathrm{Head}) + \Pr(\mathrm{Tail}) + \Pr(\mathrm{Head,Tail}) = 1.$

See also

• Dempster–Shafer theory

References

• Smets Ph. (1988) "Belief function". In: Non Standard Logics for Automated Reasoning, ed. Smets Ph., Mamdani A, Dubois D. and Prade H. Academic Press, London
• Smets Ph. (1990) "The combination of evidence in the transferable belief model", IEEE Pattern Analysis and Machine Intelligence, 12, 447–458
• 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
• Smets Ph. and Kennes R. (1994) "The transferable belief model", Artificial Intelligence, 66,191–234
• Smets Ph. and Kruse R. (1995) "The transferable belief model for belief representation" In: Smets and Motro A. (eds.) Uncertainty Management in Information Systems: from Needs to solutions. Kluwer, Boston
• Ramasso, E., Rombaut, M., Pellerin D. (2007) "Forward-Backward-Viterbi procedures in the Transferable Belief Model for state sequence analysis using belief functions", ECSQARU, Hammamet : Tunisie (2007).
• Touil, K., Zribi, M., Benjelloun, M. (2007) "Application of transferable belief model to navigation system", Integrated Computer-Aided Engineering, 14 (1), 93–105

External links

• The Transferable Belief Model
• Publications on TBM
• Software for TBM in Matlab