Transferable belief model

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 Philippe Smets propagated 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 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, Philip Smets avoided probabilistic notations, e.g. probability masses or update in the sense of probabilistic calculus, and replaced them by technical terms degrees of beliefs and transfer, whose appear in the name of his model.

Zadeh’s example in TBM 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.)

Formal Definition

Beliefs exist 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 mass function $m$ is defined such that:

$m:2^{X}\rightarrow [0,1]\,\!$

with

$\sum _{A\in 2^{X}}m(A)=1\,\!$

where the power set $2^{X}$ contains all possible subsets of the frame of discernment $X$. In contrast to the DST the mass $m$ allocated to the empty set $\emptyset$ is not required to be zero, and hence generally $0\leq m(\emptyset )\leq 1.0$ holds true. The underlying idea is that the frame of discernment is not necessarily exhaustive, and thus belief allocated to a proposition $A\in 2^{X}$, is in fact allocated to $A\in 2^{X}\cup {e}$ where ${e}$ 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 $m(\emptyset )=0$. Hence, in the TBM any two independent functions $m_{1}$ and $m_{2}$ are combined to a single function $m_{1,2}$ by:

$m_{1,2}(A)=(m_{1}\oplus m_{2})(A)=\sum _{B\cap C=A}m_{1}(B)m_{2}(C)\,\!$

where

$A,B,C\in 2^{X}\neq \emptyset \,\!$.

The degree of belief in a hypothesis $H\in 2^{X}\neq \emptyset$ is defined by a function:

$\operatorname {bel} :2^{X}\rightarrow [0,1]\,\!$

with

$\operatorname {bel} (H)=\sum _{\emptyset \neq A\subseteq H}m(A)$

$\operatorname {bel} (\emptyset )=0\,\!$.

Note that in contrast to the DST the belief is not normalized, and thus:

$\sum _{H\in 2^{X}}\operatorname {bel} (H)\leq 1.0.\,\!$

Pignistic Level:

When a decision must be made the credal beliefs are transferred to pignistic probabilities by:

$P_{Bet}(x)=\sum _{x\subseteq A\in X}{\frac {m(A)}{|A|}}\,\!$

where $x$ denote the singletons (or atoms) and $|A|$ the number of singletons $x$ in $A$. Hence, masses $m(A)$ are equally distributed among the singletons of A. This strategy corresponds to the principle of insufficient reason according to which unknown distributions most probably correspond to uniform distributions. Pignistic probability functions have the following properties:

$P_{Bet}:X\rightarrow [0,1]\,\!$

with

$\sum _{x\in X}P_{Bet}(x)=1\,\!$

$P_{Bet}(\emptyset )=0\,\!$

and thus, satisfy the probability axioms. Philip Smets called them pignistic to stress the fact that those probability functions are defined if someone is forced to place a bet based on partial belief.

Open World Example

When tossing a coin one usually assumes that Head or Tail will occur, so that $\Pr(\mathrm {Head} )+\Pr(\mathrm {Tail} )=1$. 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
• 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.
• 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
• Dempster, A.P. (2007) "The Dempster–Shafer calculus for statisticians", International Journal of Approximate Reasoning, Volume 48, Issue 2, June 2008, Pages 365-377, ISSN 0888-613X, http://dx.doi.org/10.1016/j.ijar.2007.03.004.

External links

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