Transferable belief model

According to the DST, a probability mass function ${\displaystyle m}$ is defined such that:[1]

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

with

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

where the power set ${\displaystyle 2^{X}}$ contains all possible subsets of the frame of discernment ${\displaystyle X}$. In contrast to the DST the mass ${\displaystyle m}$ allocated to the empty set ${\displaystyle \emptyset }$ is not required to be zero, and hence generally ${\displaystyle 0\leq m(\emptyset )\leq 1.0}$ holds true. The underlying idea is that the frame of discernment is not necessarily exhaustive, and thus belief allocated to a proposition ${\displaystyle A\in 2^{X}}$, is in fact allocated to ${\displaystyle A\in 2^{X}\cup {e}}$ where ${\displaystyle {e}}$ 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 ${\displaystyle m(\emptyset )=0}$. Hence, in the TBM any two independent functions ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ are combined to a single function ${\displaystyle m_{1,2}}$ by:[5]

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

where

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

In the TBM the degree of belief in a hypothesis ${\displaystyle H\in 2^{X}\neq \emptyset }$ is defined by a function:[1]

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

with

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

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

When a decision must be made the credal beliefs are transferred to pignistic probabilities by:[4]

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

where ${\displaystyle x\in X}$ denote the atoms (also denoted as singletons)[6] and ${\displaystyle |A|}$ the number of atoms ${\displaystyle x}$ that appear in ${\displaystyle A}$. Hence, probability masses ${\displaystyle m(A)}$ are equally distributed among the atoms of A. This strategy corresponds to the principle of insufficient reason (also denoted as principle of maximum entropy) according to which an unknown distribution most probably corresponds to a uniform distribution.

In the TBM pignistic probability functions are described by functions ${\displaystyle P_{\text{Bet}}}$. Such a function satisfies the probability axioms:[4]

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

with

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

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

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.[1]