Imagine sitting by a tree full of birds. You know the tree only has a Yellow Rumped Warbler ($w$), a Northern Cardinal ($c$), and an American Goldfinch ($g$). These birds are respectful in that they don’t call at the same time.

You make the following observations of bird calls: $$ obs = w, w, c, g, g, w $$ What’s the probability that we hear a Warbler next assuming the calls are independent from each other? $$ count_w = 3, total = 6, P(w) = \frac{3}{6} = 0.5 $$ This example assumes that we’re bird call experts and are able to uniquely determine each bird call. What happens if our observations are imprecise?

Dempster-Shafer Theory (DST)

DST, otherwise known as belief functions theory or the theory of evidence, looks at what happens if we allow each observation to be a disjunction ($\vee$)? $$ obs = w \vee c \vee g, c, w \vee c, w \vee g $$ There can be many reasons for this. Maybe our hearing isn’t so good. There’s additional noise around you disrupting your sensing.

Formally, let us define $\Omega$ to be our event space. In this example, this is the set of possible bird calls. $$ \Omega = \{w, c, g\} $$ The mass function $m: A \rightarrow [0, 1], \forall A \subseteq \Omega$ assigns a value between 0 and 1 to every possible subset of our event space. The set $\{w, g\}$ represents the observation $w \vee g$.

An important property is that the sum of all the masses are equal to 1. $$ \sum_{A \subseteq \Omega} m(A) = 1 $$ In order to derive this mass function, we can normalize our observations from earlier. $$ m(\{w, c, g\}) = \frac{1}{4}, m(\{c\}) = \frac{1}{4}, m(\{w, c\}) = \frac{1}{4}, m(\{w, g\}) = \frac{1}{4} $$ We assign the value $0$ to all other subsets of $\Omega$. By definition, $m(\{\}) = 0$.

The plausibility measure for a disjunctive set $A$ is the sum of all the mass values of the subets of $\Omega$ that intersect with $A$. $$ Pl(A) = \sum_{B \cap A \ne \emptyset}{m(B)} $$ In our example, $$ \begin{align*} Pl(\{w\}) &= m(\{w, c, g\}) + m(\{w, c\}) + m(\{w,g\}) + m(\{w\}) \\ &= \frac{3}{4} \end{align*} $$ The necessity measure is more restrictive in that we only look at the summation of the masses of the subsets of $A$. $$ Nec(A) = \sum_{B \subseteq A}{m(B)} $$ Consider an arbitrary event $a \in \Omega$. Then, $$ \begin{align*} Nec(\{a\}) &= m(\{a\}) + m(\{\}) \\ &= m(\{a\}) \end{align*} $$ Therefore in our example, $$ Nec(\{w\}) = 0, Nec(\{c\}) = \frac{1}{4}, Nec(\{g\}) = 0 $$ Another example, $$ \begin{align*} Nec(\{w, c\}) &= m(\{w, c\}) + m(\{c\}) + m(\{w\}) + m(\{\}) \\ &= \frac{1}{4} + \frac{1}{4} + 0 + 0 \\ &= 0.5 \end{align*} $$ The probability measure is bounded by the necessity and plausibility measures.

For a disjunctive set $A$, $$ Nec(A) \le P(A) \le Pl(A) $$ Extending probability to a range of values gives us a way to model ignorance. We say an agent is completely ignorant if $|\Omega| > 1$ and $m(\Omega) = 1$.

Consider a completely ignorant agent where $\Omega = \{w, c, g\}$.

Then, $$ \begin{align*} Nec(\{w\}) \le P(\{w\}) &\le Pl(\{w\}) \\ m(\{w\}) \le P(\{w\}) &\le m(\{w\}) + m(\{w, c\}) + m(\{w, g\}) + m(\{w, c, g\}) \\ 0 \le P(\{w\}) &\le 1 \end{align*} $$ Probability theory is a subset of Dempster-Shafer theory. In order to see this, let us look at an example of observations where there is no disjunction. $$ obs = w, w, c, g, g, w $$ Normalize our observations to derive the mass function: $$ m(\{w\}) = \frac{1}{2}, m(\{c\}) = \frac{1}{6}, m(\{g\}) = \frac{1}{3} $$ The mass function in this example is $0$ for every non-singleton subset of $\Omega$.

What is the probability range for $w$? $$ \begin{align*} Nec(\{w\}) \le P(\{w\}) &\le Pl(\{w\}) \\ m(\{w\}) \le P(\{w\}) &\le m(\{w\}) + m(\{w, g\}) + m(\{w, c\}) + m(\{w, c, g\}) \\ \frac{1}{2} \le P(\{w\}) &\le \frac{1}{2} + 0 + 0 + 0 \end{align*} $$ Therefore, $P(w) = \frac{1}{2}$ as expected in probability theory.

Conclusion

Dempster-Shafer theory is an attempt at addressing imprecise observations through disjunctive events. It extends probability theory to consider not just a single value, but a range of possible values. This allows the model to decouple uncertainty from ignorance.