4.4 KiB
| title | date | draft | tags | math | medium_enabled |
|---|---|---|---|---|---|
| Introduction to Dempster-Shafer Theory | 2024-10-29T12:38:21-04:00 | false | true | false |
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.