Subjective Logic: Fusion of Opinions

Last time, we introduced opinions as analogous to propositions held by actors, with some degree of belief and uncertainty.

This post will cover the following scenario (and more): consider a trial, and you are on the jury. A number of witnesses X, Y, and Z, all testify what they saw. We have some opinion about the truthfulness or believability of each witness \(\omega^{A}_{X}\), \(\omega^{A}_{Y}\), \(\omega^{A}_{Z}\), and then the opinions held by the witnesses concerning some shared event \(\omega^{X}_{v}\), \(\omega^{Y}_{v}\), \(\omega^{Z}_{v}\). How can we combine these opinions to form some opinion about the event v based on the evidence supplied by the testimony of the witnesses alone?

Fusion of Opinions

Now we can combine opinions, which is the main strength of subjective logic. But we have several operators combining opinions together, depending on the situation. Let's start with the decision tree to pick out which operator we'd need:

1. Is compromise possible between conflicting opinions? Are sources totally reliable? If no compromise is possible and sources are totally reliable, then Belief Constraint Fusion should be used; otherwise go to step 2.
2. Does including more opinions increase the amount of independent, distinct evidence? (For example, we use independent sensors in a science experiment as our source of beliefs. Each sensor is independent, so adding more sensors will add more independent evidence.) If so, then Cumulative Belief Fusion should be used; otherwise go to step 3.
3. Should completely uncertain opinions increase uncertainty? Does adding duplicate opinions increase belief in the fused opinion? (Example: are we on a jury listening to witness testimony?) If so, then the Averaging Belief Fusion; otherwise, go to step 4
4. How do we handle conflicting opinions (based on possibly overlapping evidence)? If we can use a weighted average to combine conflicting opinions together (e.g., doctors expressing multinomial set of diagnoses), then we should use the Weighted Belief Fusion; otherwise go to step 5.
5. So, we have conflicting opinions, possibly based on overlapping evidence, with compromise possible among the conflicts. For this case, we should use Weighted Belief Fusion with Vagueness Maximization. This is ideal when the analyst (us) wants to preserve shared opinions while transforming conflicting beliefs into uncertain opinions.

We will consider how to fuse together N sources (denoted \(C_{1}\), ..., \(C_{N}\)) who have opinions on a multinomial matter X (which we could restrict to binomial concerns as a special case). Generically we refer to the set of sources as \(\mathbb{C}\), and the domain for the opinion \(\mathbb{X}\) (for a binomial opinion, this would be \(\{x,\bar{x}\}\), for multinomial generalization this is the set of possible "rolls of the die").

Caution/warning/disclaimer/caveat: I subscribe to Cromwell's Law (we should never have probabilities of 100% or 0%), so uncertainty will never be 0 (nor 1), belief will never be 1 or 0, and our prior inclination \(a\) will never be zero or 1.

Belief Constraint Fusion

This is best expressed using the mass function. Remember, we are working with multinomial opinions, so the possible domain for an opinion is \(\mathbb{X}\) (a good intuition, for our interests, would be race ratings for elections). The mass function is defined on the power set of the opinion domain \(\mathcal{P}(\mathbb{X})\) taking some subset \(x\subset\mathbb{X}\) and assigning \[m(x) = b_{X}(x)\quad\mbox{if }x\neq\mathbb{X}, x\neq\emptyset\] \[m(\emptyset)=0\] \[m(\mathbb{X})=u_{X}\]

Given the mass function, we define the belief, uncertainty, and prior to be: \[ b^{\&(\mathbb{C})}_{X}(x) = \frac{\mathrm{Har}(x)}{1-\mathrm{Con}} \] \[ u^{\&(\mathbb{C})}_{X}(x) = \frac{\prod_{C\in\mathbb{C}}u^{C}_{X}}{1-\mathrm{Con}} \] \[ a^{\&(\mathbb{C})}_{X}(x) = \frac{\sum_{C\in\mathbb{C}}a^{C}_{X}(x)(1 - u^{C}_{X})}{\sum_{C\in\mathbb{C}}(1 - u^{C}_{X})} \] where "Har(-)" is the sum of beliefs harmonic with the argument, "Con" is the sum of conflicting beliefs: \[ \mathrm{Har}(x) = \sum_{X^{C_{1}}\cap\dots\cap X^{C_{N}}=x}\left(\prod_{C\in\mathbb{C}}m^{C}_{X}(x^{C})\right) \] And "Con" is defined as a sum over (nonempty?) subsets of the opinion domain \[ \mathrm{Con} = \sum_{X^{C_{1}}\cap\dots\cap X^{C_{N}}=\emptyset}\left(\prod_{C\in\mathbb{C}}m^{C}_{X}(x^{C})\right) \]

Cumulative Belief Fusion

Cumulative belief fusion is best when the situation satisfies: combining more opinions amounts to combining more evidence.

\[b^{\diamondsuit(\mathbb{C})}_{X}(x) = \frac{\sum_{C_{j}\in\mathbb{C}} \left(b^{C_{j}}_{X}(x)\prod_{C\neq C_{j}}u^{C}_{X}\right)}{\left[\sum_{C_{j}\in\mathbb{C}}\left((1 - u^{C_{j}}_{X})\prod_{C\neq C_{j}}u^{C}_{X}\right)\right] + \left[\prod_{C\in\mathbb{C}}u^{C}_{X}\right]}\] \[u^{\diamondsuit(\mathbb{C})}_{X} = \frac{\prod_{C\in \mathbb{C}}u^{C}_{X}}{\left[\sum_{C_{j}\in\mathbb{C}}\left((1 - u^{C_{j}}_{X})\prod_{C\neq C_{j}}u^{C}_{X}\right)\right] + \left[\prod_{C\in\mathbb{C}}u^{C}_{X}\right]}\] \[ a^{\diamondsuit(\mathbb{C})}_{X}(x) = \frac{\sum_{C_{j}\in\mathbb{C}}\left(a^{C_{j}}_{X}(x)(1 - u^{C_{j}}_{X})\prod_{C\neq C_{j}}u^{C}_{X}\right)}{\sum_{C_{j}\in\mathbb{C}}\left((1 - u^{C_{j}}_{X})\prod_{C\neq C_{j}}u^{C}_{X}\right)} \]

One property such a fusion operator satisfies is its resulting opinion's evidence vector (useful when setting parameters to the corresponding Dirichlet distribution) is the sum of the evidence vectors of its constituents: \[ r^{\diamondsuit(\mathbb{C})}_{X}(x) = \sum_{C\in\mathbb{C}}r^{C}_{X}(x). \] Hence the "cumulative" adjective to this fusion rule.

Averaging Belief Fusion

When we have a trial with testimony on the same events from different witnesses, we use the averaging belief fusion

\[b^{\underline{\diamondsuit}(\mathbb{C})}_{X}(x) = \frac{\sum_{C_{j}\in\mathbb{C}} \left(b^{C_{j}}_{X}(x)\prod_{C\neq C_{j}}u^{C}_{X}\right)}{\left[\sum_{C_{j}\in\mathbb{C}}\left(\prod_{C\neq C_{j}}u^{C}_{X}\right)\right]}\] \[u^{\underline{\diamondsuit}(\mathbb{C})}_{X} = \frac{\prod_{C\in \mathbb{C}}u^{C}_{X}}{\left[\sum_{C_{j}\in\mathbb{C}}\left(\prod_{C\neq C_{j}}u^{C}_{X}\right)\right]}\] \[ a^{\underline{\diamondsuit}(\mathbb{C})}_{X}(x) = \frac{\sum_{C\in\mathbb{C}}a^{C}_{X}(x)}{N} \]

This has the nice property that the evidence vector for the resulting opinion is the average of the evidence vectors for its constituents: \[ r^{\underline{\diamondsuit}(\mathbb{C})}_{X}(x) = \frac{\sum_{C\in\mathbb{C}}r^{C}_{X}(x)}{N}. \]

Weighted Belief Fusion

When we want uninformed opinions to not influence us (e.g., an opinion representing "This isn't my field, I have no expertise, I don't know"), we can weigh opinions based off the "confidence" of the opinion (the complement of the uncertainty).

\[b^{\widehat{\diamondsuit}(\mathbb{C})}_{X}(x) = \frac{\sum_{C_{j}\in\mathbb{C}} \left(b^{C_{j}}_{X}(x)(1-u^{C_{j}}_{X})\prod_{C\neq C_{j}}u^{C}_{X}\right)}{\left[\sum_{C_{j}\in\mathbb{C}}\left((1-u^{C_{j}}_{X})\prod_{C\neq C_{j}}u^{C}_{X}\right)\right]}\] \[u^{\widehat{\diamondsuit}(\mathbb{C})}_{X} = \frac{\left(\sum_{C\in\mathbb{C}}(1-u^{C}_{X})\right)\left(\prod_{C\in \mathbb{C}}u^{C}_{X}\right)}{\left[\sum_{C_{j}\in\mathbb{C}}\left((1-u^{C_{j}}_{X})\prod_{C\neq C_{j}}u^{C}_{X}\right)\right]}\] \[ a^{\widehat{\diamondsuit}(\mathbb{C})}_{X}(x) = \frac{\sum_{C\in\mathbb{C}}a^{C}_{X}(x)(1-u^{C}_{X})}{\sum_{C\in\mathbb{C}}(1-u^{C}_{X})} \]

The confidence vector for the resulting opinion is, unsurprisingly, the weighted sum of the evidence vectors for its constituents (weighted by confidence): \[ r^{\widehat{\diamondsuit}(\mathbb{C})}_{X}(x) = \frac{\sum_{C\in\mathbb{C}}(1 - u^{C}_{X})r^{C}_{X}(x)}{\sum_{C\in\mathbb{C}}(1 - u^{C}_{X})} \]

Vagueness Maximization

When there are multiple competing hypotheses, and only one can be right (like: multiple doctors give different diagnoses to the same patient's symptoms, and one of them is correct, but we do not know which one), then we have Vagueness or a Vague Belief.

I think I'm going to skip the sordid details, they may be found in the references below.

Trust Transitivity

We have just reviewed all the ways to combine opinions together into one. But how can we factor in the degree of trust in the sources?

We can say, if A is the analyst and B is the source of information, we have an additional opinion \(\omega^{A}_{B}\) the analyst has of the quality of the source B. The analyst evaluates the source of information, and forms a modified opinion \(\omega^{A}_{B}\otimes\omega^{B}_{x}\) the analyst would posses about the matter reported \(x\) by the source B.

There are two different ways to go about this. If the source is sincere, or if the source is deliberately lying; these paths necessitate different treatments.

Now, before continuing on, let's stress how to interpret the opinion \(\omega^{A}_{B}\). Specifically here \(b^{A}_{B}\) is the degree we believe B is an honest source (i.e., that B would tell us the truth insofar as the source understands it), \(d^{A}_{B}\) is the degree we believe the source is trying to deceive us with a lie, and \(u^{A}_{B}\) reflects the lack of familiarity in our source's expertise or knowledgeability (e.g., we have a lack of history, or our interactions have been on other topics).

Earnest Sources

When we think the source B is sincere, then the weighted evidence is denoted \(\omega^{A:B}_{x}=\omega^{A}_{B}\otimes\omega^{B}_{x}\) with components: \[b^{A:B}_{x} = \Pr(\omega^{A}_{B})b^{B}_{x}\] \[d^{A:B}_{x} = \Pr(\omega^{A}_{B})d^{B}_{x}\] \[u^{A:B}_{x} = 1-\Pr(\omega^{A}_{B})(b^{B}_{x} + d^{B}_{x})\] \[a^{A:B}_{x} = a^{B}_{x}\]

This requires care when dealing with extremely naive analysts (where \(b^{A}_{B}\approx 1\)) or very cynical ones (with \(d^{A}_{B}\approx 1\)). In such cases, the analyst should use \(\Pr(\omega^{A}_{B})=b^{A}_{B}\).

Dishonest Sources

When we think the source may know the truth and deliberately misinform us, we can still use the information supplied to aid our judgment. We just deliberately interpret the information as a lie. Specifically, we use the following modified opinion with (confusingly enough) the same notation \(\omega^{A:B}_{x}=\omega^{A}_{B}\otimes\omega^{B}_{x}\) but components: \[b^{A:B}_{x} = b^{A}_{B}b^{B}_{x} + d^{A}_{B}d^{B}_{x}\] \[d^{A:B}_{x} = b^{A}_{B}d^{B}_{x} + d^{A}_{B}b^{B}_{x}\] \[u^{A:B}_{x} = u^{A}_{B} + (b^{A}_{B}+d^{A}_{B})u^{B}_{x}\] \[a^{A:B}_{x} = a^{B}_{x}\]

Here \(d^{B}_{x}\) is the degree to which the source believes x to be false, and \(b^{B}_{x}\) is the degree to which the source believes x to be true.

Usefulness

If we get data from a source, we need to formulate that as an opinion \(\omega^{A:B}_{x}=\omega^{A}_{B}\otimes\omega^{B}_{x}\) using the appropriate rules depending on the source's reliability.

Once we have accumulated data, we combine them together using the fusion rules we discussed earlier in this post. Next time when we discuss subjective logic further, we will consider a more in-depth example.

References

1. Audun Jøsang, "The Cumulative Rule for Belief Fusion". arXiv:cs/0606066
2. Audun Jøsang, "Cumulative and Averaging Fission of Beliefs". arXiv:0712.1182
3. Audun Jøsang, Categories of Belief Fusion. Journal of Advances in Information Fusion 13 no.2 (2018)