No one likes making bad decisions. In certain fields, a bad decision can be extremely detrimental or even fatal. However, the data we rely on to make decisions is not always 100% certain. It is possible to have a report containing errors, to gather data from a sensor that is not 100% reliable, or simply to have information that is distorted before it reaches us. Assessing the relevance and reliability of information to incorporate it into our decision-making process is a task that humans perform constantly and effortlessly. Two humans may not necessarily have the same internal value of reliability for the same piece of information, nor the same value of relevance or confidence. Humans do not necessarily seek to calculate THE truth value of information but THEIR truth value, although they often confuse the two.
Of course, if we want to automate this process so that machines can make decisions automatically, it is important to use a formalism that allows us to express the complexity of the situations encountered and to combine the information received to make the best possible decision with the available data.
One formalism that comes to mind is Boolean algebra, also known as Boolean logic. It deals with truth values, TRUE and FALSE, which can be combined using various operators such as AND, OR, NOT, XOR, etc. Boolean algebra starts with absolute truth values. We KNOW that things are TRUE or FALSE, and from there, we can combine them to make a decision. However, it is rare to have data that is so reliable; sensors can make mistakes, information can be misunderstood, etc.
Probabilistic logic is an extension of Boolean algebra where we no longer deal with truth values directly but with probabilities. This allows for deductive reasoning about probabilities. This is much more interesting because now it is possible to introduce imprecision into our decision-making process. Indeed, if a sensor gives me a certain value, I can use the probability that it is correct based on its error rate. This allows me to refine my process and arrive at a more accurate decision. However, we do not necessarily know the error rate of a sensor a priori, and this rate can change over the sensor’s lifetime. Although this formalism is better suited to representing real-world situations, it is not necessarily sufficient. Indeed, probabilistic logic does not allow for the modeling of uncertainty, which can be insufficient in many situations.
Audun Jøsang developed subjective logic to model these real-life situations. Subjective logic can be seen as an evolution of probabilistic logic. It is based on both Bayesian inference and Dempster-Shafer theory, which allows reasoning in the presence of uncertainty. Subjective logic has two major contributions compared to probabilistic logic: firstly, the truth value depends on who is making the observation, which is the subjective aspect. Secondly, subjective logic incorporates uncertainty.
Since truth values are obtained from the observer’s point of view, we speak of opinions. From a more formal perspective, an opinion is a quadruplet (b, d, u, a). For simplicity, we will focus on the case of binary opinions, but subjective logic is also defined for more general frameworks.
The quadruplet (b, d, u, a) corresponds to:
- b, belief: the observer’s confidence in the truth of the variable
- d, disbelief: the observer’s disbelief in the truth of the variable
- u, uncertainty: the observer’s uncertainty about the truth value of the variable
- a, a priori: the observer’s prior belief in the truth of the variable
For a quadruplet to be valid, it must satisfy the following condition: b + d + u = 1. In summary, knowledge is modeled by the sum of confidence, disbelief, and uncertainty. The truth value of a binary variable can be obtained using the following formula: EX = b + u.a.
Of course, this formalism is more complicated than simple Boolean algebra and probabilistic logic. However, it is important to be able to use it appropriately. This more complex model is therefore more difficult to implement, and one must be sure to have the means to handle this complexity to represent data for decision-making.
The interest of this formalism is to be able to evolve the uncertainty of opinions over observations. Thus, opinions can become more precise over time, eventually eliminating uncertainty. The interest of subjective logic is precisely to be able to combine opinions even in the presence of this uncertainty.
There are several ways to combine opinions. We will review them and discuss their usefulness. Of course, there are operators equivalent to Boolean algebra operators for the opinions of subjective logic. They allow us to obtain opinions on combinations of truth values. We will focus on the operators that allow us to merge opinions from different observers.
All these operators have precise mathematical definitions that we will not present in this article. We will rather discuss the usefulness of the operators and the context in which it is relevant to apply them.
One very interesting operator is the transitivity or recommendation operator. Imagine that observer A has an opinion about observer B, and the latter has an opinion about a variable X. It is possible to combine the two opinions to obtain A’s opinion about variable X.
There are several fusion operators that can be used when multiple observers wish to determine a truth value from their different opinions. It is possible to cumulate the opinions, using the cumulative operator, to benefit from everyone’s experiences in order to reduce overall uncertainty, or to average the opinions of each observer, using the consensus operator, to obtain an average opinion. This allows us to weigh the belief and uncertainty of each observer.
A final type of operator that is very interesting and very important in subjective logic is the so-called “belief constraining” operator. This operator is to be used in cases where different observers seek to agree based on their personal experience. Here, we do not seek to merge everyone’s experiences to increase global knowledge but rather to find a choice that suits everyone. This operator is very important because it allows decisions to be made between different agents.
This formalism, along with these operators, not only allows us to model realistic situations but also provides the tools to combine data and make decisions in these uncertain contexts. Subjective logic is an excellent formalism for representing confidence, for example. This has been successfully done in the past. This formalism has also been used to design a system for distributing students into groups. Since each student’s experience is very personal, this allows us not only to base the group formation on group performance but also on each individual’s experience to assemble the most functional groups possible.
Making automatic decisions in uncertain contexts can be perilous. In this framework, it is reassuring to be able to rely on solid theoretical frameworks such as subjective logic to combine uncertain information and make the best possible decision.
Subjective logic offers a powerful framework for modeling and managing uncertainty in decision-making. By integrating concepts such as belief, disbelief, and uncertainty, it allows for the coherent combination of subjective opinions and improves the quality of decisions made in uncertain contexts. Whether in cybersecurity, risk analysis, or project management, subjective logic has proven its usefulness. For those who wish to explore this field further, implementations and resources are available, facilitating the adoption of this approach. In the future, the application of subjective logic in emerging fields such as artificial intelligence and machine learning could open up exciting new perspectives.
