# A Brief Roadmap into Uncertain Knowledge Representation via Probabilistic Description Logics
## 3 Representing Uncertain Knowledge
### Subjective Probabilities  
Consider first the case of subjective probabilities. These refer to the situation where the uncertainty is about the state of the world, and hence about the specific model under consideration. Thus, the semantics of these kinds of logics introduce a probability distribution over the class of relevant models, expressing which of them are more likely. From a syntactic point of view, it is necessary to express the likelihood of the knowledge appearing in the knowledge base. The typical approach is to associate to each axiom (that is, each constraint) a probability degree. This probability expresses the (subjective) belief that the constraint expressed by the axiom actually holds in the world [25; 26]. Intuitively, if this probability is \(p\), then the probability distribution should assign probability \(p\) to the set of all the interpretations which satisfy \(p\) and probability \(1-p\) to its complement. Unfortunately, things as not as easy as they seem at first sight, and there are many aspects to take into account. One issue is how to relate the probabilities of different axioms with each other for the construction of the probability distribution. That is, if an axiom \(\alpha\) has probability \(p\) and another axiom \(\beta\) has probability \(q\), the we question which probability should be assigned to the class of interpretations satisfying both axioms \(\alpha,\beta\). In most cases, to simplify the language and the probability computations, it is common to assume that axioms are probabilistically independent--that is, that the truth value of one does not affect the likelihood of the other. Under this assumption, the probability of satisfying both axioms becomes \(pq\). This assumption, however, is not always realistic in a knowledge representation application. For one, as knowledge bases tend to be big, knowledge engineers often rely on modelling guidelines, which specify how some specific kinds of knowledge should be represented in a given scenario. These guidelines commonly require simple axioms, which can only specify complex knowledge when combined with other axioms. If this complex knowledge is uncertain, it is unreasonable to assume that all the small pieces building it are probabilistically independent. The other side of the same coin are the normalisation steps often implicitly performed to aid reasoning. Again, in this case, many simple axioms are generated from one complex one, but they are clearly all dependent on each other. Formally, to solve this issue one would need to specify the full probability distribution of the axioms or at the very least the joint probabilities for all relevant combinations of axioms. Unfortunately, this solution requires a complex representation and slow reasoning. Some approaches have been proposed to use only partial independence assumptions [27; 28; 29]. Another approach is to consider all possible coherent assignments of probabilities in what is known as [PERSON]'s semantics [30]. However, this semantics does not generally satisfy the axioms of probability, and knowledge manipulation always increases imprecision. A second issue arises from the presence of the open world assumption. Recall that we previously said that if an axiom holds with probability \(p\), then the interpretations that do not satisfy this axiom should have a probability of \(1-p\). The issue, however, is with guaranteeing that the remaining interpretations indeed do not satisfy an axiom, and deciding what precisely this (i.e., violating an axioms) means in practice. From the open world assumption, we note that knowledge which is not explicitly stated could be true or false. When dealing with uncertain axioms, we will have some interpretations where the axiom _explicitly_ holds, and some--due to the open world assumption--where it may hold, but is not required to do so. This means that in reality, the probability stated by the axiom is only a lower bound: the likelihood of making it true may in fact be higher.