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3.3. REPRESENTATION OF UNCERTAINTY

Logic-based representations have originally binary decision opportunities only, the truth value outcomes true or false represented with 1 and 0, respectively. All uncertain possibilities lie between them, and express the measure of uncertainty. All different uncertainty-oriented philosophies, theories, calculation methods do practically the same. This preaches about the common origin of uncertainty estimation, all coming from frequency sensation, should it be hard statistics or weak intuition. The uncertainty of data is attached to that, i.e., to the definition details of an object, the values put into the slots of the frame, the nodes and branches of directed graphs, nets.


FIGURE 2 Frames and nets.

Fuzzy-type uncertainty measure, expressed in verbal form, is transferred to the same types of numerical representation; the meaning is, on the one hand, a philosophical explanation, beyond the computer representation process; on the other hand, it is a simple look-up table representation attaching the 0 to 1 values to different verbal set membership measures.

The interpretations of the nature of uncertainty, the use of different uncertainty models deviate at combination, propagation process. An attempt for a unified graph representation form is the use of probabilistic (uncertainty) networks by Pearl (1988).

The outcome of any uncertainty combination-propagation calculation is used in two ways: the first provides a "certain" resulting value used for decision. If two or more different methods or different time medical checks indicate a certain evidence, then the resulting evidence will be higher. How high depends on the uncertainty model and the related calculation algorithm. The same is the case in the multiple evidences of a legal case, a hypothesized outcome of different interventions and situations in economy, etc. This means that the earlier outlined logical structures remain the same, only this outcome chosen in a somehow arbitrary way, is embedded.


FIGURE 3 Internal control.

The second use of uncertainty combination-propagation results can be useful if different trials are feasible, either in a model calculation or in real life experimentation. In the latter case a typical application is the selection of different medical diagnostic and therapeutical strategies. Led by considerations of least expenses, least possible harm and suffering of the patient, by least risk for the doctor and an unrecoverable deterioration, different strategies can be used with different uncertain but somehow estimable effects, starting from nonintervention to an immediate radical operation, using milder or extremely radical pharmaceutics. In this representation the action branches of a decision tree are taken one after the other according to the resulting uncertainty values. The two types of outcomes are not contradictory, and can apply the same method. The application being different, influences the responsibility, risk-related estimations of the uncertainty values.

As was emphasized, the calculation method of combination-propagation is a basic representation issue, representing the general model of the nature of uncertainty. This very relevant fact is forgotten several times. To make it clear: where reliable statistics are available and the risk of a single decision is not high (moreover, if some experimentation with different possible decisions is possible), the best is a classical probabilistic model representation, which is a well-defined model on well-defined, mutually independent events having clear-cut truth values, excluding any intermediate situations. In this case, highly sophisticated estimation methods are available for the distribution of event data, confidence on them, expected failures. Problems of mass production, like quality control, are typical examples of application areas. The observations of physical and chemical phenomena also belong to this model usance.


FIGURE 4 Representation of Bayesian inference.

A classic probabilistic-related model is the Bayesian; it concerns certain classes of events within the realm of the total observation. The Bayesian model supposes also well-defined classes, reliable sampling methods, and a relevant amount of data for each class. The condition of independence belongs also to the model hypothesis. The Bayesian model establishes the well-known relation between a priori and a posteriori estimations, i.e., probability of an effect, if we know the probability of a certain cause, and vice versa, the probability of a cause based on the known probability of an effect (Figure 4). The Bayesian method established relations between a priori and a posteriori probabilities, i.e., between two events which are cause and consequence-related and vice versa. If

p(a) is the probability estimate of an event (e.g., people salting their food twice higher than the average), and
  p(b) the probability of a certain consequence (e.g., people having a high blood pressure), and
  p(b/a) the statistics of people who have high blood pressure among those who are exces- sively salting, then
  p(a/b) is the estimate of looking for a cause b in presence of a (e.g., for supposing a habit of excessive salting in a case of high blood pressure) and this will be

If there are many (n) possible reasonings (consequences), the formula for one of them will be modified to:

(in our example high blood pressure caused by salting, neurosis, kidney problems, sclerosis, etc.) as a consequence and having a statistics of those singular sets of events. The doctor looks for the cause of the certain symptom, ranking the hypotheses by this formula.

A further extension is updating, i.e., including new data into the existing statistics:

If b1, b2, ... bm are different poolings of evidences for b, then

This relation is applicable for singular causes and effects within a situation of multiple causes and effects if we have rather reliable data on all hypothetical sampling classes. The double weakness is apparent from the conditions: first, the need for a crisp definition, separation of the singular sampling classes, and second, the requirement of sufficient statistics on all possible classes. Diagnostics, statistical analysis of complex and mass effects, like demographic, diet-related search on medical data, weather research, and research on environmental effects, are typical application fields. Extension of Bayesian concepts, covering a broad field of different representations, like nonmonotonic logic, Dempster-Shafer schemes, etc., is done by the Bayesian nets of Pearl.


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