Comparative probability: Decision-making prospective

So far, we have only been concerned with CP from mathematical, logical and philosophical standpoints. Philosophy is an arena of debate, whereas science is one of research, analysis of empirical observations and application of theories to yield useful products. Therefore, it will be of value to know whether CP has managed to escape the “ballrooms” of philosophical debates and mathematical theorising into the realm of practical implementation. In section 3.1, we looked at the criteria of a good interpretation of probability, and CP should not be an exception to them. The emphasis of a good probability theory should be on its applicability as a framework is simply useless to science if it cannot be utilized in any way.

There are many frameworks in the literature regarding the formalisation of CP axioms and sometimes for developing some guidelines as to how it would be used for decision-making and inference. As much as these frameworks are packed with long mathematical formalizations, they are short on comparison to their quantative probability counterpart. Without justification for preferring CP over the de facto interpretation of probability in science, which withstood the test of time and was there during all of our scientific endeavours, why would anyone choose CP? In this section, we will be looking at some of the interesting frameworks in CP, their applications, and limitations.

Peter Fishburn is one of the well-known names in decision-making and the axiomisation of CP as an independent interpretation of probability [59, 61, 70, 79, 80]. The previous section has already presented some of his contributions to CP. The main framework of Fishburn was answering de

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Finnite’s question on the existence of an order-wise relationship that is sufficient for the existence of order-preserving probability measure [80]. The answer to the question was to introduce more basic but constraining limits on CP in order to preserve the order-wise nature of CP while solving any paradoxical objection to it [61]. Fishburn showed an example of how his version of CP can solve Ellsberg paradoxical examples of subjective probability through the introduction of the skew-symmetric function [61]. As mentioned earlier, there was no discussion on how the performance of CP compares to KP. Such a comparison, if it was in favour of CP, would prove the case of CP as an interesting alternative to KP that scientists should start to use, rather than shelve it along with the other mathematical constructs with internal inconsistencies. In addition, Fishburn did not provide us with a clear algorithm that explains in a step-by-step fashion how to use his framework to solve problems in decision-making beyond some isolated examples and more mathematical constraints.

Terrence Fine is another example of the independent interpretation approach to CP [54, 69, 81]. However, Fine’s approach seems relaxed and less constrained than Fishburn’s or Luce’s, as examples [54]. Fine developed five axioms that characterize a rational decision-making process and expectations, all in terms of comparative-like inequalities [54]. However, all of these axioms were incomplete in showing a single example of how to use them to come up with a rational decision within any context, not even a game of chance. Fine admitted the existence of the problem of measuring subjective probability or preferences and even the psychological factors

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leading to constraining the process of extracting those from decision-makers [54,p. 233]. Along with Walley, the framework seemed to be shifted toward establishing a unified framework of upper and lower probability, or imprecise probability [82, 83]. Imprecise probability refers to classes of mathematical models that deal with uncertainty and the availability of partial information [82].

Walley tried to unify many of the proposed models of imprecise probabilities, including CP itself, using the subjective framework of Bayesian networks [82].

When there is not enough data to infer a descriptive probability distribution, then upper and lower bounds are defined and the gap between the upper and lower limit is supposed to decrease as data is gathered, until the gap is closed and what was imprecise is now precise [84]. Walley’s focus was on the mathematical level of generality that will be needed to achieve such unification [82]. His framework was further applied to graphical models’ [85] belief functions [86], among others. Walley’s framework seems interesting within the context of this thesis; however, it still has the drawback we mentioned earlier, namely, no step-by-step algorithm was specified that could aid a decision-maker in making the decision and no comparison with KP was attempted.

The third example of CP framework is Andrea Capotorti, who proposed some interesting CP axioms that can be described algorithmically and implemented on computers [66, 87]. For Capotorti, the reason a decision-maker would prefer CP over the other interpretations of probability is that they are not compatible with the psychology of human preferences and sometimes even violate the axioms of KP [66], not to mention the “where are all the numbers coming from?” argument [66]. However, the same argument goes

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against CP because if it was valid for us to wonder where the frequentists take their quantities from, then by the same logic it is valid to wonder where the comparativists get their qualities from. The Capotorti algorithm works by constructing qualities and constraints that describe a situation [66]. For example, if heads was more probable than tails, then we can describe it as ℎ . The algorithm continues to use constraints and new information to update the qualities until a decision, or inference, is possible [66], although it was not clearly specified how such an update is made. In addition, it seems like interfacing with such DSS, if ever implemented, will be extremely difficult because it does not provide a quantified output, nor does it have an objective procedure for converting sensor measures into qualities. Finally, it relies heavily on expert knowledge to come up with the qualities that represent a situation.

In conclusion, this section tried to summarize the most important frameworks of CP as a tool for inference and decision-making. We have seen how CP frameworks were shifted when faced with different challenges in regard to measuring personal preferences, restricting their flexibility by the addition of more axioms and pitching for unification with other frequency-based probabilistic theories. We have also seen that the closest framework to the objectives of this thesis was that of Peter Walley, which aimed at unifying CP with the upper and lower probability model of imprecise probability. What steps are required in order to improve Walley’s axioms? Will it be possible to propose an algorithm that automates the process of inference in a way that proves more beneficial than the current conventional methods? If so, how do

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they compare? We will explore the answers to these questions in the next section.