In the analysis of the risk associated to rare events that may lead to catastrophic consequences with large uncertainty, it is questionable that the knowledge and information available for the analysis can be reflected properly by probabilities. Approaches other than purely probabilistic have been suggested, for example using interval probabilities, possibilistic measures, or qualitative methods. In the present paper, we look into the problem and identify a number of issues which are foundational for its treatment. The foundational issues addressed reflect on the position that "probability is perfect" and take into open consideration the need for an extended framework for risk assessment that reflects the separation that practically exists between analyst and decision maker. 'There is only one kind of uncertainty stemming from our lack of knowledge concerning the truth of a proposition […] [Distinctions between probabilities are] merely for our convenience in investigating complex phenomena. Probability is always a measure of degree of belief.' As for the objective/subjective distinction that is often tacitly taken to underlie the frequentist/subjective probability distinction, Winkler writes [8 p. 128]: 'To some individuals, especially those trained in a scientific environment, the subjectivity in subjective probability seems to be at odds with a scientific search for objectivity. But, in fact, objectivity in dealing with uncertainty in the real world is a goal that is elusive at best and might more realistically be viewed as unattainable. In order to apply the classical or relative-frequency interpretations of probability, certain assumptions must be made, and these are ultimately subjective assumptions (Gelman et al. 3 ).' As for measurement, Popper distinguishes three worlds (see e.g. [10]): the physical world, the personal and subjective world, and the objective world of others; in addition, at the intersection of the second and third worlds, is the shared inter-subjective world. Measurement of properties of the physical world, such as the length of an object, is different from measurement of uncertainty in the sense of lack of knowledge, expressed by subjective probability reflecting a person's degree of belief. Some (e.g. [10, 11] ) would refer to this as measurement of hard and soft systems, respectively. Probabilistic analysis is the predominant method used to handle the uncertainties involved in risk analysis, both of aleatory and epistemic types. It is also used in the quantification of margins and uncertainties in analyses that use computational models to predict complex system behaviour [12] . It seems correct to say that probability is indeed perfectly suited to describe aleatory
Recent directions of development to represent uncertainty in risk assessment are provided in REF .
Concerns, challenges, and directions of development for the issue of representing uncertainty in risk assessment.
Risk analysis : an official publication of the Society for Risk Analysis
Risk analysis : an official publication of the Society for Risk Analysis
Engineering Medicine