Beyond the Domain of Direct Observation: How to Specify a Probability Distribution That Represents the "State of Knowledge" About Uncertain Inputs. F. Owen Hoffman, SENES Oak Ridge, Inc., 102 Donner Dr., Oak Ridge, TN 37830; Stan Kaplan, 3678 Vigilance Dr., Rancho Palos Verdes, CA 90275; and Bob Tardiff, 1423 Trapline Court, Vienna, VA 22182
Uncertainty is inherent in all exposure and risk assessments in which mathematical models are used to extrapolate information beyond the domain of direct observation. Uncertainty exists because models are mimics of reality. In addition, the data available for inputs are seldom ever directly relevant to the defined assessment endpoint. The use of classical statistics to summarize the variability of direct observations is usually inappropriate for most exposure and risk projections. Classical statistics should be restricted to instances when data are obtained from either a random or stratified random design, appropriately averaged in space and time, and directly relevant to the target individuals or populations of interest. These situations are rare in human health and ecological risk assessments for which extrapolations are made in space and time and from non-human species to sensitive subgroups of the human population. Sometimes, analysts "assume" a data set to be directly relevant in order to justify using the familiar tools of classical statistics. This procedure is what we call being "subjectively objective." Its practice can result in misleading statements of uncertainty and can misdirect the need for additional information. Our preference, however, is to be "objectively subjective" and to use all sources of information available to quantify the present state of knowledge. We recommend treating most data published in the literature as indirect evidence. We recommend combining the information contained in these data with all other sources of information to form the basis for a written rationale from which a subjective probability distribution is specified. As we practice uncertainty analysis, probability distributions represent alternative realizations of true outcomes that are obtained from our current state of knowledge given the present body of evidence. These subjective probability distributions are equivalent to the "informative prior" in Bayes’ Theorem and represent the degrees of belief of an individual analyst or a group of experts. These informative prior distributions are then propagated through the assessment model to obtain a prior distribution for the exposure or risk. Each value in this distribution represents an alternative realization of the true value of exposure or risk. Subjective confidence intervals are obtained from these distributions, and decisions are made to either accept or reject an activity or to get additional information to reduce uncertainty.