Management of uncertainty in Statistical Reasoning: The case of Regression Analysis

AbstractStatistical Reasoning is affected by various sources of Uncertainty: randomness, imprecision, vagueness, partial ignorance, etc. Traditional statistical paradigms (such as Statistical Inference, Exploratory Data Analysis, Statistical Learning) are not capable to account for the complex action of Uncertainty in real life applications of Statistical Reasoning. A conceptual framework, called “Informational Paradigm”, is introduced in order to analyze the role of Information and Uncertainty in these complex contexts. Regression Analysis is taken as the reference problem for developing the discussion. Three basic sources of Uncertainty are considered in this respect: (1) uncertainty about the relationship between response and explanatory variables; (2) uncertainty about the relationship between the observed data and the “universe” of possible data; (3) uncertainty about the observed values of the variables (imprecision, vagueness). Some of the available methods for coping with these different types of Uncertainty are discussed in an orderly way, from the simpler cases where only one source at a time is dealt with, to the more complex ones where all sources act together. Probabilistic and Fuzzy-Possibilistic tools are exploited, in this connection. In spite of the recent relevant contributions in this domain, the weaknesses and deficiencies of the current procedures for managing Uncertainty in Regression Analysis, as well as in other areas of Statistics, are emphasized. The elements of a generalized system of Statistical Reasoning, capable to deal with the various sources of Uncertainty, are finally introduced and the lines for future investigation in this perspective are indicated