Professor Forman

Ratio Scale Likelihood Measurement Methods

Ratio scale measurements for likelihoods can be based on historical data as well as human judgment.

If we have historical ratio scale data, such as how many times an event occurred in the past, and if it is reasonable to assume that the future will be similar to the past, then the historical data can be used. However, either or both of the ‘ifs’ are not typical and methods such as the following can be used:

  • Pairwise Comparisons: Pairwise comparisons of the likelihoods of a cluster of sources or events using the fundamental verbal scale of the Analytic Hierarchy Process (AHP). Likelihoods are derived from the principle, normalized right hand eigenvector of the comparisons.
    (Google employed this method to become the leader in pageranking).
    The pairwise process also includes a measure for the inconsistency of comparisons so that a variety of errors can be detected and corrected where necessary.
    To see how this method accurately translates human judgment in verbal form to ratio scale priorities, click Area Validation Exercise
  • Pairwise of Probabilities: Pairwise comparisons of a range of probabilities of the source or event likelihoods resulting in a distribution and expected value for the source or event likelihood
  • Pairwise with given likelihood: The relative likelihoods derived from pairwise comparisons are translated to absolute likelihoods based on a given likelihood for one of the sources or events.
  • Rating Scales with ratio scale intensities: Verbal intensities, such as Almost certain, Probable, Fifty-fifty, Probably not, Almost certainly not, are compared using pairwise comparisons or pairwise comparisons with a given likelihood to derive ratio scale probabilities for each which in turn are used as a rating scale for sources or events.
  • Utility Curve to translate data to likelihoods: historical data can be transformed to ratio scale likelihoods using a utility curve which could be linear increasing, linear decreasing, non-linear increasing concave or convex, or non-linear decreasing concave or convex with a curvature elicited from subject matter experts.
  • Step Function to translate data to likelihoods: A series of data points that are subjected to pairwise comparisons for their likelihoods. A piecewise linear interpolation of such points is applied similar to a utility curve but based on likelihoods derived from pairwise comparisons rather than having to guess at the curvature.
  • Bayesian Updating: New information that is related to a previously estimated likelihood is used to revise the estimate using Bayes rule.

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