are represented as information granules described by linguistic terms and modeled as triangular fuzzy numbers.
(14) What is less well understood is that a world of complete markets is isomorphic to a world in which subjective probabilities
can be assigned to all states of the world.
However--like other biases and heuristics--the anti-inference heuristic also gives rise to systematic error; it strongly affects decisions even when the objective and subjective probabilities
of the pertinent occurrence are equal according to circumstantial and direct evidence.
With this philosophy it would be easy, for example, to allow a child to give quite different answers to symmetric, experimental and subjective probabilities
of obtaining a six, and to make no pedagogical use of the discrepancy between the answers.
As is well known, subjective probabilities
are applied to an event or outcome that is unique or occurs once.
Rather, the assessment is one of subjective probabilities
, assessed in light of rational bases for inference, an entirely different concept.
(1) Yet the new work has tended to remain either too wedded or overly hostile to subjective probabilities
for evaluating evidence (2) and to Bayes' theorem for combining evidence, (3) and so caused the debates to become "unproductive and sterile." (4) In any event, the debates have left unsolved some troubling problems and paradoxes in our law on proof.
And since, as was seen above, probability must be defined subjectively if the world is causally-deterministic, this means that the relative frequency method is only capable of producing subjective probabilities
. That is, the method is only capable of generating numerical measures of human uncertainty.
We check this explanation by using data on beliefs elicited as subjective probabilities
and a rich set of other variables from the Health and Retirement Study.
The theories of subjective probabilities
advocated by eighteenth-century English mathematician and theologian Thomas Bayes and by twentieth-century Italian statistician Bruno de Finetti are very applicable today when we assign likelihood to any future conditions or outcomes.