Information-Gap Decision Theory

Rating:
Summary: Ben-Haim is ignorant of the work of J M Keynes and R Carnap
Review: Yakov Ben-Haim(BH)presents another method for solving problems where the information base is(a) incomplete or(b) nonexistent.BH correctly categorizes these types of decision problems as "Decision under Severe Uncertainty".J M Keynes categorizes these two types of problems as(a) decision making under uncertainty in his General Theory(1936;GT)(or low weight of the evidence in the A Treatise on Probability(1921;TP) in chapter 26)and(b) complete or total uncertainty in the GT and ignorance in the TP.BH has developed a solutions technique called the geometric ellipsoidal information-gap model.It is based on a quadratic form,W.W is a real,symmetric positive definite matrix which determines the shape of the family of ellipsoids of uncertain probability vectors,given a particular level of uncertainty which I will denote as *.The general goal of the decision maker is to obtain a satisfactory objective function output(satisfice)while maximizing his immunity to uncertainty(subject to a minimum uncertainty level).BH shows that his mathematical model can deal completely with the Ellsberg and Popper paradoxes.Thus,the modeling is successful.The major criticism is his failure to compare his approach to other models designed to handle the risk versus uncertainty issue in formal,decision theoretic terms.There is no mention or comparison of the BH model with the Ellsberg,Gardenfors-Sahlin,Levi,Kyburg,L J Cohen,Hogarth-Einhorn,J M Keynes or Carnap models.BH appears to think that only Frank Knight grasped the clear distinctions between risk and uncertainty.BH makes the following incorrect claim:"Where Keynes,Carnap and virtually all modern philosophers have erred,however, is in believing that probability is the only("only" is underlined by BH for emphasis)means of handling uncertainty..."(Ben-Haim,p.290).He then quotes Kyburg:"...some novel procedure could be used in a decision theory that is based on some non-probabilistic measure of uncertainty."(ibid.,p.290).Of course,John Maynard Keynes did precisely that back in 1921 in his TP.In fact,this area of Keynes's work in the TP is nearly identical to the results that appear in his unaccepted 1907 Cambridge fellowship thesis and in his successful 1908 Cambridge Fellowship thesis.BH presents his reader with two irrelevant quotations taken from page 3 and pages 281-282 of the TP.The relevant material is contained in chapters 6 and 26 of the TP.In these chapters,Keynes develops his concept of the weight of the evidence.Weight of the evidence is completely independent of probability,however defined.Only in the case where the relevant evidence is made up completely of statistical evidence accumulated over many years/experiments will their be a connection with the standard error of the estimate,which could then serve as a practical measure of the weight.In chapter 26,Keynes normalizes the variable,w,where w denotes weight of the evidence,on the unit interval[0,1],where 0<=w<=1.Keynes then defines his conventional coefficient of weight and risk,c.The goal of the decision maker is to maximize cA,where A is some outcome, as opposed to probability based decision rules such as the expected value rule,maximize pA,where p is the probability of success(p+q=1)and the expected utility rule,maximize pU(A),where U is a utility function.c is thus a decision weight,not a probability.c=p/(1+q)[2w/(1+w)].I have deducted one star for BH's failure to apply Ockham's razor to his model and another star for his apparent ignorance of basic philosophical literature.Both of these deficiencies could be easily remedied in a revised edition.A five star rating would then be merited.I recommend the purchase of this book to any reader who has the appropriate technical training in decision and optimization theory.

Rating:
Summary: An extraordinary insight
Review: Prof. Ben-Haim has defined a statistic with broad ranging application from ecology to economics to health care. It may be too aggressive to say that this book reveals a fundamental truth but it certainly provides a valuable tool for understanding uncertainty. It is not for the mathematically faint of heart.