# Dilation for Sets of Probabilities (17 pages)

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Dilation for Sets of Probabilities Teddy Seidenfeld Larry Wasserman The Annals of Statistics Vol 21 No 3 Sep 1993 pp 1139 1154 Stable URL http links jstor org sici sici 0090 5364 28199309 2921 3A3 3C1139 3ADFSOP 3E2 0 CO 3B2 V The Annals of Statistics is currently published by Institute of Mathematical Statistics Your use of the JSTOR archive indicates your acceptance of JSTOR s Terms and Conditions of Use available at http www jstor org about terms html JSTOR s Terms and Conditions of Use provides in part that unless you have obtained prior permission you may not download an entire issue of a journal or multiple copies of articles and you may use content in the JSTOR archive only for your personal non commercial use Please contact the publisher regarding any further use of this work Publisher contact information may be obtained at http www jstor org journals ims html Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission The JSTOR Archive is a trusted digital repository providing for long term preservation and access to leading academic journals and scholarly literature from around the world The Archive is supported by libraries scholarly societies publishers and foundations It is an initiative of JSTOR a not for profit organization with a mission to help the scholarly community take advantage of advances in technology For more information regarding JSTOR please contact support jstor org http www jstor org Tue Mar 4 10 39 53 2008 The Annals of Statistics 1993 Vol 21 No 3 1139 1154 DILATION FOR SETS OF PROBABILITIES Carnegie Mellon University Suppose that a probability measure P is known to lie in a set of probability measures M Upper and lower bounds on the probability of any event may then be computed Sometimes the bounds on the probability of an event A conditional on an event B may strictly contain the bounds on the unconditional probability of A Surprisingly this might happen for every B in a partition a If so we say that dilation has occurred In addition to being an interesting statistical curiosity this counterintuitive phenomenon has important implications in robust Bayesian inference and in the theory of upper and lower probabilities We investigate conditions under which dilation occurs and we study some of its implications We characterize dilation immune neighborhoods of the uniform measure 1 Introduction If M is a set of probability measures then P A sup P A and A inf P A are called the upper and lower probability of A respectively Upper and lower probabilities have become increasingly more common for several reasons First they provide a rigorous mathematical framework for studying sensitivity and robustness in classical and Bayesian inference Berger 1984 1985 1990 Lavine 1991 Huber and Strassen 1973 Walley 1991 and Wasserman and Kadane 1992 l Second they arise in group decision problems Levi 1982 and Seidenfeld Schervish and Kadane 1989 l Third they can be justified by an axiomatic approach to uncertainty that arises when the axioms of probability are weakened Good 1952 Smith 1961 Kyburg 1961 Levi 1974 Seidenfeld Schervish and Kadane 1990 and Walley 1991 l Fourth sets of probabilities may result from incomplete or partial elicitation Finally there is some evidence that certain physical phenomena might be described by upper and lower probabilities Fine 1988 and Walley and Fine 1982 l Good 1966 1974 in response to comments by Levi and Seidenfeld Seidenfeld 1981 and Walley 1991 all have pointed out that it may sometimes happen that the interval A P A is strictly contained in the interval P AIB P A I B for every B in a partition 9 In this case we say that 9 dilates A It is not surprising that this might happen for some B What is surprising is that this can happen no matter what B E 9 occurs Consider the following example Walley 1990 pages 298 2991 Received October 1991 revised September 1992 supported by NSF Grant SES 92 089428 2 upported by NSF Grant DMS 90 05858 AMS 1991 subject classifications Primary 62B15 secondary 62F35 Key words and phrases Conditional probability density ratio neighborhoods contaminated neighborhoods robust Bayesian inference upper and lower probabilities 1139 1140 T SEIDENFELD AND L WASSERMAN Suppose we flip a fair coin twice but the flips may not be independent Let H refer to heads on toss i and T tails on toss i i 1 2 Let M be the set of p 5 all P such that P H l P H i and P H I and H p where 0 I Now suppose we flip the coin Then P H i but 0 P H JH P H H H P H H 1 and o P H IT P H B H P H T I We begin with precise beliefs about the second toss and then no matter what happens on the first toss merely learning that the first toss has occurred causes our beliefs about the second toss to become completely vacuous The important point is that this phenomenon occurs no matter what the outcome of the first toss was This goes against our seeming intuition that when we condition on new evidence upper and lower probabilities should shrink toward each other Dilation leads to some interesting questions For example suppose the coin is tossed and we observe the outcome Are we entitled to retain the more precise unconditional probability instead of conditioning See Levi 1977 and Kyburg 1977 for discussion on this To emphasize the counterintuitive nature of dilation imagine that a physician tells you that you have probability i that you have a fatal disease He then informs you that he will carry out a blood test tomorrow Regardless of the outcome of the test if he conditions on the new evidence he will then have lower probability 0 and upper probability 1 that you have the disease Should you allow the test to be performed Is it rational to pay a fee not to perform the test The behavior is reminiscent of the nonconglomerability of finitely additive probabilities For example if P is finitely additive there may be an event A and a partition 39 such that P A say but P AIB for every B E 39 See Schervish Seidenfeld and Kadane 1984 A key difference however is that nonconglomerability involves infinite spaces whereas dilation occurs even on finite sets dilation cannot be explained as a failure of our intuition on infinite sets A key similarity is that both phenomena entail a difference between decisions in normal and extensive form Seidenfeld 1991 l It is interesting to note that Walley 1991 regards nonconglomerability as incoherent but he tolerates dilation The purpose of