Dilation for Sets of ProbabilitiesTeddy Seidenfeld; Larry WassermanThe 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-VThe 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 athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe 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 athttp://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 printedpage of such transmission.The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academicjournals 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 takeadvantage of advances in technology. For more information regarding JSTOR, please contact [email protected]://www.jstor.orgTue Mar 4 10:39:53 2008The 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 prob- ability of A, respectively. Upper and lower probabilities have become increas- ingly more common for several reasons. First, they provide a rigorous mathe- matical 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 probabili- ties [Fine (1988), and Walley and Fine (1982)l. Good (1966, 1974), in response to comments by Levi and Seidenfeld, Seiden- feld (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(AIB)] 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 9occurs. 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. 11391140 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 all P such that P(Hl)= P(H,) = i,and P(HI and H,) =p where 0 Ip 5 $. 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 physi- cian 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) = a, say, but P(AIB,)= $ for every B, E 39. See Schervish, Seidenfeld