SUBJECT: THE HILL ABDUCTION CASE FILE: UFO 2707 PART 6 ---------------------------------------------------------------------- REPLY: By David R. Saunders Last month, Steven Soter and Carl Sagan offered two counterarguments relating to Terence Dickinson's article, "The Zeta Reticuli Incident" (ASTRONOMY, December 1974). Their first argument was to observe that the inclusion of connecting lines in certain maps "is what a lawyer would call 'leading the witness'." This was used as the minor premise in a syllogism for which the major premise was never stated. Whether we should consider "leading the witness" a sin or not will depend on how we conceive the purpose of the original article. The implied analogy between ASTRONOMY magazine and a court of law is tenuous at best; an expository article written for a nonprofessional audience is entitled, in my opinion, to do all it can to facilitate communication -- assuming that the underlying message is honest. Much of what we call formal education is really little more than "leading the witness", and no one who accepts the educational goals objects very strongly to this process. In this context, we may also observe that Soter's and Sagan's first argument provides another illustrative example of "leading the witness"; the argument attacks procedure, not substance -- and serves only to blunt the reader's possible criticism of the forthcoming second argument. This paragraph may also be construed as an effort to lead the witness. Once we have been sensitized to the possibilities, none of us needs to be further misled! The second argument offered by Soter and Sagan does attack a substance. Indeed, the editorial decision to publish the original article was a responsible decision only if the issues raised by this second line of possible argument were fully considered. Whenever a statistical inference is made from selected data, it is crucial to determine the strenuousness of that selection and then to appropriately discount the apparent clarity of the inference. By raising the issue of the possible effects of selection, Soter and Sagan are right on target. However, by failing to treat the matter with quantitative objectivity ( by failing to weigh the evidence in each direction numerically, for example), they might easily perform a net disservice. In some situations, the weight of the appropriate discount will suffice to cancel the clarity of a proposed inference -- and we will properly dismiss the proposal as a mere capitalization on chance, or a lucky outcome. (It is abundantly clear that Soter and Sagan regard the star map results as just such a fortuitous outcome.) In some other situations, the weight of the appropriate discount may be fully applied without accounting for the clarity of the inference as a potentially valid discovery. For example, if I proposed to infer from four consecutive coin tosses observed as heads that the coin would always yield heads, you would properly dismiss this proposal as unwarranted by the data. However, if I proposed exactly the same inference based on 40 similar consecutive observations of heads, you would almost certainly accept the inference and begin looking with me for a more systematic explanation of the data. The crucial difference here is the purely quantitative distinction between 4 and 40; the two situations are otherwise identical and cannot be distinguished by any purely qualitative argument. When Soter and Sagan use phrases such as "some subset that resembles", "free also to select the vantage point", "simple matter to optimize", and "freedom to contrive a resemblance", they are speaking qualitatively about matters that should (and can) be treated quantitatively. Being based only on this level of argument, Soter's and Sagan's conclusions can only be regarded as inconclusive. A complete quantitative examination of this problem will require the numerical estimation of at least three factors, and their expression in a uniform metric so that wee can see which way the weight of the evidence is leaning. The most convenient common metric will be that of "bits of information", which is equivalent to counting consecutive heads in the previous example. One key factor is the degree of resemblance between the Hill map and the optimally similar computer-drawn map. Precisely how many consecutive heads is this resemblance equivalent to? A second key factor is the precise size of the population of stars from which the computer was allowed to make its selection. And a third key factor is the precise dimensionality of the space in which the computer was free to choose the best vantage point. If the first factor exceeds the sum of the other two by a sufficient margin, we are justified in insisting on a systematic explanation for the data. The third factor is the easiest to deal with. The dimensionality of the vantage-point space is not more than three. A property of the metric system for weighing evidence is that each independent dimension of freedom leads us to expect the equivalent of one more consecutive head in the observed data. Three dimensions of freedom are worth exactly 3.0 bits. In the end, even three bits will be seen as relatively minor. The second factor might be much larger than this, and deserve relatively more discussion. The appropriate discount for this selection will be log2C, where C is the number of distinct combinations of stars "available" to the computer. If we were to agree that C must represent the possible combinations of 46 stars taken 14 at a time, then log2C would be 37.8 bits; this would be far more than enough to kill the proposed inference. However, not all these combinations are equally plausible. We really should consider only combinations that are adjacent to one another and to the sun, but it is awkward to try to specify exactly which combinations these are. The really exciting moment in working with these data came with the realization that in the real universe, our sun belongs to a closed cluster together with just six of the other admissible stars -- Tau Ceti, 82 Eridani, Zeta Tucanae, Alpha Mensae, and Zeta 1 and Zeta 2 Reticuli. The real configuration of interstellar distances is such that an explorer starting from any of the seven should visit all of them before venturing outside. If the Hill map is assumed to include the sun, then it should include the other members of this cluster within an unbroken network of connections, and the other connected stars should be relatively adjacent in the real universe. Zeta Reticuli occupies a central position in all of the relatively few combinations that now remain plausible. However, in my opinion, the adjacency criteria do leave some remnant ambiguity concerning the combination of real stars to be matched against the Hill map -- but only with respect to the region farthest from the sun. The stars in the closed cluster and those in the chain leading to Gliese 67 must be included, as well as Gliese 86 and two others from a set of five candidates. Log2C for this remnant selection is 3.9 bits. we must also notice that the constraint that Zeta Tucanae be occulted by Zeta Reticuli reduces the dimensionality of the vantage-point space from 3.0 to 1.0. Thus, the sum of factors two and three is now estimated as only 4.9 bits. The first factor is also awkward to evaluate -- simply because there is no standard statistical technique for comparing points on two maps. Using an approximation based on rank-order correlation, I've guessed that the number we seek here is between 11 and 16. (This is the result cited by Dickinson on page 15 of the original article.) Deducting the second and third factors, this rough analysis leaves us with an empirical result whose net meaning is equivalent to observing at least 6 to 11 consecutive heads. (I say "at least", because there are other factors contributing to the total picture -- not discussed either by Dickinson or by Soter and Sagan -- that could be adduced to enhance this figure. For example, the computed vantage point is in good agreement with Betty Hill's reported position when observing the map, and the coordinate system implicit in the boundaries of the map is in good agreement with a natural galactic coordinate system. Neither have we discussed any quantitative use of the connections drawn on the Hill map, which were put there in advance of any of these analyses.) In the final interpretation, it will always be possible to argue that 5 or 10 or even 15 bits of remarkable information simply isn't enough. However, this is a matter for each of us to decide independently. In deciding this matter, it is more important that we be consistent with ourselves (as we review a large number of uncertain interpretations of data that we have made) than that we be in agreement with some external authority. I do believe, though, that relatively few individuals will continue a coin-tossing match in which their total experience is equivalent to even six consecutive losses. In scientific matters, my own standard is that I'm interested in any result that has five or more bits of information supporting it -- though I prefer not to stick my neck out publicly on the basis of less than 10. Adhering to this standard, I continue to find the star map results exceedingly interesting. Dr. David R. Saunders is a Research Associate at the University of Chicago's Industrial Relations Center. ********************************************** * THE U.F.O. BBS - http://www.ufobbs.com/ufo * **********************************************