164 lines
10 KiB
Plaintext
164 lines
10 KiB
Plaintext
SUBJECT: THE HILL ABDUCTION CASE FILE: UFO2709
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PART 8
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REBUTTAL: To David Saunders and Michael Peck
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By Carl Sagan and Steven Soter
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Dr. David Saunders last month claimed to have demonstrated the
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statistical significance of the Hill map, which was allegedly found on
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board a landed UFO and supposedly depicted the sun and 14 nearby
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sunlike stars. The Hill map was said to resemble the Fish map -- the
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latter being an optimal two-dimensional projection of a three-
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dimensional model prepared by selecting 14 stars from a positional list
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of the 46 nearest known sunlike stars. Saunders' argument can be
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expressed by the equation SS = Dr -(SF + VP), in which all quantities
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are in information bits. SS is the statistical significance of the
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correlation between the two maps, DR is the degree of resemblance
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between them, SF is a selection factor depending on the number of stars
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chosen and the size of the list, and VP is the information content
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provided by a free choice in three dimensions of the vantage point for
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projecting the map. Saunders finds SS = 6 to 11 bits, meaning that the
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correlation is equivalent to between 6 and 11 consecutive heads in a
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coin toss and therefore probably not accidental. The procedure is
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acceptable in principle, but the result depends entirely on how the
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quantities on the right-hand side of the equation were chosen.
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For the degree of resemblance between the two maps, Saunders claims
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that DR = 11 to 16 bits, which he admits is only a guess -- but we will
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let it stand. For the selection factor, he at first takes SF = log2C =
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37.8 bits, where C represents the combinations of 46 things taken 14 at
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a time. Realizing that the size of this factor alone will cause SS to
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be negative and wipe out his argument, he makes a number of ad hoc
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adjustments based essentially on his interpretation of the internal
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logic of the Hill map, and SF somehow gets reduced to only 3.9 bits.
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For the present, we will let even that stand in order to avoid becoming
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embroiled in a discussion of how an explorer from the star Zeta
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Reticuli would choose to arrange his/her/its travel itinerary --a
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matter about which we can claim no particular knowledge. However, we
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must bear in mind that a truly unprejudiced examination of the data
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with no a priori interpretations would give SF = 37.8 bits.
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It is Saunders' choice of the vantage point factor VP with which we
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must take strongest issue, for this is a matter of geometry and simple
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pattern recognition. Saunders assumes that free choice of the vantage
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point for viewing a three-dimensional model of 15 stars is worth only
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VP = 3 bits. He then reduces the information content of directionality
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to one bit by introducing the "constraint" that the star Zeta Tucanae
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be occulted by Zeta Reticuli (with no special notation on the Hill map
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to mark this peculiarity). This ad hoc device is invoked to explain the
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absence of Zeta Tucanae from the Hill map, but it reveals the circular
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reasoning involved. After all, why bother to calculate the statistical
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significance of the supposed map correlation if one has already decided
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which points represent which stars?
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Certainly the selection of vantage point is worth more than three
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bits (not to mention one bit). Probably the easiest circumstance to
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recognize and remember about random projections of the model in
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question are the cases in which two stars appear to be immediately
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adjacent. By viewing the model from all possible directions, there are
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14 distinct ways in which any given star can be seen in projection as
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adjacent to some other star. This can be done for each of the 15 stars,
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giving 210 projected configurations -- each of which would be
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recognized as substantially different from the others in information
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content. And of course there are many additional distinct recognizable
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projections of the 15 stars not involving any two being immediately
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adjacent. (For example, three stars nearly equidistant in a straight
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line are easily recognized, as in Orion's belt.) Thus for a very
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conservative lower bound, the information content determined by choice
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of vantage point (that is, by being allowed to rotate the model about
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three axes) can be taken as at least equal to VP = log2(210) = 7.7
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bits. Using the rest of Saunders' analysis, this would at best yield SS
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= zero to 4.4 bits -- not a very impressive correlation.
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There is another way to understand the large number of bits involved
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in the choice of the vantage point. The stars in question are separated
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by distances of order 10 parsecs. If the vantage point is situated
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above or not too far from the 15 stars, it need only be shifted by
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about 0.17 parsecs to cause a change of one degree in the angle
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subtended by some pair of stars. Now one degree is a very modest
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resolution, corresponding to twice the full moon and is easily detected
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by anyone. For three degrees of freedom, the number of vantage points
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corresponding to this resolution is of order (10/0.17) cubed ~ (60)
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cubed ~ 2 X 10 to the fifth power, corresponding to VP = 17.6 bits.
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This factor alone is sufficient to make SS negative, and to wipe out
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any validity to the supposed correlation.
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Even if we were to accept Saunders' claim that SS = 6 to 11 bits
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(which we obviously do not, particularly in view of the proper value
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for SF), it is not at all clear that this would be statistically
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significant because we are not told how many other possible
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correlations were tried and failed before the Fish map was devised. For
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comparison, there is the well-known correlation between the incidence
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of Andean earthquakes and oppositions of the planet Uranus. It is
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unlikely in the extreme that there is a physical causal mechanism
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operating here -- among other reasons, because there is no correlation
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with oppositions of Jupiter, Saturn or Neptune. But to have found such
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a correlation the investigator must have sought a wide variety of
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correlations of seismic events in many parts of the world with
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oppositions and conjunctions of many astronomical objects. If enough
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correlations are sought, statistics requires that eventually one will
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be found, valid to any level of significance that we wish. Before we
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can determine whether a claimed correlation implies a causal
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connection, we must convince ourselves that the number of correlations
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sought has not been so large as to make the claimed correlation
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meaningless.
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This point can be further illustrated by Saunders' example of
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flipping coins. Suppose we flip a coin once per second for several
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hours. Now let us consider three cases: two heads in a row, 10 heads in
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a row, and 40 heads in a row. We would, of course, think there is
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nothing extraordinary about the first case. Only four attempts at
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flipping two coins are required to have a reasonable expectation value
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of two heads in a row. Ten heads in a row, however, will occur only
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once in every 2 to the tenth power = 1,024 trials, and 40 heads in a
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row will occur only once every 2 to the fortieth ~ 10 to the twelfth
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power trials. At a flip rate of one coin per second, a toss of 10 coins
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requires 10 seconds; 1,024 trials of 10 coins each requires just under
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three hours. But 40 heads in a row at the same rate requires 4 X 10 to
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the thirteenth power seconds or a little over a million years. A run of
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40 consecutive heads in a few hours of coin tossing would certainly be
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strong prima facie evidence of the ability to control the fall of the
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coin. Ten heads in a row under the circumstances we have described
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would provide no convincing evidence at all. It is expected by the law
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of probability. The Hill map correlation is at best claimed by Saunders
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to be in the category of 10 heads in a row, but with no clear statement
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as to the number of unsuccessful trials previously attempted.
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Michael Peck finds a high degree of correlation between the Hill map
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and the Fish map, and thereby also misses the central point of our
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original criticism: that the stars in the Fish map were already
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preselected in order to maximize that very correlation. Peck finds one
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chance in 10 to the fifteenth power that 15 random points will
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correlate with the Fish map as well as the Hill map does. However, had
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he selected 15 out of a random sample of, say, 46 points in space, and
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had he simultaneously selected the optimal vantage point in three
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dimensions in order to maximize the resemblance, he could have achieved
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an apparent correlation comparable to that which he claims between the
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Hill and Fish maps. Indeed, the statistical fallacy involved in "the
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enumeration of favorable circumstances" leads necessarily to large, but
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spurious correlations.
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We again conclude that the Zeta Reticuli argument and the entire Hill
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story do not survive critical scrutiny.
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Dr. Steven Soter is a research associate in astronomy and Dr. Carl
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Sagan is director of the Laboratory for Planetary Studies, both at
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Cornell University in Ithaca, N.Y.
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**********************************************
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* THE U.F.O. BBS - http://www.ufobbs.com/ufo *
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********************************************** |