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REBUTTAL: To David Saunders and Michael Peck
By Carl Sagan and Steven Soter

Dr. David Saunders last month claimed to have demonstrated the
statistical significance of the Hill map, which was allegedly found on
board a landed UFO and supposedly depicted the sun and 14 nearby
sunlike stars. The Hill map was said to resemble the Fish map -- the
latter being an optimal two-dimensional projection of a three-
dimensional model prepared by selecting 14 stars from a positional list
of the 46 nearest known sunlike stars. Saunders' argument can be
expressed by the equation SS = Dr -(SF + VP), in which all quantities
are in information bits. SS is the statistical significance of the
correlation between the two maps, DR is the degree of resemblance
between them, SF is a selection factor depending on the number of stars
chosen and the size of the list, and VP is the information content
provided by a free choice in three dimensions of the vantage point for
projecting the map. Saunders finds SS = 6 to 11 bits, meaning that the
correlation is equivalent to between 6 and 11 consecutive heads in a
coin toss and therefore probably not accidental. The procedure is
acceptable in principle, but the result depends entirely on how the
quantities on the right-hand side of the equation were chosen.

For the degree of resemblance between the two maps, Saunders claims
that DR = 11 to 16 bits, which he admits is only a guess -- but we will
let it stand. For the selection factor, he at first takes SF = log2C =
37.8 bits, where C represents the combinations of 46 things taken 14 at
a time. Realizing that the size of this factor alone will cause SS to
be negative and wipe out his argument, he makes a number of ad hoc
adjustments based essentially on his interpretation of the internal
logic of the Hill map, and SF somehow gets reduced to only 3.9 bits.
For the present, we will let even that stand in order to avoid becoming
embroiled in a discussion of how an explorer from the star Zeta
Reticuli would choose to arrange his/her/its travel itinerary --a
matter about which we can claim no particular knowledge. However, we
must bear in mind that a truly unprejudiced examination of the data
with no a priori interpretations would give SF = 37.8 bits.

It is Saunders' choice of the vantage point factor VP with which we
must take strongest issue, for this is a matter of geometry and simple
pattern recognition. Saunders assumes that free choice of the vantage
point for viewing a three-dimensional model of 15 stars is worth only
VP = 3 bits. He then reduces the information content of directionality
to one bit by introducing the "constraint" that the star Zeta Tucanae
be occulted by Zeta Reticuli (with no special notation on the Hill map
to mark this peculiarity). This ad hoc device is invoked to explain the
absence of Zeta Tucanae from the Hill map, but it reveals the circular
reasoning involved. After all, why bother to calculate the statistical
significance of the supposed map correlation if one has already decided
which points represent which stars?

Certainly the selection of vantage point is worth more than three
bits (not to mention one bit). Probably the easiest circumstance to
recognize and remember about random projections of the model in
question are the cases in which two stars appear to be immediately
adjacent. By viewing the model from all possible directions, there are
14 distinct ways in which any given star can be seen in projection as
adjacent to some other star. This can be done for each of the 15 stars,
giving 210 projected configurations -- each of which would be
recognized as substantially different from the others in information
content. And of course there are many additional distinct recognizable
projections of the 15 stars not involving any two being immediately
adjacent. (For example, three stars nearly equidistant in a straight
line are easily recognized, as in Orion's belt.) Thus for a very
conservative lower bound, the information content determined by choice
of vantage point (that is, by being allowed to rotate the model about
three axes) can be taken as at least equal to VP = log2(210) = 7.7
bits. Using the rest of Saunders' analysis, this would at best yield SS
= zero to 4.4 bits -- not a very impressive correlation.

There is another way to understand the large number of bits involved
in the choice of the vantage point. The stars in question are separated
by distances of order 10 parsecs. If the vantage point is situated
above or not too far from the 15 stars, it need only be shifted by
about 0.17 parsecs to cause a change of one degree in the angle
subtended by some pair of stars. Now one degree is a very modest
resolution, corresponding to twice the full moon and is easily detected
by anyone. For three degrees of freedom, the number of vantage points
corresponding to this resolution is of order (10/0.17) cubed ~ (60)
cubed ~ 2 X 10 to the fifth power, corresponding to VP = 17.6 bits.
This factor alone is sufficient to make SS negative, and to wipe out
any validity to the supposed correlation.

Even if we were to accept Saunders' claim that SS = 6 to 11 bits
(which we obviously do not, particularly in view of the proper value
for SF), it is not at all clear that this would be statistically
significant because we are not told how many other possible
correlations were tried and failed before the Fish map was devised. For
comparison, there is the well-known correlation between the incidence
of Andean earthquakes and oppositions of the planet Uranus. It is
unlikely in the extreme that there is a physical causal mechanism
operating here -- among other reasons, because there is no correlation
with oppositions of Jupiter, Saturn or Neptune. But to have found such
a correlation the investigator must have sought a wide variety of
correlations of seismic events in many parts of the world with
oppositions and conjunctions of many astronomical objects. If enough
correlations are sought, statistics requires that eventually one will
be found, valid to any level of significance that we wish. Before we
can determine whether a claimed correlation implies a causal
connection, we must convince ourselves that the number of correlations
sought has not been so large as to make the claimed correlation
meaningless.

This point can be further illustrated by Saunders' example of
flipping coins. Suppose we flip a coin once per second for several
hours. Now let us consider three cases: two heads in a row, 10 heads in
a row, and 40 heads in a row. We would, of course, think there is
nothing extraordinary about the first case. Only four attempts at
flipping two coins are required to have a reasonable expectation value
of two heads in a row. Ten heads in a row, however, will occur only
once in every 2 to the tenth power = 1,024 trials, and 40 heads in a
row will occur only once every 2 to the fortieth ~ 10 to the twelfth
power trials. At a flip rate of one coin per second, a toss of 10 coins
requires 10 seconds; 1,024 trials of 10 coins each requires just under
three hours. But 40 heads in a row at the same rate requires 4 X 10 to
the thirteenth power seconds or a little over a million years. A run of
40 consecutive heads in a few hours of coin tossing would certainly be
strong prima facie evidence of the ability to control the fall of the
coin. Ten heads in a row under the circumstances we have described
would provide no convincing evidence at all. It is expected by the law
of probability. The Hill map correlation is at best claimed by Saunders
to be in the category of 10 heads in a row, but with no clear statement
as to the number of unsuccessful trials previously attempted.

Michael Peck finds a high degree of correlation between the Hill map
and the Fish map, and thereby also misses the central point of our
original criticism: that the stars in the Fish map were already
preselected in order to maximize that very correlation. Peck finds one
chance in 10 to the fifteenth power that 15 random points will
correlate with the Fish map as well as the Hill map does. However, had
he selected 15 out of a random sample of, say, 46 points in space, and
had he simultaneously selected the optimal vantage point in three
dimensions in order to maximize the resemblance, he could have achieved
an apparent correlation comparable to that which he claims between the
Hill and Fish maps. Indeed, the statistical fallacy involved in "the
enumeration of favorable circumstances" leads necessarily to large, but
spurious correlations.

We again conclude that the Zeta Reticuli argument and the entire Hill
story do not survive critical scrutiny.


Dr. Steven Soter is a research associate in astronomy and Dr. Carl
Sagan is director of the Laboratory for Planetary Studies, both at
Cornell University in Ithaca, N.Y.

Jerry
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