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### Bode's Law and the Processes of Statistical and Scientific Inference

#### I. J. Good

#### 1968

At the Spring 1968 meetings in Blacksburg I gave an invited paper, sponsored by the Institute of Mathematical Statistics, The American Statistical Association, and the Biometric Society, on statistical and scientific inference. Instead of giving a survey in general terms I decided to consider four specific examples, namely (i) a Bayesian significance test for multinomial distributions, (ii) a sharpened form of Ockham's razor, (iii) the Bode-Titius law, and (iv) a test for near-rationality of numbers. These topics are distinct but related. The treatment of the first three was Bayesian, but not the last. I did not say much about the first two topics since my work on the first one had already been published [1] and that on the second one was in the press. [2] The first and third of the topics made use of computer programs that were written at the Atlas Computer Laboratory by Alan Tritter, Tim Gover, Esther Litherland, Peter Hallowell, and myself.

It is rather interesting that the method of treating the first topic depended on the use of an electronic computer although in principle the problem can be regarded as philosophical. Another example of this kind of thing is the use of computational linguistics for attacking problems in the philosophy of language. I shall say no more about the first two topics in this account.

The main topic is the Bode-Titius law, or *Bode's law* for
short (see the Appendix). This I treated in a thoroughly subjective manner,
but not merely by means of a snap judgment which is the method that has been
used in the past. Various astronomers have contradicted one another,
and sometimes an astronomer has contradicted himself. All judgments are
subjective but some are more subjective than others.

Bode's law is a piece of physical numerology that provides an
approximate formula for the semi-major axes of the orbits of the planets.
It breaks down for the planet closest to the sun (Mercury), when expressed
honestly, and also for the two furthest out (Neptune and Pluto). Since the orbit of
Pluto passes inside that of Neptune it is usually thought that Pluto was
once a satellite of Neptune. Since Neptune threw Pluto away it is reasonable
for us to do the same. That is, we can reasonably restrict our attention to the
bodies that were presumably planets from the start. My excuse for ignoring Mercury
and Neptune is that, on a nebular hypothesis, it is not too unreasonable to assume
that the edges of the proto-planetary nebula were somewhat irregular. Moreover in the
earlier millenia of the solar system the sun might very well have been far from
spherical and this would have tended to upset the orbit of Mercury. Further
evidence for this is that Mercury has very high eccentricity and also a high
angular deviation from the plane of the ecliptic.
For the privilege of ignoring Mercury, Neptune and Pluto, I think it is reasonable
to concede a factor of 5 on the Bayes factor in favour of the hypothesis
that Bode's law is *causal*.

By saying that a law is *causal* I mean that it must have some
fairly simple physical explanation, whether that explanation is known or not.
This is of course not the same as saying that the law is *correct*.
For example, the inverse square law is causal, in that it can be explained by
General Relativity, but it is known not to be quite correct.
A correct law explains, a causal one demands an explanation.
A causal law can be used for prediction, but is not as reliable as a *correct* law.
For example, if Bode's law is causal, then it predicts something about
other planetary systems. Of course there are no clear-cut distinctions between
correct, approximately correct, and causal laws.

The conclusion I reached was that my subjective Bayes factor in favour of
Bode's law being not a mere accident is between 300 and 700. More tentatively, the *law*
seems to provide a factor of about 20 in favour of a non-cataclysmic origin of the
solar system and hence in favour of there being an extremely large number of
planetary systems in our galaxy. The urgent practical implication is an increase
in the *evolving* probability [2] that a galactic government already exists,
in which case the world is a zoo. [3] This justifies the presentation of the paper to
the Biometric Society. (An *evolving* probability is one that changes in the
light of reasoning alone, without the intervention of new empirical evidence.
It is used, for example, by all chess players, at least implicitly. [4])

The main connection between topics (i) and (iii) is that both use Bayesian methods, and both make use of the log-Cauchy distribution which is a very convenient distribution to use when we wish to approximate the improper Jeffreys-Haldane distribution, while maintaining propriety.
The connection between topics (ii) and (iii) is that the *Sharpened Razor* is
required for the detailed logic in the discussion of Bode's law.

There have been various refinements in Bode's one of which is due to Miss Blagg. (See Ref. 5, drawn to my attention by Dr. R. F. Churchhouse.) I did not deal in complete detail with Blagg's law in my paper because it is very complicated and difficult to evaluate.

There is another law mentioned by Chambers [6], p.67. He says that various dabblers
in astronomy have suggested modifications of Bode's law and specifically mentions
one that, in 1889, applied not too badly to the eight known satellites of Saturn.
It was ungracious of Chambers not to mention the name of the originator,
and I call the *law* Dabbler's law. Like Bode's law, it is of the
form a + 2^{n}b and applied for n = 1, 2, . . ., 8. I now find that Dabbler's
law works well for n = 10 also, and I predict that a further satellite, or a
belt of quasi-asteroids, will fit the formula well for n = 9. Also it seems to me that
Dabbler's law is further support for the causal nature of Bode's law,
but further programming would be required for an evaluation of the support.
We must lose a little support for not having a similar law for Neptune,
but surely this is more than compensated by Dabbler's law.

Part of the argument concerning Bode's law is related to the
so-called *commensurabilities* that are found between the periods of
some of the satellites of some of the planets. In order to apply a statistical test
to the data it is convenient to have a measure of the *degree of irrationality*
of a real number, x. The measure first chosen, for reasons connected with the theory of
numbers, was M(x) = min n^{2}|x - m/n|. Note that M(1/x) = M(x). This leads to an
approach that seems to me to be more satisfactory than the approach of
Roy and Ovenden [7], but I cannot give further details in this account.

#### Appendix. A slight modification of Bode's law and the calculations done on Atlas for its evaluation

Bode's law states that the *mean distances* of the planets from the
Sun are roughly proportional to 4 + 2^{n}3 where n = -infinity, 0, 1, 2, etc.,
but the first of these values is artificial, and the fair value to associate with
Mercury is n = -1. A modified law, B, is considered in this paper in which the
approximation is assumed only for seven consecutive values of n. If B is *causal*
then so is Bode's law, but a penalty must be accepted for not including
Mercury and Neptune. In order to evaluate B by Bayes' theorem some further assumptions
must be made, and this introduces some further inevitable subjectivity into the evaluation.
Without going into details I mention here only that the evaluation depends on the
evaluation of some triple integrals of the form

where

with similar definitions for L_{2} and L_{3}. Although the integrand
is always positive the numerical integration is somewhat tricky: it is a triply
infinite integral which also needs care near rho = 0.

#### References

1. Good, I. J. 'A. Bayesian significance test for multinomial distributions', J. Roy. Statist. Soc. B, 29 (1967), 399-431.

2. Good, I. J. 'Corroboration, explanation, evolving probability, simplicity, and a sharpened razor', Brit. J. Phil. Sc., 19 (1968), 123-143.

3. Good, I. J. 'Life outside the earth', The Listener, 73 (3rd June, 1965), 815-817.

4. Good, I. J. 'A five-year plan for automatic chess', in Machine Intelligence, II (ed. by E. Dale and D. Michie, Edinburgh, Oliver and Boyd, 1968),89-118.

5. Roy', A. E. 'Miss Blagg's formula', J. Brit. Astron. Assn. 63 (1953), 212-215.

6. Chambers, George F. (1889), A Handbook of Descriptive and Practical Astronomy, I. The Sun, Planets, and Comets. (Oxford: Clarendon Press).

7. Roy, A. E. and Oven den, M. W., (1954), 'On the occurrence of commensurable mean motions in the solar system', Monthly Notes Roy. Astron. Soc. 114, 232-241.