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Bode's Law and the Processes of Statistical and Scientific Inference
I. J. Good
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  and that on the second one was in the press.  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  that a galactic government already exists, in which case the world is a zoo.  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. )
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 , 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 + 2nb 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 n2|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 , 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 + 2n3 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
with similar definitions for L2 and L3. 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.
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.