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SCIENTIFIC CORRESPONDENCE

= stretch(pref(X)) on which his values the prior for most reasonable prior dis­ When the lensing has an ellip­ prob(X) obey prob (A) + prob(notA) = tributions when a useful amount of tical surface density, the 1, prob(A,B) = prob(A) prob (B given observational or experimental data is degrades into four images that are lo­ A). available. cated symmetrically along the two prin­ These rules define the only general It is possible that some non-bayesians cipal axes7 as observed8,9 in the case of calculus for patterns containing up to (anti-bayesians?) have a conceptual diffi­ Q2237+0305, where the lensing galaxy three bits. Larger patterns can be built culty with the idea of the probability of a has a zL =0.0394 and a rotational up by investigating their bits sequen­ probability, which could lead to the velocity of - 260 km s-1 (ref. 8), and the tially, during which the rules remain above approach being ignored. Fisher's images have z5 = 1.695. consistent as well as being demanded. misinterpretation of what to do about The predicted angular separation be­ Thus a generally applicable calculus can ignorance of the values of probabilities tween opposite images (roughly the exist, proving existence and uniqueness. in a bayesian argument, is at the root of diameter of the ring) is A0=1.85", in Cox's stretched values are probabili­ many arguments against the use of baye­ very good agreement with the ground­ ties, and the product rule immediately sian methods in expert systems, and as based observations8•9 (A0=1.75"±0.10"), gives Bayes' theorem, the required tool we see, this argument is invalid. and with recent observations from the for inference. RON lARHAM (~0=1.78"± Edwards discards axiom 1, because he 37 Sancroft Road, 0.05"). The system Q2237+0305 was has improperly started in an infinite Harrow Weald, highly advertised as an evidence that a space where he cannot recover a single Middlesex HA3 7NU, UK high-redshift quasar lies very near the element by double negation. But in my 1. Edwards, A. W. F. Nature 352, 386--387 (1991). 2. Howson, C. & Urbach, P. Nature 350, 371- 374 (1991). centre of a low redshift galaxy. How­ finite computer, _n_o patterns occupy less ever, this would produce an angular than 1 part in 2b,lhon of the space. Nega­ separation between opposite images tion is the complement operator, which Quasar redshifts smaller than that observed by at least remains reversible for any finite compu­ SIR - Evidence in support of the cos­ three orders of magnitude. ter: thus 2billion _ (zbillion _ 1) = 1. mological origin of quasar redshifts, One mail also compare the observed Bayesian probability calculus is rock which comes mainly from the absorption time delay o,u of 410±10 days between solid and, as Edwards admits, "statistical features in quasar spectra does not seem the two images~f Q0957+561 at angular theory and statistical practice [can there­ to have settled the "quasar ~stances 10 .J =5.24"±0.5" and by] be greatly simplified" . controversy"1, as evident from the 16 al =l.00"±0.5" from the centre of the JOHN SKILLING 'Hypothesis' by Arp et al. 2 , who argue lensing galaxy at redshift Z! = 0.36 and Dept of Applied Mathematics and that the lie at much smaller the predicted t~e delay1 AtA 8 = Theoretical Physics, distance than their redshift distance, and 41r(l+zd(l0 Al-JS sl)(ow'c)2(Dodc). A University of Cambridge, thus that quasar redshifts are largely recent measurement of the line-of-sight Cambridge CB3 9EW, UK intrinsic in origin and not cosmological. velocity dispersion in the fiant galaxy 1 . Edwards, A. W. F. Nature 352, 386-387 (1991). However, recent observations on gravi­ gave13 or (303±50) km s- . Using the 2. Cox, R. T. Am. J. Phys. 14, 1--13 (1946). tational lensing of quasar images provide 1 1 3. Cox, R. T. The Algebra of Probable Inference (Johns best value H0 = 67± 8 km s- M pc- for Hopkins, Baltimore, 1961). clear-cut evidence that quasars lie at the Hubble parameter, found in a recent their redshift distances. review14, the predicted time delay is SIR - The example of Fisher's quoted Einstein's theory of 416±130 days, in agreement with the by Edwards1 is a straw man to discredit predicts that a galaxy with a radially observed time delay. However, if the the opposing view. Only in this type of symmetric surface density that happens quasar lies in or near the lens, the argument would a bayesian practitioner to lie on or very near the line-of-sight to deflection angle, and consequently the confronted with the black mouse give a a distant quasar, forms in the sky a ring expected time delay, would be smaller probability of the mouse being homozy­ image3 (Einstein ring) of the quasar with by more than four orders of magnitude gous of one half when s/he knew nothing an angular diameter4 A6= 41r(DLs/ than the observed values. about any possible relationship between D05)(vci,/c)2, where Dos and DLS are the In all other known cases of gravita­ colour and genotype. In practice, a real angular distances from the observer to tional lensing of quasar images by galax­ bayesian practitioner would observe that the source (quasar) and from the lens ies (and clusters of ) the angular s/he knew nothing about the genotype (galaxy) to the source, respectively, and separations between the multiple images and so would assign a uniform prior llcir is the rotational velocity in the lens­ are of similar magnitude15. Moreover, distribution over the interval 0,1 to the ing galaxy at the bending point. Such an the number of lensed quasars that are unknown probability. Informally, this Einstein ring, MG1654+ 1306, of a radio observed is that expected4 if the quasars means that the bayesian thinks that the lobe of a quasar at redshift z5 = 1. 75 was lie at their redshift distance. probability of the mouse being homo­ discovered by Langston et al. 5•6 . It is ARNON DAR zygous lies between O and 1, while the formed by a bright at Department of Physics, knowledgeable scientist believes it is redshift zL=0.254 and it has an angular Israel Institute of Technology, one-half. The seems to be a satisfactory diameter of A0=1.97"±0.04". If the Technion City, 32000 Haifa, Israel situation, given their respective states of quasar and the galaxy lie at their redshift 1. Arp, H. C. Quasars, Redshifts And Controversies (lnter- knowledge and belief. distances then DLs/D05= 0.73 and the steller Media, Berkeley 1987). 2. Arp, H. C. et al. Nature 346, 807 (1990). The prior distribution used is subjec­ expected angular diameter of the ring 3 . Einstein, A. Science 84, 506 (1936 ). tive, as stated by Howson and Urbach in from the estimated5 circular velocity in 4 . Turner, E. L., Ostriker J. P. & Gott Ill, J. R. Astrophys, J. 1 284, 1 (1984). their Commentary2; however most statis­ the lensing galaxy (vci,= 330±20 km s· ) 5. Langston, G. I. et at. Astron. J. 97, 1283 (1989). tical procedures contain subjective or is A0= 2.09" ±0.27", in good agreement 6 . Langston, G. I. et al. Nature 344, 43 (1990). 7. Blandford . R. D. et at. Science 245, 824 (1988). arbitrary elements (the significance with the observed value. On the other 8. Schneider, D. P. et al. Astron. J. 95, 1619 (1988). levels for rejecting hypotheses, for ex­ hand, if the quasar lies in or near the 9. Yee, H. K. C. Astron. J. 95. 1331 (1988). ample). Here at least the assumptions lensing galaxy (that is, if DLs ,s; 50 kpc) 10. Vanderriest, C. et al. Astron. Astrophys. 215, 1 (1989). 11. Schield, R. E. Astrophys. J. 100, 1771 (1990). have to be stated explicitly. In practice, then D0 s= DoL, DLslD0 s ,s; 50 kpc/ 12. Borgeest, U. Astrophys. J. 309, 467 (1986). of course, this is not a real difficulty as DoL ,s; 10-4 and A0 would be less that 2 13 . Rhee, G. Nature (in the press). 14. van den Bergh, S. Astron. Astrophys. Rev. 1, 111 posterior distributions generally show x 10-4 arcs, which vastly contradicts the (1990). only weak dependence on the form of observations. 15. Blandford, R. D. Q. JI R. astr. Soc. 31, 305 (1990). 708 NATURE · VOL 353 · 24 OCTOBER 1991 © 1991 Nature Publishing Group