Volume 16 Number 81 Produced: Thu Nov 24 9:16:57 1994 Subjects Discussed In This Issue: Is God a Bayesian - II (hopefully the last) [Meylekh Viswanath] Sources re Osteoporosis [Richard Schwartz] Tetrahedron and Modern Orthodoxy [Stan Tenen] ---------------------------------------------------------------------- From: Meylekh Viswanath <pviswana@...> Date: Tue, 22 Nov 1994 20:48:10 EST5EDT Subject: Is God a Bayesian - II (hopefully the last) The problem with some Bayesians (not all; after all I myself am a Bayesian) is their concentrated (and almost degenerate priors), which leads them to ignore evidence. A taste of this is perhaps visible in the above misspelling of my name. Presumably, Mechy has heard of the Indian name Vishnu and hence concluded that my name must be Vis_na_wath and not Vis_wa_nath. His strong prior beliefs led him to ignore the evidence (i.e. my name as given in my posting). > 1. Considering the problem of a shor mooad, Meylekh's suggestion that > perhaps in this instance He was working from the sample mean doesn't > work since that would in fact leave us with a shor tam, not a (3-gore) > shor mooad at all. In any event, three points is a rather poor base > from which to talk of means, variances, and such like. Most of the > useful frequentist theorems work well only for large n. I have absolutely no idea what Mechy is talking about here. Let me try and explain in greater detail what my original point was. Suppose we need to know the mean of a certain distribution. In general, if we know only, say, that the distribution is normal, and do not know its mean and variance, a sample of three points would be quite insufficient for a classical statistician to infer the mean with much confidence. However, if the statistician knew that the variance was small, then a small sample, even of size 3, would be enough. To give an extreme example, suppose we wish to know the average number of pages in a certain statistics textbook in a bookstore. If we are confident that all the copies of that textbook in the store have the same number of pages (variance = 0), we would simply take a sample of size one, and infer from that small sample, the average number of pages in copies of that textbook. There is nothing in classical statistics that prevents this procedure. Classical statistics certainly does not prevent us from taking certain kinds of information into account in inferring things from a given sample. For example, most statistics textbooks of the classical variety would give an example of statistical hypothesis testing by assuming something about the parent distribution, e.g. that it is normal. Clearly, this is information. What classical statistics does not like is the notion that there are probability distributions for population parameters, which clearly are not random quantities. Thereby, classical statisticians lose out on a great method of incorporating prior information that the researcher might have. In our case of a shor tam and a shor muad, let's assume that Pt(x) = Prob(# gorings=x|shor tam; alpha) and Pm(x) = Prob(# gorings=x|shor muad; alpha), where alpha is some other conditioning parameter, give the conditional probability functions for tam and muad oxen. The unconditional distribution of the # of gorings can be obtained by integrating over the distribution of alpha. Assume that these unconditional distributions of the # of gorings are normally distributed with given means Mm and Mt for the muad and tam oxen. For convenience, assume that the variances are known and are equal to some small number. Let the hypothesis that the shor is tam be the null (this is natural, given the principle 'ha motsi mekhaveyro, alav ha raaya'). A classical statistician would have no problem coming up with a test of the null hypothesis that a given shor is tam, with the alternative hypothesis being that the shor is muad (perhaps a likelihood ratio test). It is quite conceivable that an observation of 3 gorings in three tries would result in a test statistic that would reject the null in favor of the alternative, for given levels of type I and type II error. > 2. It is also not true that with only three points God would need to > know the prior distribution pretty well (though of course He would. It > seems faintly sacriligious to attribute to God an uninformed prior.) to > form a sharp posterior conclusion. The trick is in the likelihood > function. It is clear to me that when considering shors, as we all often > do, God meant us to steer clear of those otherwise popular binomials and > stick to more rapidly convergent likelihood functions. bideedee hava > oovda, and I can testify that things will shapen up pretty smartly even > with sparse data if the likelihood is carefully chosen. That's the same thing that I said upstairs, clothed in Bayesian language. > 3. The question of choice of priors is important. Non-withstanding my > above claim that I can whip even relatively uniform priors into > reasonable posterior shape early on, there is no a priori reason why the > priors should not be reasonably "informed'. After all, He has lots of > information at His disposal, and who are we to questions His Judgement > (-al distribution assignments). Indeed, choice of priors may in some > cases lead bayesians to radically different conclusions than > frequentists. c.f. Lindley paradox. As was pointed out in a previous m.j. by another poster, it is very strange to talk about God being a statistician at all, of whatever persuasion. The only way to make sense of this discussion, therefore, is not in terms of what God is and isn't but rather in terms of what model it is, that halakha follows. > 4. Finally, in contradistinction to unconvincing frequentist contortions > to demonstrate relevance to these onesy-twosy data bases, I'd like to > emphasize the fundamental naturalness of the Bayesisn paradigm to such > issues as chazaka and sparse data sets. The bayesian assumes we start > off with some initial model, or picture of the world. This shor is a > tam, that physics model is true, etc. Our initial model, or judgement, > may be based on accumulated wisdom/data to date, or even on no data at > all. By investing additional effort (perform an experiment, pay a spy, > etc.) we may acquire new data, reducing our uncertainty, and allowing us > to refine our initial judgement. The quantitative Bayesian methodology > then instructs us us precisely how, and to what degree to modify our > original hypothesis as data is serially accumulated - even at low rates. > This of course is how we all really do it. We observe the shor take an > additional pot shot or two and begin to suspect that we have a four > footed ax murderer on our hands. This revision of of judgement and > incremental fine tuning as incremental information is acquired thus > precisely mimics the real world serial accumulation of experience and > formation of updated judgements. If we were arguing whether the Bayesian approach is better or the classical approach, we could discuss the above question, and a lot of other questions that statisticians for lack of practical work spend their time discussing. However, that is not the point under discussion here. The question is whether we can infer from the halakha that "a shor being observed to gore on three consecutive occasions renders it a muad" that the principle relies on Bayesian statistical principles. I think it is pretty obvious that one cannot make any such inference. > So, while I make no attempt to deny that frequentists have made many > glorious contributions to civilization, such as the central limit > theorem and political pollsters, my faith that God is a Bayesian remains > unshaken. Mechy has finally got to the crux of the question: it is a matter of faith. And as one Bayesian to another, I hope he never loses the faith, while at the same time desisting from trying to convince people on _logical_ bases that halakha follows Bayesian rules. P.V. Viswanath, Rutgers University Graduate School of Management, 92 New St, Newark NJ 07102 Tel: (201) 648-5899 Fax: (201) 648-1233 email: <pviswana@...> ---------------------------------------------------------------------- From: Richard Schwartz <RHSSI@...> Date: Wed, 23 Nov 94 13:56:12 EST Subject: Sources re Osteoporosis In Volume 16, No. 74, Doni Zwotofsky asked for sources for the assertion in my previous posting that it is excessive protein in the diet, rather than a lack of calcium, that is the main cause of osteoporosis. So. here are some sources: 1. Johnson, S., et al, "Effect of Level of Protein Intake on Urinary and Fecal Calcium and Calcium retention . . .", Journal of Nutrition, 100:1425, 1970. 2. Allen, L., et al, "Protein Induced Hypercaluria: A Longer Term Study<" Ameri can Journal of Clinical Nutrition, 32:741, 1979. 3. Altchuler, S., "Dietary Protein and Calcium Loss: A Review", Nutritional Research, 2:193, 1982. 4. Hegsted, M., "Urinary Calcium and Calcium Balance in Young Men as Affected by Level of Protein and Phosphorus Intake", Journal of Nutrition, 111:553, 1981 5. Walker, R., "Calcium Retention in the Adult Human Male as Eggected by Protein Intake," Journal of Nutrition, 102:1297, 1972. As can be seen, these studies go back in some cases over 20 years, which makes current misconceptions even more scandalous, in view of the great harm that is being done by high protein diets. As I previously indicated, the great loss of calcium in urine can easily be checked by physicians. Happy Chanukah, Richard ---------------------------------------------------------------------- From: Stan Tenen <meru1@...> Date: Thu, 10 Nov 1994 11:40:01 -0800 Subject: Tetrahedron and Modern Orthodoxy Warning: non-technical persons read this at their own risk. <big smile> In m-j 16 No. 44 Jonathan Rogawski states that my tetrahedron scheme is not quite correct. I think that may be true in a sense. It is very hard to be specific, describe geometric forms, and be (at least reasonably) understandable in a short posting. It is also important to remember that mathematical qualities, such as symmetry, may have been measured differently in the ancient world than they are now. There may have been different conventions about how to look and what to look for. The symmetry that a form appears to have when projected and viewed in one particular dimension can be different - or can appear different - than when the same form is viewed in a different dimension. A cube can look like a hexagon in 2-D projection, for example. Actually, I was asked a similar question via email. I will post an edited version of my email response below: Thanks for taking my posting seriously enough to think about it. I appreciate that. There is more than one way to look at a tetrahedron. If you look at a 2-D projection across the centers of opposite edges of a _transparent_ 3-D tetrahedral frame, you will see a square shaped outline with BOTH diagonals showing. Thus, the tetrahedron viewed as a "square with diagonals" exhibits 4-fold symmetry - when it is considered as a transparent frame. Generally this is not what we do now. That is why Jonathan Rogawski and the previous emailer are correct also. Lest you think that this is just so much "fudge", consider the following: The tetrahedron is formally represented by the E6 symmetry group, and the E6 group, in turn, can be represented by the 3,4 torus knot. The 3,4 knot also exhibits a kind of 3-fold and 4-fold symmetry: 3-layers with 4-lobes or 4-layers with 3-lobes. The 3,4 torus knot is extraordinarily interesting. A tetrahelical column consisting of exactly 33-tetrahedra (ask about the 30 vs 33 problem, a la Buckminster Fuller) has its 3-edges twisted in a braid that when connected end to end produces the 3,4 knot. Connecting the braid end to end is the equivalent of identifying the insphere and the outsphere of the "Dirac String Trick" (which illustrates the quaternions in the motions of the human hand and in spin 1/2 fermions - supercomputer video of part of this is available, just ask) as a single hypersphere. It is a radius of this hypersphere, pointing in one of the 27-directions specified by the general cubic equation, that determines the shapes and meanings of the Hebrew letters. And the 33-tetrahedron column makes exactly one full turn over its length. It is the UNIT column. (In my theory, nearly every major element in Kabbalistic models represents an aspect of Unity or Wholeness because that is a mathematical way to help to illustrate the exquisite Unity and Wholeness of Hashem.) Each of the 3-ribbons (of triangles) that makes up the (twisted) faces of the tetrahelical column displays 22-triangles - one each for each of the 22-letters of the alphabet without finals. Then the Unit column produces 3-unit alphabet strings. In the email query I was also asked about the relationship between the tetrahedron and discussion above and the initial Bet of B'reshit: The Bet of B'reshit defines the first logical distinction possible, the distinction between inside and outside. This Bet is the "house of creation". The next several letters of B'reshit tell us more about this "house." B'reshit: Bet-Resh-Aleph-Shin-Yod-Tov It is "woven of net" Reshet, net, (Resh-Shin-Tov). It starts with the mind, Resh, "head" which holds ALL, Aleph. It contains the "fire of consciousness" Aish (Aleph-Shin) formed of "seeing" RA (Resh-Aleph) And all of the preceding is "housed" (Bet) in a tetrahedral tent, the "six" -sided "thorn", Shith (Shin-Yod-Tov.) Thus the Bet specifies the archetypal container and it is further refined as a Shith ("six" or "thorn"), which in modern language may be what we call a tetrahedron because a tetrahedron is shaped like a pointy thorn and has six sides. This tetrahedron is the archetypal container in geometry (and 4-pole logic.) Later, when Torah is no longer dealing with the mathematical idealizations necessary to specify "continuous creation" and has evolved to human stories and the "creation" of Am Israel, the archetypal tetrahedron becomes identified with the "Tent of Meeting". The "Tent of Meeting" has a "light" inside. These represent the "light" Resh-Aleph and Aleph-Shin, in the middle of the word B'reshit - which belongs in the middle of the "Meeting Tent." In a cultural sense the 3-fathers and 4-mothers represent the "house" or the "tent" of Israel. (The 3-fathers can be said to "geometrically sit" within the 4-mothers.) They form the "walls and floor" of the space of Am Israel. There are two different metaphors at two different levels, but they both have the same abstract image. B'reshit forms a "Light in a Meeting Tent" and so does the 3-fathers and 4-mothers, except one Tent (and light) is abstract, the tetrahedron, and the other is cultural and historical, the fathers and mothers. I hope this helps. <smile> B'Shalom, Stan Tenen ----------------------------------------------------------------------

End of Volume 16 Issue 81