Bayesian Inference

UPDATE: I have corrected an inaccuracy in my probabilistic analysis of the Gemara, and I have added to it an example using concrete numbers.

The Mishnah states:

האשה שנתארמלה או שנתגרשה היא אומרת בתולה נשאתני והוא אומר לא כי אלא אלמנה נשאתיך אם יש עדים שיצאת בהינומא וראשה פרוע כתובתה מאתים1

The Gemara analyzes:

וכיון דרוב נשים בתולות נישאות כי לא אתו עדים מאי הוי

אמר רבינא משום דאיכא למימר רוב נשים בתולות נישאות ומיעוט אלמנות וכל הנשאת בתולה יש לה קול וזו הואיל ואין לה קול איתרע לה רובא

אי כל הנשאת בתולה יש לה קול כי אתו עדים מאי הוי הנך סהדי שקרי נינהו

אלא אמר רבינא רוב הנשאת בתולה יש לה קול וזו הואיל ואין לה קול איתרע לה רובא2

This is a classic example of Bayesian inference, albeit expressed in qualitative, rather than quantitative, terms:

Bayesian inference uses aspects of the scientific method, which involves collecting evidence that is meant to be consistent or inconsistent with a given hypothesis. As evidence accumulates, the degree of belief in a hypothesis ought to change. With enough evidence, it should become very high or very low. …

Bayesian inference uses a numerical estimate of the degree of belief in a hypothesis before evidence has been observed and calculates a numerical estimate of the degree of belief in the hypothesis after evidence has been observed. …

Bayes’ theorem adjusts probabilities given new evidence in the following way:

Baye's theorem

where

  • H represents a specific hypothesis, which may or may not be some null hypothesis.
  • P(H) is called the prior probability of H that was inferred before new evidence, E, became available.
  • P(E | H) is called the conditional probability of seeing the evidence E if the hypothesis H happens to be true. It is also called a likelihood function when it is considered as a function of H for fixed E.
  • P(E) is called the marginal probability of E: the a priori probability of witnessing the new evidence E under all possible hypotheses. It can be calculated as the sum of the product of all probabilities of any complete set of mutually exclusive hypotheses and corresponding conditional probabilities:

    Summation

  • P(H | E) is called the posterior probability of H given E.

The factor P(E | H) / P(E) represents the impact that the evidence has on the belief in the hypothesis. If it is likely that the evidence E would be observed when the hypothesis under consideration is true, but unlikely that E would have been the outcome of the observation, then this factor will be large. Multiplying the prior probability of the hypothesis by this factor would result in a larger posterior probability of the hypothesis given the evidence. Conversely, if it is unlikely that the evidence E would be observed if the hypothesis under consideration is true, but a priori likely that E would be observed, then the factor would reduce the posterior probability for H. Under Bayesian inference, Bayes’ theorem therefore measures how much new evidence should alter a belief in a hypothesis.

[One application of Baysian analysis is to the notorious Monty Hall problem, as explained here.]

In our Gemara:

  • H is the hypothesis that the woman was a virgin at her marriage
  • E is the absence of a קול
  • P(H) is רוב נשים בתולות נישאות
  • P(E | H) is low, since רוב הנשאת בתולה יש לה קול
  • P(E) is not stated
  • P(H | E) is הואיל ואין לה קול איתרע לה רובא

The Gemara’s point, expressed in the language of Bayesian probability, is that in the absence of a קול [E], the likelihood of בתולה נשאת [P(H | E)] is significantly lower than our initial assumption of the same [P(H)], given that the likelihood of the absence of a קול had she been a virgin [P(E | H] is low. [This is true regardless of the value of as long as P(E | H) is significantly lower than P(E), which it clearly is, since the absence of a קול is certainly much more likely if our hypothesis of her virginity is false than if it is true.] In the הוה אמינא of the Gemara, P(E | H) was actually zero, and we would therefore reject even the testimony of witnesses to her virginity, normally the gold standard of evidence in Halachah, having no choice but to conclude that they are liars.

For concreteness’s sake, here is the logic expressed using specific, albeit somewhat arbitrary numbers. Let us assign the value of 75% to any רוב, and let us assume that the rate of false positives, i.e., the presence of a קול for a woman who is not a virgin, is quite low, say 5%. So:

  • P(H) is .75
  • P(E | H) is .25
  • P(E) is (.75 * .25) + (.25 * .95) = .425

And plugging these values into Baye’s theorem tells us that P(H | E) = .25 * .75 / .425 ≆ .441

As the Gemara says, איתרע לה רובא.

The principle of this Gemara arose during a presentation that I heard yesterday. The question being discussed was the status of a בעלת תשובה vis-a-vis marrying a Cohen. The problem is that many Poskim rule that an apostate who returns to observance is assumed to be טמאה, with her protestations to the contrary not believed, based on the principle of רוב גוים פרוצים בעריות3, and it would seem to follow that the same should apply to an irreligous Jew, since she has moved in non-Jewish society and followed its norms.

One (out of several) arguments raised by the presenter was as follows: He noted that he had been informed, by a Posek with much experience in these matters, that 95% of בעלות תשובה actually admit that they are טמאות, and even though we have no idea of what percentage of the remaining 5% are telling the truth, it is nevertheless indubitably clear that רוב טמאות admit to their history4. He therefore reasoned, by analogy to the above Gemara, that whenever a woman insists that she is טהורה, the argument against her purity that רוב גוים פרוצים בעריות is weakend by the opposing, supporting argument that רוב טמאות admit to their status, and she does not.

The reaction to this suggestion by the audience was mild pandemonium. While I staunchly defended the plausibility of the idea, nearly everyone else who expressed an opinion, as well as a couple of my colleagues to whom I later related it, seemed to find it at least dubious, if not downright preposterous. I did not, however, hear anyone articulate a clear objection. Of course, one can simply deny the claim of 95% of בעלות תשובה admitting to being טמאות, but if, for the sake of argument, we grant it, I challenge the reader to either concede the point, or clearly articulate why it is incorrect!

  1. כתובות ריש פרק שני []
  2. שם דף ט”ז ע”א – ע”ב []
  3. See שולחן ערוך אה”ע סימן ז’ סעיף י”א בהגה []
  4. We are tacitly assuming that the rate of false positives, i.e. admissions of טומאה by טהורות, is very low. Obviously, if we assume that admission or denial of טומאה is completely independent of her true status, then we can derive nothing at all from her denial. []
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11 Responses to Bayesian Inference

  1. wolf2191 says:

    I can think of two objections, neither very strong.

    1 – Why have earlier posekim not thought of this idea? Is mots admitting a new “metzias” and how can this be established?

    2 – Rabbi Soloveitchik’s rejection of Rackman proposal as precedent.

    ” We must not
    tamper, not only with the halachos, but even with the chazakos, for the
    chazakos of which chazal spoke rest not upon transient psychological
    behavioral patterns, but upon permanent ontological principles rooted in
    the very depth of the human personality, in the metaphysical human
    personality, which is as changeless as the heavens above. Let us take for
    example the chazaka that I was told about: the chazaka tav l’meisiv tan du
    mil’meisiv armalo [20] has absolutely nothing to do with the social and
    political status of women in antiquity. ”

    http://mail-jewish.org/rav/talmud_torah.txt

  2. wolf2191 says:

    Obviously the above is regards to psychological mindsets, and this is in regards to metzios (Rov), but there is still a similarity.

    • Yitzhak says:

      I have long been utterly baffled by Rav Soloveitchik’s famous magisterial insistence that “the chazakos of which chazal spoke rest not upon transient psychological behavioral patterns, but upon permanent ontological principles rooted in the very depth of the human personality, in the metaphysical human personality, which is as changeless as the heavens above” – this is simply inconsistent with Halachah as it has actually developed!

      A favorite counterexample of mine: Rav Hamnuna famously rules (נדרים דף צ”א ע”א) that a woman who tells her husband to his face that he has divorced her is believed, דחזקה אין אשה מעיזה פניה בפני בעלה, and yet, many Poskim rule that this does not apply today:

      וי”א עוד דבזמן הזה דנפישי חוצפא ופריצותא אינה נאמנת אלא לחומרא דאיתרע חזקה דאינה מעיזה – הגהת הרמ”א אה”ע סימן י”ז סעיף ב

      Rav Soloveitchik’s “permanent ontological principles rooted in the very depth of the human personality, in the metaphysical human personality, which is as changeless as the heavens above” apparently do not stop the Poskim from “tamper[ing] … with the chazakos of which chazal spoke”!

      I do agree with the first point that you raise. I am a conservative thinker, and I’m always reluctant to blithely dismiss precedent with the airy claim that “times have changed”. In the post itself I noted that I am merely accepting the factual assumption for the sake of argument

  3. Gedalia Walls says:

    I think you can back this up with some decent teshuvos mentioned in EH siman 7 in the Pischei Teshuvah relating to couples who go off the derech together. However, the inference “95% of BT” seems somewhat ad hoc, but let’s assume that there is still a rov in his favor. Without a clear rey’usa, a woman should still get a shot to claim her virginity. Only women who are known to have been violated (e.g. rape) or captured lose the ability to claim they are still virgin. Plain and simple (even in the days of the gemara) virginity was a milsa d’avidi ligluyei such as by sitting atop a wine barrel. It’s only against the claim of the husband and the lack of a document stating she was a virgin at the time of marriage that the mishna sides with the husband because he is muchzak and she is motzei mamon. Otherwise, it is fair for anyone to take a BT girl at her word, write the kesubah, etc. and even have her marry a Kohein

  4. Dudi Mendel says:

    It could be tennable to reject the applicability of probability in the case of the B”T insisting on her purity — Her admission or denial is in her hands, while the Kol in the gemaras case is not in her control.

    This approach is in line with my understanding of the gemara’s applying probablity in “kol deparish meruba parish” while ignoring probablity in “kol kavua k’mechtza amechtza dami.” see Ketubot 15A

    • Gedalia Walls says:

      That is referring to the status of the MAN in those cases- if she is in Tzippori in a house- it’s a safeik on the MAN if he is kasher or pasul. She has certainly had intercourse in that case. Also important is the fact that not all of the Rishonim held that she had anything to add either (i.e. no ta’ana), which is the approach of Rashi and Rambam to permit her based on two rovs and maybe even one, while Rosh and Tosafos hold at least a ta’anas bari.

  5. Dovid Kornreich says:

    The claim she would want to make to permit herself to a Kohen is not about virginity, but that she only had relations with a man with whom she could theoretically marry within halacha. (Only a kohen gadol is required to marry a virgin–not your average kohen)
    Thus she does not have to claim virginity, which is somewhat far-fetched in today’s secular culture, but that she only cohabited with another Jewish man. This is a more reasonable claim.

  6. Rentsy says:

    Yes. It makes it much more likely that we should take a ba’alat teshuva at her word. Remember, virginity isn’t a big deal, having relations with someone you could never marry is.

    CHIDDUSH:

    Furthermore, she isn’t a goy. If “chazakos of which chazal spoke rest not upon transient psychological behavioral patterns, but upon permanent ontological principles rooted in the very depth of the human personality, in the metaphysical human personality” it is WRONG to apply the chazakot of a goy to a ba’alat teshuva, because ontologically she is a Jew, regardless of what transient psychilogical behavior patterns you may force upon her.

    Points for Rentsy?

    • Yitzhak says:

      > Yes. It makes it much more likely that we should take a ba’alat teshuva at her word. Remember, virginity isn’t a big deal, having relations with someone you could never marry is.

      But the starting point for the potential problem of the בעלת תשובה is the ruling that a repentant apostate is prohibited to marry a כהן, and the issue there is also not one of virginity, but of her having consorted with a non-Jew.

      > … it is WRONG to apply the chazakot of a goy to a ba’alat teshuva, because ontologically she is a Jew, regardless of what transient psychilogical behavior patterns you may force upon her.

      But as I’ve mentioned, the Poskim most definitely do apply the חזקה in question to apostates, apparently on the grounds that they conduct themselves as do non-Jews.

      I do admit that It is certainly possible to distinguish between apostates and non-observant Jews; a strong basis for this is Resp. Avnei Nezer (EH 32:3 – link), who argues that the stringency in the case of the apostate is a גזירה מדרבנן, and it is therefore limited to the exact case of the apostate, and may not be extended to a woman who apparently suffered from nymphomania.

      • Rentsy says:

        Rambam, in Hilchot Teshuva 3:9, defines a mumar l’kol haTorah kul as someone who says, “What do I gain by attaching myself to Israel, who are lowly and beaten down, I should join those whose strength is better.”

        It seems to me that along with being an apikorus is a conscious rejection of Judaism, and also an embrace of gentile society and their ways.

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