Statistics and Law: Legal and Probabilistic Reasoning (Column 226)
With God's help
This morning someone sent me an article from the Alaxon website by David Papineau that deals with the concept of 'knowledge' and its relation to concepts of belief (not in the religious sense). This can be discussed from several angles, especially regarding the meaning of belief in God and its relation to information and to facts of various kinds. But here I wanted to address a particular angle that arises in the article, one touching on evidence in the legal world (and in Jewish law). I suggest that anyone who wants to get the most out of this first read that article, and then continue.
Statistical Evidence
I want to focus on the following example, and I will present it in Papineau's own words:
Imagine one hundred prisoners exercising in the prison yard, and suddenly 99 of them attack the guard, in a premeditated assault in which the hundredth prisoner takes no part. Now one of the prisoners is sitting in the dock. There is no additional evidence. The chance that he is guilty is 99 percent, and the chance that he is innocent is 1 percent. Should the court convict him? Everyone's initial reaction is: obviously not. The court has no evidence that rules out the possibility that the defendant is the innocent prisoner. A person cannot be convicted solely on the basis of statistical evidence.
Today, thank God, courts know that eyewitness testimony may be mistaken, and so they examine it carefully and make sure it is reliable. Even so, courts do not demand one-hundred-percent certainty, and are satisfied if no doubt can be cast on the testimony
The commotion in the prison yard is a fictional example. But the same issue often arises in real courts, and there the law accords with ordinary intuition. Statistical evidence alone is insufficient. In both civil and criminal trials, defendants can be found guilty only if the evidence relates directly to them and does not replace them with some general category on the basis of which it is likely that they are guilty.
This may be an intuition, but the fact that the law forbids the use of statistical evidence is surprising. Think of a person who is convicted because an eyewitness saw him steal a necklace. Today, thank God, courts know that eyewitness testimony may be mistaken, and so they examine it carefully and make sure it is reliable. Even so, courts do not demand one-hundred-percent certainty, and are satisfied if no doubt can be cast on the testimony—and from cases in which judges have been persuaded to speak in numbers, one can infer that we are dealing with 95 percent certainty.
In other words, we are often willing to convict on the basis of eyewitness testimony, but never on the basis of statistical evidence. You may wonder why we think that eyewitnesses whose reliability is assessed at 95 percent are less likely to mislead us than statistical data whose reliability is 99 percent.
It is indeed troubling. On the face of it, there is no logic in preferring eyewitness testimony to statistical evidence of similar, or even greater, significance. More generally, why is one considered statistical evidence and the other not? At first glance, both seem to be statistical evidence.
Later on he proposes an explanation for this phenomenon through the concept of knowledge. Information that reaches us through eyewitness testimony is regarded by us as some kind of knowledge about reality, whereas in the case of the prisoners we have no knowledge but only belief. He argues that this explanation may describe our way of thinking quite well, but it does not really offer a substantive justification for the distinction we make. Why rely on knowledge and ignore beliefs that have even higher reliability? What difference does it make whether we call it knowledge or belief?! In the final analysis, in both cases we rely on statistical assumptions, merely giving them different names.
Additional Examples from Jewish Law
In Jewish law there are similar phenomena, and perhaps they appear there in even sharper form. According to most opinions, halakhic criminal law relies solely on the testimony of two witnesses, and not on circumstantial evidence (see Maimonides, Laws of Sanhedrin, beginning of ch. 20, as compared with the beginning of ch. 24 there. These two laws were cited in Column 224). One should understand that circumstantial evidence is not necessarily less reliable. This can be seen from the example that the Talmud itself (Shevuot 34a) brings for this principle:
For it was taught: Rabbi Shimon ben Shetach said, "May I not see consolation if I did not once see a man running after another into a ruin. I ran after him and found a sword in his hand, with blood dripping, and a murdered man still convulsing. I said to him: Wicked man, who killed this one—either I or you! But what can I do, for your blood is not entrusted to my hand, since the Torah says, 'By the testimony of two witnesses or three witnesses shall the dead man be put to death.' Rather, the Omnipresent will exact punishment from you." They said: they had not moved from there before a snake bit him and he died.
Shimon ben Shetach saw a man running after his fellow into a ruin, and when he entered it he found the pursuer holding a sword dripping blood, while the pursued man was convulsing and bleeding. In such a case, is there any doubt that the pursuer is the one who killed him? This is circumstantial evidence, but it appears conclusive. Yet in a case like this, where we have no witnesses (because the act took place inside the ruin and no one saw it with his own eyes), the murderer cannot be convicted. Note that in such a situation the certainty is not necessarily lower than that of the testimony of two witnesses (which, as noted above, is not certain either), and nevertheless this evidence is not accepted in order to impose punishment on the murderer. Here too we have a distinction between information and statistical belief.[1]
Jewish law has several additional examples in which we distinguish between different kinds of evidence even though they do not differ in the degree of certainty they provide us. Close relatives are disqualified from testimony even though the Talmud says (and so too Maimonides and the Shulchan Arukh) that there is no concern regarding their reliability (There is a presumption that a person does not sin when no benefit accrues to him.). We distinguish between majority and presumption (for example in monetary law), between migo and presumption (with respect to extracting money), we do not accept self-incrimination even though there is no concern at all about its reliability[2], and so on.
Two Concepts of Plausibility: Legal Rationales
In examples like these there is a tendency to speak of a 'scriptural decree,' that is, a rule innovated by the Torah for which there is not necessarily any rationale. But at least in some of these cases the distinctions are made without any clear source in the Torah, and therefore some logic must lie behind them.
In the final part of my booklet on migo on the site, there is an appendix dealing with legal rationales. Legal rationales are rationales that are not based on reliability, that is, on plausibility and probability, yet they are still not scriptural decrees in the sense described above. There is logic behind them, even though it is not probabilistic logic. Thus, for example, self-incrimination is not accepted although, as noted, according to most opinions there is no reliability problem there. On the other hand, there is an intuitive logic to refusing incriminating testimony from a person against himself. There is a sense that a person has a right not to trip himself up, and that his words should not be used against him. Another example is what the later authorities call 'migo as force of argument' (or 'power of credibility'; see the above booklet). There too there is a non-probabilistic rationale, but one that has its own logic. In that booklet I explained that presumptive possession of money is also such a rationale (we leave the money in the hands of the possessor even though it is not more probable that it is his, especially since this happens even when there is a migo against him).[3]
My conclusion there is that there are two kinds of reasoning that can ground legal conduct in the laws of evidence: plausibility in the probabilistic sense (which is not necessarily based on calculation, but whose foundation is the reliability of the evidence; one may speak of probability or plausibility), and legal plausibility (that this is how one ought to proceed legally, even though no probabilistic reliability consideration is involved).
Back to the Prisoners Example
In the prisoners example there is a clear sense that it is unjustified to convict the person. Papineau is right that this is not based on the chance of error, which, as noted, is very small (and indeed smaller than the chance of a defect in eyewitness testimony). And yet there is something in the situation that evokes the sense that it is wrong to convict in such a case. Apparently this is what I called a 'legal rationale,' but as we shall see there may even be a consideration of probabilistic plausibility here (that is, of reliability or of our degree of certainty). I will go on to argue that the distinction between these two kinds of explanation is not as sharp as it seems.
Two Kinds of Majority[4]
In Jewish law a distinction is drawn between two kinds of majority: rov de'ita kaman and rov d'leita kaman. Rov de'ita kaman is a majority that is before us, such as most shops in a city selling kosher meat. In such a case, if we find a piece of meat in the street, the assumption is that it is kosher because we follow the majority, and in Talmudic language: Anything that separates is presumed to have come from the majority. (whatever separates is assumed to have come from the majority). By contrast, there is rov d'leita kaman, which is basically a name for characteristics of the natural world. For example, most women give birth after nine months (and not after seven), or most women are not infertile, and so on. These are laws of nature or phenomena rooted in the nature of the world.
In the passage at Chullin 11a they try to find a source for the law of following the majority. They find it for rov de'ita kaman (from the verse Follow the majority.) but not for rov d'leita kaman. The Talmud clearly assumes that these are two different kinds of majority, and therefore one cannot be derived from the other.[5] What exactly is the difference between them? On the face of it, this is a matter of probability, so why should it matter whether the majority is before us or not?
In Column 79 I presented this distinction between the two kinds of majority and explained the difference between them. My claim there was that rov d'leita kaman is based on a generalization from a sample. We saw a sample of women, and most of them were not infertile or gave birth after nine months, and we inferred that this is true of most women in the world. The assumption is that the sample we encountered is representative, and therefore one can infer from it a conclusion about women in general. This is an inference from a collection of cases that we encountered at random. Sometimes one can conduct a controlled experiment on a sample and observe the result, and infer from it a general conclusion. That is usually how science works. By contrast, in rov de'ita kaman, as with the shops, there is no way to examine a sample and no way to perform a deliberate experiment, and so in this case the general rule (= the majority) is not the result of generalization. Here we are dealing with a priori reasoning, and not with the result of measurement and probabilistic inference. In a generalization based on a sample we assume that the sample is representative, and there are even ways of trying to ensure that it is representative. In a conclusion based on rov de'ita kaman we have no way of doing that.
In rov de'ita kaman the 'probabilistic' conclusion is somewhat arbitrary. We assume that the chance the piece came from each shop is equal (in the absence of knowledge), and therefore we assume that the chance it is kosher is 90 percent. But that is not really probability in the strict, rigorous sense of the term. There is no sample here and no well-defined event space, so it is hard to call this conclusion probability. The '90 percent probability' that the piece is kosher merely describes the degree of persuasion we have, and not the result of some objective calculation. It is important to understand that even in ordinary probabilistic calculations we make certain assumptions, and of course there too the information is partial (for otherwise we would not need a probabilistic calculation), but the calculation itself is objective, and therefore its result can be treated as probability. That is not the case with rov de'ita kaman. In the case of the shops, for example, we have a total lack of information. We have no idea which shop is the source of the meat, and so in the absence of information we assume that the probability for each of them is equal. But that is merely an assumption and not the result of a calculation. What we have here is a somewhat arbitrary quantification of one of our own rationales. The quantification gives us no information or measure beyond our a priori assumption, and in this case simply beyond our veil of ignorance (= lack of information).
We can now propose a first explanation for Papineau's puzzle, one that is specifically probabilistic in character.
A Probabilistic Explanation
First, we must note that the case of the hundred prisoners is rov de'ita kaman. We have before us one hundred prisoners, one of whom did not commit the crime, but we do not know which one. Now a prisoner comes before us to stand trial for that crime. We have no way to convict him, because it is not correct to say that the probability that he did not commit the crime is 1/100. We have a complete lack of information, and so our null hypothesis is that all the people are equiprobable. But that is merely a hypothesis, and one cannot really speak here of probability. It may be plausibility, and even that only on the basis of our fundamental ignorance in the matter. It is no surprise that the explanation offered for this (as Papineau reported) is that in such a case we have no information, and the legal principle is that conviction requires information.
By contrast, when there are eyewitnesses to some event, and let us assume for the sake of the discussion that there really is a chance (say 5 percent) that a person's vision is defective. This is a case of rov d'leita kaman, because it involves a natural characteristic of human beings in the world, namely that sometimes their sight is distorted and unreliable. This is a generalization based on a sample of cases that we or others have encountered. In this case we have information about the criminal, though it is not certain. Here one can speak of probability, and therefore one can convict.
It is important to understand that this is not a formal-legal-conceptual distinction, but a probabilistic rationale, that is, a rationale concerning the reliability of the evidence (or our degree of certainty). My claim is that in rov de'ita kaman we have no information, and the probabilistic assumptions do not really describe something in reality itself. They describe a state of uncertainty or ignorance on the part of us, the observers. It is true to say that in such a case we have no information about the matter, but not because of a merely conceptual distinction, as Papineau assumed, but because in fact here we have no way of knowing anything about the world and about the facts themselves. The plausibility we describe in such a case concerns us and not the world (which is why it is better to call it plausibility rather than probability). This is a subjective state (I am in such-and-such a state of certainty) that has no necessary connection to what is happening or happened in the world itself. Merely because I lack information and assume that all possibilities are equal, I speak of a 99 percent probability. That is not information about the world, and rightly one cannot convict on its basis. Therefore this is not a conceptual explanation (what I called above a 'legal rationale') but a probabilistic one. The evidence in the prisoners case is genuinely weaker.
'We Do Not Follow the Majority in Monetary Cases'
The legal explanation can be presented through a rationale suggested by Rabbi Shimon Shkop. In the Talmud the Amoraim dispute whether we follow the majority in monetary cases or not. The source is the passage in Bava Kamma 46a-b (see also Bava Batra 92):
As it was stated: If one sold an ox to another and it was found to be a goring ox, Rav said: this is a mistaken sale; and Shmuel said: he can say to him, "I sold it to you for slaughter"…
Rav said: this is a mistaken sale—go by the majority, and the majority of people buy for plowing. And Shmuel said: he can say to him, "I sold it to you for slaughter," and we do not follow the majority. For we follow the majority in matters of prohibition, but in monetary matters we do not follow the majority; rather, the burden of proof rests on the one seeking to extract money from another.
The case concerns a person who sold his fellow an ox and it turned out to be a goring ox. The buyer wants to rescind the transaction, and the Amoraim dispute whether this can be done. According to Rav, this is a mistaken sale, since most people buy an ox for plowing (and a goring ox is of no use for that). According to Shmuel, it is not a mistaken sale, because the buyer can claim that he bought it for slaughter and not for plowing. True, most people buy for plowing, but in Shmuel's view we do not follow the majority in monetary matters. In practice, the law follows Shmuel (as it usually does in monetary law), that we do not follow the majority in monetary matters.
Now, in several places we find that nevertheless we do follow the majority in monetary matters. For example, in a religious court where the judges disagree, we follow the majority even in monetary cases. One should remember that the source of the rule of following the majority in the laws of evidence is the verse Follow the majority., which is said about a majority within a court. It is therefore hard to understand how in a court we follow the majority even in monetary cases, but in the laws of evidence that are derived from there the majority is ineffective in monetary matters. The medieval authorities indeed raised this difficulty. Some of them explained that the rule is that even in monetary matters we do follow the majority, and the case of Rav and Shmuel is exceptional. Others explained that we do not follow the majority in monetary matters, and precisely the case of a majority of judges in a court is the exceptional one.
In Tosafot, s.v. 'Dinei Mamonot,' Sanhedrin 3b (see a similar but not identical move in Tosafot, s.v. 'Ka Mashma Lan,' Bava Kamma 27b):
"Monetary cases—how much more so": This is puzzling, for at the beginning of the chapter HaMokher Peirot (Bava Batra 92b), Shmuel says that in monetary matters we do not follow the majority. Why, then, should we not derive it by an a fortiori argument from capital cases, as is said here? And one cannot say that even in capital cases themselves we do not follow a majority unless it is a majority present before us, but not a statistical majority, for at the beginning of the chapter Ben Sorer Umoreh (below 69a) it implies that everywhere we follow the majority in capital cases—for example, the majority of women give birth after nine months, and the majority of people err in calculating the month’s intercalation. It must therefore be said that the majority that people buy for plowing is not considered like those other majorities; therefore we do not rely on this majority in monetary law.
Underlying Tosafot's words are two difficulties: first, how can it be that in capital law, which is more severe, we follow the majority, whereas in monetary law we do not? Second, how is it that in a court we do follow the majority even in monetary cases? Tosafot raises the possibility of distinguishing between rov de'ita kaman and rov d'leita kaman (see above), but rejects it. In conclusion, he explains that the particular majority in the case of the sale of the ox is exceptional, and therefore specifically there we do not follow the majority, but in general we really do follow the majority as in capital law. In this way he resolves both difficulties together.
However, it is not clear what is special about this particular majority as opposed to other cases in which we do follow the majority even in monetary matters, and for some reason Tosafot does not trouble himself to explain it. Rabbi Shimon Shkop offers the following explanation (Shaarei Yosher, gate 3, ch. 3, sec. 47—in the Responsa Project edition):[6]
It may be said that the reason for this is based on the words of Nachmanides in Milchamot, chapter 2 of Kiddushin, cited in Shev Shema'tata, discourse 4, chapter 6. He wrote regarding the majority of cases in which gifts are sent and only afterward betrothal takes place, that it is difficult why we are concerned about the gifts and do not follow the majority. His language is: "But the answer to this question is that this majority is not comparable to the dispute between Rabbi Meir and the Sages, for there the majority is one of obligation and nature, and it cannot be otherwise. But here it is only a matter of custom, and many times a person follows the custom of the minority; therefore, in a case involving the prohibition of a married woman, they were stringent." End quote. It is evident from his words that by Torah law such a majority is effective even with respect to the prohibition of a married woman, but the Sages were stringent. According to what we have written, the point is that such a majority is a Torah-recognized majority and not a rational majority, and it is effective by Torah law, except that the Sages were stringent. Therefore, also the majority that people buy for plowing is not a rational majority, for if this particular person needed it for slaughter, that would not in any way contradict the laws of nature and ordinary practice; rather, it is a Torah-recognized majority, and it is not effective in monetary law or capital law for the reasons we have written. For in monetary matters we certainly require a rational majority, as we have written, because it is based on reasoning; and in capital cases, because it is written, "and the congregation shall judge," as we have written.
His claim is that in the dispute about the ox, the discussion is not about the nature of the world but about a person's decision (whether he wanted to buy for slaughter or for plowing). A person who buys an ox for slaughter is a completely normal person who acts in an entirely ordinary way; it is simply that usually there is more demand for oxen for plowing than for slaughter. Therefore, if a person claims that he belongs to the minority—that is, that he wanted to buy an ox for slaughter—one cannot raise the claim of the majority against him. Even if most of the world tends to buy oxen for plowing, does that in any way prove that this person did not want to buy for slaughter? After all, there are people who buy oxen for slaughter, and it depends on their decision. That person is not making a strange or illogical claim. Here belonging to the minority is entirely logical and reasonable, and therefore the majority has no evidentiary standing. The fact that most people do not act this way does not prove that his claim is incorrect.
Similarly, if I claim that my height is 1.95 meters, one cannot argue against me that most of the world is shorter. Is that an argument showing that I am wrong? I claim that I belong to the minority, and there is indeed such a minority, with nothing at all illogical about it.[7] The fact that I happened to wind up in a situation in which I belong to a minority group proves nothing about me (that my claim is illogical).
When the dispute concerns human decisions, we do not follow the majority in monetary cases. A person has the right to claim that he belongs to the minority, and the fact that the majority act otherwise says nothing about him.
An Explanation for Papineau's Question
If we now return to our question about the prisoners, we can see that in that case too we are not dealing with the nature of the world but with human decisions. A person claims that he did not participate in the assault on the guard, and we know that there was indeed such a person. How can one convict him of the assault by force of a probabilistic consideration? True, 99 did attack, but there was also one who did not. Therefore every one of the prisoners who stands trial can claim that he was the one who did not participate, and in order to convict him one must prove that he did participate. The existence of a majority cannot serve as a consideration that proves guilt in such a case. The fact that the person happened to find himself in a situation in which 99 other people decided to assault the guard proves nothing about him himself.
By contrast, when we discuss the chance that the witnesses saw incorrectly, we are dealing with a question about the nature of the world (people sometimes see incorrectly). In such a case the fact that one usually sees correctly is important. Here there is definitely room to apply probabilistic majority considerations.
The Relation Between the Two Explanations: Is the Second Explanation Legal or Probabilistic?
There is room to discuss the character of the second explanation. At first glance, this is a legal explanation, because probabilistically there is still a 1/100 chance, except that we have no right to make that claim against him. A person has the right to say that he belongs to the minority and not to be judged in light of the environment into which he happened to fall (by mistake, according to his claim). On the face of it, this is a legal and not a probabilistic explanation.
But in light of what we saw in the previous explanation, it seems that this explanation too can be presented in language similar to the previous one: there is no way here to perform a probabilistic calculation, and therefore it is not correct to apply a probabilistic consideration here. The probability in this case does not represent something about the world but rather our lack of knowledge. Therefore, when a person claims that he belongs to the minority, his claim purports to fill in our knowledge, and in any event there is no logic in applying a probabilistic consideration and rejecting his claim.
Here, admittedly, I do not attribute this to the fact that the majority among the prisoners is rov de'ita kaman, but rather to the fact that this is a majority concerning the conduct and decisions of human beings. But the logic is quite similar. In such a situation the result cannot be treated as a probabilistic calculation.
Third Explanation: Two Kinds of Doubt
To complete the picture, I will offer here another explanation that on its face seems legal rather than probabilistic. Jewish law distinguishes between two kinds of doubt: safek de-ikva issura and safek de-lo ikva issura. To understand this, let us take an example. Before me is a piece that looks like animal fat, and I do not know whether it is forbidden suet (ḥelev, which is forbidden to eat) or ordinary permitted fat. This is a case of doubt involving one piece, and if I ate the piece I would have committed a safek de-lo ikva issura (there is no fixed prohibition in the situation). According to the law, in such a case I am not obligated to bring an asham taluy (the offering brought for a doubtful prohibition whose inadvertent violation would require a sin-offering). By contrast, in a situation where there are two pieces before me and I know that one is forbidden suet and the other is ordinary permitted fat, and I ate one of them and do not know which one, this is a safek de-ikva issura, because in such a case there is a clear prohibition present in the situation, except that I do not know whether what I ate was the prohibited item or the permitted one. In such a case I must bring an asham taluy for the doubtful prohibition.
If we return to Papineau's question, in the case of the prisoners it is clear that there was one person who did not participate. In such a case, one cannot convict him on the basis of the majority, for there is clearly an innocent person in the situation.
The Character of the Third Explanation
In the two states of doubt that I described, the chance of stumbling into a transgression is 50 percent, and yet Jewish law does not regard them as identical. One requires an asham taluy and the other does not. Likewise, between the two situations in Papineau's question there is the same level of probabilistic uncertainty (indeed, the scale even tips in favor of the prisoners), and nevertheless the legal ruling is different. On the face of it, this is a legal and not a probabilistic difference, since the probability is identical in both cases. In the halakhic context, perhaps one could attribute this to the offender's experience (he ate a piece when it was clear that there was a prohibition in the situation, and therefore he is a graver offender on the psychological plane).
But in light of what I explained above, here too there is room to wonder whether this is a legal or a probabilistic explanation. In the doubt of one piece out of two, the chance that I took the forbidden piece is 50 percent. That is the result of a simple calculation (one possibility out of two). But from where does the number 50 percent come in the case of doubt about a single piece? Only because there are two possibilities (either the piece is forbidden or it is permitted). Our ignorance is what dictates the chances that we attach to the different possibilities. We have no positive indication that the chance that the piece is kosher or forbidden is really 50 percent. There are two possibilities and we have no idea at all about them, so we declared this an evenly balanced doubt. Put differently, the distribution is unknown here (because we do not know how this piece got here, and therefore we have no way to weight the chances that it is kosher or forbidden). This is not the result of a calculation but of reasoning born of ignorance, exactly as we saw above. If so, it may be that the doubt in the situation of a single piece is indeed lighter on the probabilistic plane. One cannot say that the doubt here has a real basis, and therefore it is harder to act and convict a person on its basis. The fact that I am uncertain between several possibilities has nothing to do with the question of the defendant's guilt. The evenly balanced doubt is in me and not in the world itself.
Nevertheless, Back to the Legal Plane
If we return to the distinction between ikva and lo ikva issura, my main claim was that in the case of the prisoners (as in the case of one piece out of two pieces) there is before us, with certainty, one person who will be convicted unjustly. In this mechanism, if we follow the majority then all one hundred will be convicted, even though we know that there was one there who did not participate. In this way there is not merely a chance of error but certainty that one error will occur. In such a situation, people are not convicted on the basis of a majority consideration.
One might argue that in the case of eyewitnesses as well there is a 5 percent chance that their sight was distorted, and therefore there too, among the total set of convictions, there will be some convictions that are unjustified. If we look at the entirety of the trials decided on the basis of eyewitness testimony, it is clear that there too there will be people convicted unjustly. So what, nevertheless, is the difference between the cases?
The difference is that in the case of the prisoners we have before us a situation in which one person will certainly be convicted unjustly. By contrast, in the context of eyewitnesses, statistically it does seem that there will be unjustified convictions, but we do not have, and cannot have, certainty about this. That is only a probabilistic consideration, and we do not take it into account. To be sure, probabilistically it is likely that full acceptance of two witnesses will lead to false convictions in about 5 percent of cases, but we do not have before us a case in which a false conviction will certainly occur. Such a concern is not one to which we pay attention.
Note that this explanation may perhaps return us somewhat to the legal plane. After all, probabilistically there are false convictions both here and there. Why are we concerned about false convictions only in the case of the prisoners (where it is clear that there will be a false conviction) and not in the second case? This is a legal distinction and not a probabilistic one.
A Note on the Difference Between Rov De'ita and D'leita Kaman
I once explained in this way the difference between rov de'ita kaman and rov d'leita kaman. Maimonides' view is that rov d'leita kaman is stronger (contrary to the simple reading of the passage in Chullin. See Shaarei Yosher, beginning of gate 3, at length). I argued that in rov d'leita kaman there is a law of nature that most women are not infertile, and therefore with respect to every woman who comes before us we decide that she is not infertile, without exceptions. By contrast, in rov de'ita kaman the differing minority is before us, and so although if one woman comes before us we will follow the majority, if we had to decide about all of them we certainly could not follow the majority with respect to all the women.[8]
This is connected to what I once called the 'David Levy effect.' David Levy argued at a meeting of the Likud central committee that none of the members of his faction received a position in the party institutions. They argued against him that everything was determined by democratic voting, and therefore it was not clear what his complaint was. David Levy explained to them the stupidity of that argument. Let us suppose, for the sake of discussion, that 30 percent of the committee members belonged to David Levy's faction. For every position there is a vote, and thus in each and every vote the 70 percent majority votes for the candidate of the majority faction. The result is that 100 percent of the positions are staffed by members of the faction that has some majority, as small as you please. This is exactly the situation in rov d'leita kaman. Every case that comes before us is decided according to the majority, even though it is clear that there are cases in which this does not hold. But that is clear statistically and not with certainty, unlike rov de'ita kaman.
Summary: The Relation Between the Two Types of Explanation
In the final analysis, it seems that explanations that appear to us legal and not probabilistic can also be connected to a consideration that concerns probability, that is, the degree of certainty the evidence provides us. But the reverse is also true: explanations that appear probabilistic have, at their base, a legal layer. The boundary between these two kinds of explanation is not sharp at all.
Link to a judgment on the issue.
[1] Tosafot do disagree with Maimonides, and in their view this is circumstantial evidence that is not conclusive. I will not go into the matter here.
[2] Maimonides does have one formulation from which it emerges that the issue is a concern that the confessor has gone mad (that is, a reliability problem), but these are puzzling words that run contrary to the plain sense of the Talmud, and Maimonides himself also contains contradictions on the matter. All the other medieval and later authorities agree that there is no reliability problem here but rather inadmissibility.
[3] There are such legal rationales in ordinary legal systems as well, except that there they usually provoke questions. Thus, for example, the distinction in the severity of the offense between attempted murder and murder, even though the offender's guilt and blameworthiness are the same in both cases. Or alternatively, the distinction between failing to rescue and active murder. People look for rational explanations for these principles because they are unwilling to accept the intuitive feeling that I called here a legal rationale. Much more could be said on this subject, but this is not the place.
[5] However, in the conclusion of the passage, according to one interpretation in Rashi on 9b, one is indeed derived from the other.
[6] And see a different formulation in his novellae to Bava Kamma, sec. 34. There he deals with the above Tosafot in Bava Kamma, whereas here he explains the view of Tosafot in Sanhedrin.
[7] Here, admittedly, this is a claim not connected to human decisions, and therefore it may be that Rabbi Shimon Shkop's distinction should be formulated differently. So long as the claim I make is not illogical, my belonging to a minority group proves nothing.
[8] On the basis of a similar rationale, several halakhic decisors wrote that at least rabbinically (and perhaps even at the Torah level) one cannot be lenient regarding all the pieces in a dry mixture of solids; rather, one should leave one or two pieces aside and not eat them.
Discussion
A halakhic example that may be somewhat similar.
Bava Metzia 112a explains that every laborer takes a risk in his work when he climbs trees, and this is permitted according to halakhah.
("Why did this one climb a ramp and hang from a tree and expose himself to death?")
It is clear that certain levels of risk are permitted.
And nevertheless, it seems obvious as a matter of halakhah that if there is a mixture of one hundred cups, one of which contains poison, it is forbidden to drink because of the uncertainty.
Even though, percentage-wise, there is a greater chance of falling from a tree than of drinking the poisoned cup out of a hundred cups.
A geshmak post.
You explained so well why a rova de-leita kaman is more plausible, that it is hard to understand our common assumption that a rova de-ita kaman is preferable. It seems that the explanation lies in the fact that, in resolving uncertainties, there is priority to a reality that is before us over a reality that is not before us. In fact, this gets further support from what you wrote at the end of your remarks: that it is impossible to judge on the basis of statistics, because then in Papino's case we would have to punish all the prisoners, and that cannot be. If so, then in a rova de-leita kaman too, it would follow that all the women before us are not pregnant, and that cannot be; whereas in a rova de-ita kaman, where we are discussing only the reality before us, it is easier to determine that this meat came from the majority of the stores.
If I understood correctly, you are claiming that Papino was talking nonsense, because as you wrote, it is impossible to convict everyone based on statistics. I did not read Papino's article, but from what you quoted in his name, he did not raise this argument. If so, the whole discussion is puzzling from the outset. May God enlighten his eyes.
Thank you very much for the fascinating article!
Two comments regarding your first explanation of the difference between eyewitness testimony and statistical evidence (and between rova de-ita kaman and rova de-leita kaman):
You claim that with statistical evidence and rova de-ita kaman, the probabilistic conclusion is "somewhat arbitrary," and that this is "not really probability in the strict sense," because "there is neither a sample nor a verified event-space here."
In my opinion, it is still not clear what exactly you mean by "probability in the strict sense."
If you mean (as is somewhat implied by your requirement of a sample and a verified event-space) that probability is nothing but the relative frequency of a certain event within a set of similar events, then one cannot speak at all about the probability of a single event (not even of a particular eyewitness testimony), but only about probability as a feature of a set of similar events (a reference class). And then eyewitness testimony too would not suffice, and every rova de-leita kaman would also lack "probability in the strict sense" so long as we are referring to a particular case.
On the other hand, if you mean that probability is also a feature of a single event (for example, some kind of physical property that determines what the frequency distribution would be if we ran many experiments of the same type), then one can also speak of such probability in statistical testimony and in rova de-ita kaman (if we repeatedly recreated the story of the meat found on the street while preserving the features of the story that are known to us now, then in most cases it would turn out after the fact that the meat came from the kosher stores).
Another point is that, in my humble opinion, it is not correct to say that in statistical evidence the probabilistic conclusion is arbitrary or subjective. On the contrary, the assumption that the probability that the meat came from each of the stores is equal is the only non-arbitrary assumption in this case, and it is דווקא the unreasoned assumption that there is a greater chance that the meat came from one particular store that is arbitrary.
Likewise, this is not a subjective conclusion, because we would expect anyone who has the same knowledge we have to reach the same conclusion we reached. The conclusion may perhaps be subjective in the sense that it describes relations between propositions rather than features of the world, but in that sense logical relations too are subjective relations.
There are many more examples of this sort, but here this is not necessarily a good example. Perhaps when you are a laborer you are also allowed to drink one cup out of a hundred if that is your job? There are permissions for someone whose work involves risk.
Read the article. Papino accepts this as well, but in his view it is only a convention.
As I explained, the definition is relative frequency, but also if it is determined by generalization on the basis of a sample (regarding cases not included in the sample itself). That cannot be done with stores, and not in court. With eyewitness testimony it definitely can be (you can test the reliability of testimony on a representative sample of people).
What you described is that same a priori reasoning I was talking about. And it is still a piece of reasoning, as good as it may be.
If the definition of probability is relative frequency, then the difference between statistical evidence and direct eyewitness testimony is still unclear to me.
Just as with eyewitness testimony you claim that the testimony is admissible evidence because it can (theoretically) be tested on a representative sample of people and yield some relative frequency, so too regarding stores, for example, one could argue that the claim that there is a majority of kosher stores is admissible evidence because it is possible (theoretically) to test it on a representative sample of cases in which meat separated from stores and obtain a relative frequency. What is the difference?
Try describing to me an experiment that would test the relative frequency of meat coming from stores, or of an error in a court ruling. Alternatively, an impression not based on a planned experiment but on cases you have encountered.
You ask everyone who finds meat in a place where there are nine kosher stores and one non-kosher store to report the find to us.
Then you conduct an intensive investigation and discover, based on forensic evidence, what the source of the meat was.
It turns out that in nine out of ten cases, the forensic evidence indicates that the meat came from the kosher stores.
The conclusion is that even in a place where we did not examine forensic evidence, or where for technical reasons it is unavailable, the frequency is nine out of ten.
It is a bit hard to carry out this experiment, but that is not the point; the main thing is that it is possible in principle.
As you yourself understand, there is no way to carry out such an experiment. That is precisely why this is a non-probabilistic majority. Once you do such an experiment and generalize its results to future cases, then of course it will indeed become a rova de-leita kaman.
It seems to me that a simpler formulation of your explanation would be this one (and it is entirely probabilistic):
The court's goal is to convict a person under 2 conditions:
1. The probability that he committed the crime is above a certain threshold.
2. The probability that he committed the crime is higher than that of the other people (or of the other people minus the number of people who committed the crime, in the case of a crime involving multiple participants).
In Papino's case, the second condition simply is not met, whereas in the case of an eyewitness it is met.
I didn't understand. Why is the probability not higher?
I do not understand on my own that there is no way to do the experiment.
To tell the truth, I actually understand the opposite—namely, that almost always such an experiment can indeed be performed, although admittedly it would be very, very difficult. But the difficulty of carrying it out is irrelevant; what matters is the mere possibility of carrying it out.
If you insist that an experiment must actually be performed, then in my opinion you will run into the following difficulty:
Even in a rova de-leita kaman, usually no experiment is actually performed, and we make do with the theoretical possibility of performing the experiment together with assumptions that predict what its results would be. For example: most women give birth at nine months—was such an experiment done for the group of women from a certain country, or for the group of women who ate baby carrots during pregnancy? Presumably not. The claim that the probability for the group of women who ate carrots to give birth at nine months is equal to that of the group of all women is an a priori assumption with no factual basis. And if you really require an actual experiment, you turn all these cases into rova de-ita kaman.
One could explain Papino on the basis of “woe to the wicked and woe to his neighbor.” On the other hand, it is hard to convict 100 prisoners when you know for certain that one of them is innocent.
Many thanks, Rabbi Michi.
The majority of a court and the case of 9 stores are seemingly a stronger majority than an ordinary rova de-ita kaman, because in this type of majority we actually know who the majority is and who the minority is, and the implication concerns something outside the group, not a particular item within the majority/minority group. Therefore, seemingly, there is no need to explain why the majority among the prisoners is a weak rova de-ita kaman.
In the prisoners' dilemma, one could argue that if the punishment were, for example, an additional 100 days in prison, then all the prisoners would share in the punishment, and it would come out to 99 days in prison per person.
What I am trying to say is that one can follow a statistical majority as a methodological consideration, though clearly the ruling is not entirely a ruling of truth—but perhaps that is not the goal either; it seems to me that this is what Papino means. When the punishment is very severe, like execution by the court, capital cases, then if a prisoner is mixed in with those sentenced to severe death, the majority will not help.
After all, Rabbi Meir, who is stringent and takes the minority possibility into account, agrees that there is a rationale for following the majority heikha de-lo efshar (where no alternative is possible). So one can understand that if there is a dispute between statisticians and halakhic-philosophers, this is an example and a kind of parallel to that dispute between Rabbi Meir and the Sages, seemingly.
The issue of statistical decisions reminds me of a very heated discussion in psychology and economics. It is said there as well that much of what happens, even if you are a great expert, is the product of luck. Statistical average calculations by a computer regarding the market are more accurate than expert economists, and likewise regarding data about human personality, statistical calculations are on average more accurate than clinical psychologists.
But why, despite this, do people still go to investment advisers and psychologists? Because they believe in the minority. Because they believe in free choice, which underlies Rabbi Meir's reasoning.
Essentially, as I understand it, Rabbi Meir limits the validity of laws in favor of the theological position. One could say that there is possibility and free choice in every case, and therefore he always imposes full responsibility. And similarly, the reasoning that one who says “it is permitted” is considered intentional is that he should have taken even the minority possibility into account, and therefore Rabbi Meir is stringent regarding women who do not have a regular cycle.
I received a link to an article by David Enoch on the subject:
http://portal.idc.ac.il/he/schools/law/about_us/documents/seslvk,%207.11.pdf
I hope to get to read it and comment later on.
I meant to write a response very similar to what Phil wrote—indeed, every rov de-leita kaman can ultimately be a rov de-ita kaman, since there are never really two completely identical cases, and then the generalization to the specific case is not necessarily correct. And every rov de-ita kaman can be a rov de-leita kaman if one performs an experiment and tries to generalize the specific case to similar cases (and one almost always can, certainly in the case of the meat stores and the prisoners).
In the end this connects somewhat to the philosophy of science. The information on which one relies in order to accept a scientific theory—even if the case is not completely identical—would count as a rov de-leita kaman, while what lacks convincing information for creating a theory would count as a rov de-ita kaman and would be handled through the partial information (we would not expect a court to create a theory for every case presented to it and test it through experiments and refutations).
This fits with the definition of probability—what proportion of a certain result would occur in infinitely many trials. If we have accepted a scientific theory, then we have a hypothesis about the answer; if we do not have a theory, then we do not have a scientific “hypothesis,” but rather a lower level of guesswork.
Accepting two witnesses is equivalent to a 0.25% chance of error if each witness has a 5% chance of misleading the judge(s)
My dear Stav (or dear female Stav). First of all, there is a mistake here in multiplying the numbers. Assuming (falsely) that the probability of error of the two witnesses is independent, then we have here 0.05X0.05 = 0.0025. But of course even this calculation is incorrect; see article 145:
https://mikyab.net/%d7%93%d7%95%d7%92%d7%9e%d7%90%d7%95%d7%aa-%d7%94%d7%9c%d7%9b%d7%aa%d7%99%d7%95%d7%aa-%d7%9c%d7%98%d7%a2%d7%95%d7%aa-%d7%91%d7%a9%d7%99%d7%9e%d7%95%d7%a9-%d7%91%d7%94%d7%a1%d7%aa%d7%91%d7%a8%d7%95/
Regarding what you wrote about a case of fixed prohibition (safek de-ikva issura): does this also refer to a situation where one piece out of three is definitely forbidden? Or only where it is one piece out of two? Because if in a situation of 1 out of 3 there is no obligation to bring a provisional guilt-offering, since we follow the majority, then that would be comparable to the case of the prisoners, where we should likewise follow the majority (because it is one piece out of 100).
Another point I thought about concerns our ability to generalize in matters related to free choice. For example, if I conducted an experiment in which I presented a moral dilemma to a representative sample of people, and suppose that 80% chose the good option and 20% the bad one. I think one cannot generalize the results of the experiment into a general law that any person who faces this moral dilemma has an 80% chance of choosing good and a 20% chance of choosing bad. Presumably that is because our power of choice is not deterministic, and therefore it does not obey the laws of generalization that apply to deterministic phenomena. For example, a person's height is a deterministic datum not subject to choice, and therefore it obeys the laws of generalization and is statistically distributed like the distribution of a representative sample.
In order to generalize from a sample to a general group, there has to be similarity between the items in the sample and the items in the general group. But with regard to the dimension of human choice, it may be impossible to liken one person to another, and each one is completely different from his fellow in his capacity for choice—just as their faces are not alike, so their opinions are not alike. This also explains Samuel's statement: "He can say to him: I sold it to you for slaughter." Since this is something that depends on a person's choice, one cannot generalize to him from the results of the representative sample of the population (because every person is essentially different from every other in the dimension of choice). And this also explains why one cannot generalize from the sample of prisoners to a random prisoner.
On the other hand, I recall that you previously mentioned that even in matters of choice, there are statistics that work and one can predict how people will respond. But perhaps the alignment with statistics in these matters can be explained by saying that these are matters that only appear to depend on choice, but in fact perhaps they do not depend on choice. For example, if we take the percentage of violent offenders in society, we see a kind of alignment with statistics. But perhaps when a person turns to violence, he does not necessarily do so out of choice; sometimes he is born in a certain way, not dependent on him, such that he has a greater tendency toward violence, and the statistics merely teach the percentage of people born with an increased tendency to violence, not the percentage of people who chose to be violent.
What do you think of these ideas?
With one piece out of three there is nullification by majority or following the majority (according to the Rashba, in dry-with-dry these two are the same thing). Clearly the case of the prisoners cannot be decided on the basis of the rules of doubt, for here there is a majority. But the distinction between whether the prohibition was fixed or not is also correct with regard to majority. That is all I said.
As for statistics and choice, you correctly noted that I discussed this in my book The Science of Freedom. It turns out that statistics work even in matters that depend on choice. Psychology is built on this. And there is no need to assume that these are natural processes. The topographical outline determines the distribution of choices, just as the fairness of the die determines the distribution of outcomes even though each individual roll is random.
And beyond all this, statistics almost never deal with truly random reality, since as we know in physics there is no such thing (except for the quantum level). Hence, if with regard to deterministic processes you are prepared to accept statistical analysis, I do not see why not with regard to people's choices?!
I read your article, and also an article dealing with the same subject and the same example, whose conclusion is different:
I think this legal question is directly connected to the goals of punishment (which you did not address).
The approach that speaks to me most is the utilitarian one, which says that punishment is intended to create deterrence and thereby protect society.
When I am judging and there is a person sitting before me in the defendant's chair, I have to balance between the need to punish the person if he committed a crime, and the need to protect him (as part of society) if he did not commit a crime. For that purpose I construct a statistical scale and set a minimal alpha level such that only one who passes it is convicted.
When my motive is utilitarian, I do not care why there is a 5 percent chance that he is innocent. That 5 percent is an iron threshold, and whoever reaches that threshold is acquitted on grounds of doubt.
I would be glad if you would respond to the link, and to my remarks.
Are you sure you read what I wrote? That article is linked at the beginning of my piece, and it is what I am responding to.
I am not sure I understood your claim. If I did understand, you are basically saying that there is no room to distinguish between the cases, and what should determine the matter is the probability. All legal systems in the world do not agree with this, and a great deal of literature has been written about it (you can see references in the article linked in the previous comment: https://mikyab.net/%d7%a1%d7%98%d7%98%d7%99%d7%a1%d7%98%d7%99%d7%a7%d7%94-%d7%95%d7%9e%d7%a9%d7%a4%d7%98-%d7%a1%d7%91%d7%a8%d7%95%d7%aa-%d7%9e%d7%a9%d7%a4%d7%98%d7%99%d7%95%d7%aa-%d7%95%d7%94%d7%a1%d7%aa%d7%91%d7%a8/#comment-24394
You ask me to respond to your remarks, when my entire article responds to them. I did not understand what further explanation you expect. What in my remarks do you not accept?
Somehow it seems you simply did not read this article and are responding blindly. Strange…
Thank you for the response.
I read your article on the day it was published, whereas I received the Alaxon article (as an Alaxon subscriber) about a week later. So I remembered that you had discussed the same issue, but I did not remember that you had referred directly to that very article.
Let me sharpen my question:
You write:
"If we now return to our question regarding the prisoners, we can see that in that case too, we are not dealing with the nature of the world but with human decisions. A man claims that he did not take part in the assault on the guard, and we know that there was indeed such a person. How can one convict him of assault by virtue of a probabilistic consideration? True, there were 99 who assaulted him, but there was also one who did not. Therefore, each of the prisoners who stands trial can claim that he was the one who did not participate, and in order to convict him it must be proven that he did participate. The existence of a majority cannot serve as a consideration that proves guilt in such a case. The fact that the man happened to be in a situation where 99 other people decided to attack the guard proves nothing at all about him himself."
To punish, you need a "consideration that proves guilt." But from a purely utilitarian perspective, is it worthwhile for society to punish in such a doubtful situation? In my opinion, yes. That way, next time the criminals will fear carrying out a revolt, because they will know that the lone righteous person in Sodom will not save them.
I did not understand exactly why you linked there.
The most worthwhile thing would be to punish the whole world, because that is the most deterrent. Or at least when there is still significant doubt, it is still worthwhile to punish. This is an absurd argument. Incidentally, Enoch in the article I linked raises a similar argument (in which he shows the opposite—that your approach leads to over-criminality), and I will deal with it in a follow-up article that will, God willing, go up in the coming days.
It seems to me that in the argument about what is most deterrent, the solution, in my humble opinion, is to punish proportionally as I suggested above. Something akin to dividing the liability in monetary matters, and perhaps also in lashes. Of course this is not always possible.
I did not understand, and it seems to me that you put words in my mouth that I did not claim.
I did not ask to "punish the whole world." I set a statistical threshold of 95%, the same statistical level that exists in eyewitness testimony.
For that matter, I can also agree to a threshold of 99.99 percent (for doubts of both kinds equally).
My claim is that when we seek to punish solely for utilitarian reasons, the only relevant criterion is the power of the test and the size of alpha.
Only when we are driven in law by moral considerations can we refrain from punishment because of the claim that "one cannot punish on the basis of a probabilistic consideration," as you say.
I did not put words in your mouth; I showed you what follows from your approach. You ignore the considerations I wrote according to which there is no justification for punishment, and write that the only consideration for punishment is the utilitarian one (deterrence). Therefore I wrote to you that according to your approach, it follows that we should punish the whole world just to be safe, regardless of level of guilt and certainty altogether. Why do we need the rules of evidence at all?
This is a dialogue of the deaf.
According to my view, one should punish any person who happens to be in a situation where statistically (very, very highly) he committed a crime—not out of 'justification for punishment,' but out of 'the advisability of punishment.'
You need the rules of evidence in order to raise the statistical probability. What is so unclear?
I am not claiming there are no other understandings. I am only presenting my position, and claiming that it is consistent and coherent.
Indeed, a dialogue of the deaf. Raise the probability of what? That deterrence will be achieved? What connection is there between the rules of evidence and deterrence? According to your deterrence logic, the best thing is always to punish everyone. The rules of evidence are there to provide moral justification for the punishment (to prove that the person being punished is guilty), not to ensure deterrence.
I really do not understand what is unclear here.
The rules of evidence raise the probability that the person really did commit the crime:
When we punish a person for an offense, for example murder, we need almost absolute certainty that he in fact murdered.
If we punished a person who did not murder, our gain would come out in our loss.
When we punish without due inquiry and investigation, the impression created is that the connection between offense and punishment is random. Since the public knows that it is possible that the real murderer is walking free, deterrence is not achieved. Moreover, the murderer himself or others learn that murder pays. You can kill another person and escape punishment—the court will easily find a scapegoat.
Again—the deterrence is achieved only if there is confidence that the punishment was given to the real criminal. And all the rules of evidence are intended to increase that confidence in the eyes of the public.
According to your logic, one could punish randomly and merely create an appearance of guilt (publicize that it was proven the man committed the crime). This is absurd. The rules of evidence are the justification for punishment, not merely verification of its effectiveness.
Incidentally, in the next article (228) I will address your view in more detail, namely that one should punish solely according to the level of certainty.
An important correction to the second explanation, and in general to the explanation of the sugya of “one who sold an ox and it turned out to be a gorer”:
You wrote: "According to Samuel this is not a mistaken transaction, because the buyer can claim that he bought it for slaughter and not for plowing." It is not the buyer who claims this, but the seller!
If the buyer were claiming that he belonged to the minority, then even Rav would agree that we do not follow the majority—as explained in Tosafot, Bava Kamma 27a. Here the point is that the seller claims regarding the buyer that the buyer belongs to the minority.
This correction is important also for the proper understanding of the quotation from Rabbi Shimon Shkop—he does not mean to say that a person can claim against the majority facing him that there is also a minority and that he belongs to the minority; that is just the plain meaning of Samuel's words (and, as stated, even Rav agrees when a person claims something about himself). Rather, he means to say that only with regard to a majority that stems from custom—and is not fixed in reality (like the matter of heights below 1.95 that you mentioned)—can the seller claim against the buyer that the buyer belongs to the minority.
The practical difference is that I cannot (according to everyone) claim regarding a person that he belongs to the minority whose height is over 1.95, but I can claim regarding him that he is among those who customarily buy for slaughter (according to Samuel). And my being able to claim that I myself am over 1.95 is agreed even according to Rav, and that is not Rabbi Shimon Shkop's meaning at all.
Indeed.
Today I thought that Papino's prisoner example is similar to racist generalizations in an airport, job interviews, and matchmaking. The common flaw is that if everyone were to generalize in that way, there would always be a minority group that gets harmed, and therefore it is proper to act in a way that takes them into account as well.
There is similarity, but there are also significant differences. First, in airport screening we are trying to prevent serious harm, and that justifies drastic measures. In Papino's case we are talking about punishment, not prevention. Second, a search is not as drastic an act as punishment by imprisonment. Third, in screening they really do check everyone, whereas in punishment not all one hundred always stand trial (the discussion is about one person, and the argument is that if all were tried an innocent person would be punished). Admittedly, in a job interview and matchmaking too it is not about everyone. Fifth, in airport screening there are two alternative possibilities: not to check everyone or to check everyone. In punishment too one must weigh the two alternatives (to punish everyone or to punish no one).
You write that the distinction between a case where we have no information about the individual person and therefore assume an equal distribution, as in the prisoners example, and a case where there is concrete information about the individual under discussion, as in testimony, is not merely a legal-formal distinction but a probabilistic one. But I do not understand what you mean. After all, if we take for example 100 cases of the prisoners example, where in each case there are 99 guilty out of 100, and in all the cases we convict the defendant who came from the group of prisoners, statistics say that out of the 100 cases—in 99 of them our conviction is correct. By contrast, if we take 100 cases of conviction on the basis of testimony whose reliability is 95% (that is, a 5% chance the witnesses are mistaken or lying)—only in 95 cases will the conviction be correct, whereas in 5 cases there will be error. So reliance on the general statistics in the prisoners example yields five times as many correct decisions as reliance on testimony. If so, in what sense is testimony stronger probabilistically?!
No. I am claiming that this will not necessarily yield five times more convictions. If you put all the prisoners on trial, then of course one of them is guilty. But think of a situation where you selected one prisoner from among them and put him on trial. Is the probability 1:100? Not necessarily. Each prisoner can claim that he personally is not guilty, because he is a person of exalted character. You are applying to him the assumption that the chance he committed the crime is equal to that of all the others. But that is your assumption, and he denies it. I think I brought the Ramban's example of sivlonot and the majority for irrigation in monetary law.
I really do not understand. When we draw a random prisoner from the group of prisoners, knowing that 99 of them are guilty and only 1 is innocent, then the chance that the random prisoner before us is guilty is 99/100, i.e. 99%. Just as if we took a random ball from a bucket with 99 red balls and 1 blue ball, the chance that the random ball we picked is red is 99%. The fact that the prisoner claims he belongs to the minority group should not change the probability, since it is obvious that even if he were guilty he would make that claim. (And even if we assume that some of the guilty would confess, so that when he claims he is innocent the chance that he is guilty decreases, that is also true in the case of testimony, so in comparing the cases this has no effect.) In my innocence this seems like simple statistics; I do not understand why you do not accept it?
If you randomly drew one prisoner out of one hundred, the chance that you drew the guilty one is one percent. But when you examine one particular prisoner's guilt, the probability that he is guilty is not one percent. That is not the same statistical problem.
You mean that the probability that he is not guilty is one percent (as noted, 99 out of 100 are guilty and 1 is not guilty).
In any case, perhaps I do not understand statistics, but I truly do not understand the difference between the two formulations you presented. When I draw a random prisoner and ask what the chance is that I drew a guilty prisoner, I am in fact asking what the chance is that the prisoner drawn is guilty. These are two dependent claims: if I drew a guilty person—the prisoner drawn is guilty; if I drew a non-guilty person—the prisoner drawn is not guilty. And if indeed the chance that I drew a guilty prisoner is 99% (as you admit), that means the chance that the prisoner drawn is guilty is 99%.
Let me try again.
When there are two witnesses who see a murder, everything depends on the reliability of their testimony. If the reliability is 95%, then you have a 95% chance that the defendant murdered. This has nothing to do with the uniqueness of the person involved. If they saw him murder, then even if he is a proven righteous man we will convict him.
By contrast, in the case of the prisoners, if you draw a prisoner at random from the hundred, the chance that he is a murderer is indeed 99%. But you cannot convict on the basis of the draw, because in that way one of those convicted will certainly be innocent. And that is because there is no direct evidence against him, unlike testimony. Therefore, in the case of the prisoners one has to conduct a proceeding with evidence. But when we bring against him the statistical evidence of the draw in such a proceeding, he can defend himself and say that it was not him, and you have no way of knowing with 99% certainty that it was him. He can always claim that the distribution is not relevant to him because he is an exception. This is exactly as I explained regarding the case of the majority for irrigation. I explained there that when we are dealing with a person's behavior, the assumption that the distribution is uniform for all defendants is merely a result of our ignorance. When he is accused, he can claim that his character is different from the others', and you have no way to rule that out. Therefore it is not correct to say that the probability that he is a murderer is 99%. It is correct to say that out of 100 draws, in 99 of them the persons drawn will be murderers. The move from an expected value to a specific number is the problematic one. The number is indeed 99, but that does not mean that the probability for each individual is 0.99.
In other words: the probabilistic comparison is correct if you convict on the basis of the draw as such, because the draw is a blind matter and only one of those drawn is not a murderer. But in a proceeding in which the probability is brought as evidence, this is not correct. There is no 99% probability that the person standing before you is a murderer. There is no definition at all for such a probability, because murder is not a random action (even though the drawing is).
Read again my explanation of the majority for irrigation and the Ramban's case of sivlonot.
I think this is also connected to the distinction concerning the direction of probabilistic dependence. Briefly: when you are told that a certain judge is of 99% quality, this means that out of 100 cases he will rule correctly in 99. Does it follow from this that if he ruled that Reuven murdered, there is a 99% chance that he indeed murdered? Certainly not. The probability can be tiny. I explained this in the article in Assia and in the posts on conditional probability. You can search the site for a discussion of Munchausen syndrome by proxy.
Here too there is a difference between the question of the probability that the murderer will be drawn from the group, and the question of the probability that this specific one who was drawn is a murderer. It is easier to understand this in a case where there is one murderer out of the hundred. But the principle is also true for 99.
As I understand it, there is no connection between our issue and your argument regarding Munchausen syndrome by proxy (or the reliability of a test for a rare disease). There the fallacy stems from the fact that the prevalence of the phenomenon is extremely low relative to the reliability of the test. Similarly, it is indeed correct to say that one cannot infer from the fact that a judge rules correctly in 99% of cases that if he ruled that the defendant murdered, then there is a 99% chance that he indeed murdered. But again, that is only because of the assumption that a large majority of defendants truly are not murderers, and therefore it is enough that one percent of the innocent are declared murderers in court for the group of innocents among all those declared murderers to be much larger than the group of murderers declared murderers in court (and this despite the fact that 99% of murderers are declared such in court, because even so the quantity of 99% of the murderers does not equal the quantity of one percent of the innocents). But none of this has any connection to our issue. In the prisoners case, it is known that 99% of the prisoners are guilty. Therefore the fact that the percentage of guilty people in the overall population is very small has no significance, since the prisoner before us comes from the group of prisoners in which, as stated, the ratio is 99% guilty.
This can be compared to a test that always returns a positive result (guilty) for a disease whose prevalence in the tested population is 99%. We may now ask: what is the probability that someone who tested positive (all of them) is truly infected with the disease (guilty)?
According to the formula for conditional probability, one must divide the probability of being both sick and testing positive by the total probability of testing positive (that is, among those who tested positive, how many are truly sick). Since the probability of testing positive is 100% (everyone tests positive), the probability of being both sick and positive is in effect equal to the probability of being sick (because all the sick test positive). As stated, that probability is 99%. We now divide this by the total probability of testing positive, which as stated is 100%, and again we get 99% (0.99 divided by 1 equals 0.99). Therefore, the probability that someone who tested positive is truly sick is 99%. Applying this to our case: if we convict every prisoner who comes from a group that is 99% guilty (even if he claims innocence, assuming that all the prisoners—even the guilty ones—will claim innocence), the probability that the convicted person is indeed guilty is 99%.
[Of course, there is really no need to get entangled with conditional probability, because we convict the defendant in every such case, and therefore the probability of being convicted and truly guilty is equal simply to the probability of being guilty. But because you compared this to Munchausen syndrome by proxy, I wanted to show that even if we use conditional probability, this is still the result.]
You are talking about Bayes' principle, which indeed does not apply here. But I do not think the rabbi meant that at all. The idea here is the difference between statistical evidence and concrete evidence (which the rabbi divided into leita and ita kaman). A prisoner cannot have a 99% degree of guilt; either he is guilty or he is not, and we lack information. So the treatment cannot be based on his belonging to the prisoner group, for surely we would not convict the whole group because each one is 99% guilty. There is an innocent person here, and each one can claim that he should be judged a priori as a person according to the statistics of an ordinary person (rov de-leita kaman).
This is a slippery argument, because it mixes mathematics with legal ethics. Note: the claim is not purely mathematical.
Yaakov, I now see that there is a long back-and-forth between you, and I only read the last response, so perhaps I completely missed your point. My apologies.
All this is known. Therefore I wrote at the end that it is worth thinking about the case if there were only one murderer and not 99. And still you see that those two are not the same thing. My claim is that everything depends on the probability that he is a murderer (in Bayes' formula, as you wrote), except that you assume that probability is 99%, whereas I say it is not. That is the probability of drawing the murderer, not the probability that this specific defendant is the murderer as a result of the proceeding.
It can also be formulated this way. There is complete symmetry among all the prisoners. Each one may be innocent (despite the low probability), and therefore when you choose one of them, you need evidence that singles him out as a criminal (we have already established the idea that the number 99% means nothing; there is symmetry in that number among all the suspects, and you cannot convict them all). This is group probability, not individualized information, and here Bayes' principle does enter in. When you choose one prisoner and put him on trial, he asks: why did you take דווקא me out of all 100? What evidence (perhaps a rov de-leita kaman) distinguishes me from the group?
To Michi,
Nice, that last line made it clearer for me.
I enjoyed it. Thank you!