On Statistical and Deterministic Implication (Column 402)
A few days ago I was asked in the Q&A on the site the following question:[1]
| With God's help I wanted to ask what the rabbi thinks about the atheist argument from divine hiddenness/lack of intervention, etc. To the extent that there is a God, it is not plausible that there would be hiddenness. There is hiddenness, and therefore it is plausible that there is no God. |
The logic of this argument is based on a simple and well-known logical rule: if we are given X -> Y (X entails Y, or: if X is true then Y must be true), then it necessarily follows that also Y -> X (that is, if Y is not true, this necessarily entails that X is not true). The proof is by contradiction. Assume that X were in fact true; that would entail Y (by the assumption), but we are given not-Y. We arrive at a contradiction, and therefore the assumption that led to the contradiction (that X is true) must be rejected. This rule is called in logic 'modus tollens' (Modus Tollens), or in short MT (denial of the consequent).
Thus, precisely according to my own view—which holds that there is no active divine involvement in the world—the conclusion that there is probably no God would indeed seem to follow. I’ll note that this is likely also what underlies the insistence of many good and worthy people on assuming that there is divine involvement in the world (despite the lack of any indication for it, and when all of reality and our straightforward perception of it say the very opposite). I estimate that this stems, among other things, from the fear that otherwise they will find themselves forced to conclude that there is no God. If so, at least for me this is no less a reason to refute this failed atheist-frum (religious) argument.
In my reply to this question I answered as follows:
| A weak claim, for several reasons:
1. Why is intervention plausible? If He created a world with fixed laws, presumably He wants the world to run that way. 2. There is strong evidence for His existence, independent of this (see in my notebooks and in my book ‘The First Being’). 3. Conditional probability involves quite confusing illusions. Even if you accept that the probability that, if there is a God, there will be intervention is high, that does not necessarily mean that the probability that, if there is no revelation, there is no God is high. Unlike deterministic implication (if there is a God there is revelation, and therefore if there is no revelation there is no God—which is of course a valid argument), when you move to probabilistic implication you discover a different phenomenon. You can make use of Bayes’ formula and see what the relevant factors are. (One of them is the a priori chance that God exists. I addressed this in point 2.) |
Since this question rests on very common and very confusing logic (it recurs in various forms quite often even here on the site; see for example here), I thought it appropriate to explain my answer a bit more.
Preliminary analysis: between hard implication and soft implication
For the purpose of this discussion I will ignore point 1 in my answer and assume that indeed it is plausible that if God exists He would be involved in the world. The logic of the atheist’s argument in the question ostensibly relies on the MT rule. So what is problematic there after all? The key word here is “plausible.” The logic of the MT rule described above deals with deterministic implication, i.e., hard implication (if A then necessarily B). The atheist’s question, by contrast, deals with statistical implication, i.e., soft implication (if A then it is likely that B). Surprisingly, as we shall now see, that makes all the difference. To that end, I’ll offer a few simple preliminaries from probability theory. I’ll note that in several previous columns I already used Bayes’ formula and conditional probabilities, but here I will try to explain them in simple and clear language because that is the focus of the matter.
Conditional probability
To clarify the concept, let us take the following question as an example:
- What is the chance that a fair die lands on face 5?
The answer is of course 1/6.
Now I ask a different question:
- Given that the result is odd, what is the chance we get a 5?
Here the answer is of course 1/3.
What is the difference between the two formulations? The second formulation deals with a conditional question: given some datum, what is the chance of that event. The additional information in the datum may change the answer, and that is indeed what happens here. This is what is called “conditional probability.”
We can understand this if we recall that probability is based on counting the possibilities under consideration out of all possible events. In the first formulation there are a total of 6 possible events (6 outcomes of rolling the die), and the examined possibility is only one of them: the outcome 5. Therefore the chance is 1/6. In the second formulation the number of possibilities drops to 3 (there are only three odd outcomes), and therefore the chance of getting one of them rises to 1/3. I’ll note that additional information (the extra datum) always reduces the total number of possibilities, and therefore also increases the probability of the final result. Hence a conditional probability is always greater than an absolute probability.
A current example
Just this morning (Mon) I saw a headline about the NBA Finals series presently taking place between Phoenix and Milwaukee (highly recommended viewing). At one stage Phoenix led 2:0, and last night Milwaukee narrowed it to 2:1. Commentators said that in the entire history of NBA Finals only four teams have come back from a 0:2 deficit to win the series, and if Milwaukee manage to do so it would be a unique and impressive achievement. If there were 100 Finals series, ostensibly we’re talking about a 4/100 chance. But that’s not right. These things were written after Milwaukee had already won once and narrowed it to 2:1. In such a situation the estimate of the chance that this will happen should change. We have additional information. If we want to estimate now their chance of turning the series in their favor, we must examine how many teams have come back from 2:1—not how many came back from 2:0. In other words, out of the 100 Finals series held to date there were 4 teams that were down 2:0 and won. But there were cases in which the series never reached 2:0. Suppose for the sake of discussion that out of the 100 Finals there were 30 in which one team led 2:0. Of those, 4 teams turned the series around. Thus the chance is 4/30, not 4/100.
But that’s not all. Of those 30 series, there were 10 in which the score progressed from 2:0 to 2:1. If we assume that all four teams that turned the series are included in those 10, the chance rises to 4/10 (that’s almost 1/2—admit it already looks far less surprising and impressive). But even that is not the whole story, because there were also several other teams that were down 2:1 without passing through 2:0. Perhaps we should include them too in the calculation?
In short, estimating the chance of advancing in this case is not at all simple. For our purposes, what matters is that given that we are now already at 2:1, it is incorrect to estimate the chance of reversing the result as 4/30. The relevant probability is the conditional one, for a team that was down 2:0 and narrowed to 2:1, and now we must ask how many of those teams eventually also reversed the series. We saw that this chance is much higher. As noted, when there is additional given information (that the series has already reached 2:1) the number of possibilities is smaller, and therefore the conditional probability is greater.[2]
The connection between the conditional probability and the probability of the datum
It follows clearly that there is a connection between the ordinary probabilities of the additional information given to us (such as the probability that the result will be odd, which is 1/2) and the conditional probabilities related to it. For example, anyone can understand that the connection between the probabilities in the two formulations I presented will somehow depend on the probability that the result is odd. We can see this easily if we examine a third formulation:
- Given that the result is greater than 4, what is the chance we get a 5?
Here the number of possibilities is 2 (either the result 5 or 6), and therefore the chance of one out of the two is 1/2. The chance of getting 5 here is greater than that obtained in formulation B, because the probability that the result is greater than 4 is only 1/3—i.e., smaller than the chance that the result is odd (which is 1/2). We see here an inverse relationship between the ordinary probabilities of the datum and the conditional probabilities of the result (the larger the given probability, the smaller the conditional probability). At least in this example this also sounds very reasonable.
For what follows I will now denote the different probabilities as follows: P(A) denotes the probability of event A. The conditional probability of event A given B will be denoted: P(A/B). For example, if the datum B is that the result is greater than 4, then: P(B)=1/3, and the conditional probability of getting the result 5 (this is event A) is: P(A/B)=1/2. [Note that in denoting the conditional probability I use a slash, but with a different meaning. I will use it in this way from here on when it appears inside probability parentheses. In any other case, it is an ordinary division slash.]
Just for comparison, in the question in formulation B the given event B is that the result is odd, and its probability is: P(B)=1/2. The conditional probability obtained for this case is: P(A/B)=1/3. The connection to the analysis of formulation C seems simple and clear, but don’t be too impressed. This is a very simple and intuitive case. In other cases the connection is more complex, but still the basic logic is always similar to what I presented here.
Bayes’ formula
So far this sounds fairly simple. What confuses people about conditional probability is that in an imprecise formulation of a given problem (in life, in philosophy, or in any other field) it is not always easy to notice that we are dealing with a conditional probability rather than an ordinary probability. And even if you do notice, it is not easy to understand what is the datum and what is the examined case. The distinction between these two is truly critical, and many confusions in diverse questions from various fields hinge on it. This is the place to recall that in a few columns in the past (144–145 and also 176) I invoked Bayes’ formula (or: the formula of total probability), and in all of them the goal was to remove such confusion. In all of those cases the crux was understanding that we are dealing with conditional probabilities, and the difficulty of identifying which variable is conditioned and which variable is the condition. As we shall see below, this is also the situation in the atheist’s question presented above.
Bayes’ formula deals with the relation between conditional probabilities and the relation between them and ordinary probabilities. It is a very useful formula, and you can see in the above columns and elsewhere on the site and in general how necessary it is to dispel confusions and pseudo-problems. Bayes’ formula addresses complex cases involving many events, each of which can be conditioned on others. But for our purposes here it suffices to deal with just one special case, namely the case of two events only, denoted A and B. As noted, the formula determines the relation between the probabilities of the two events and their conditional probabilities upon each other.
To understand the formula, we must preface the probability of obtaining both events together, denoted: P(A^B). If the two events A and B are independent of each other, then the probability of obtaining both is the product of the absolute probabilities of each of them, P(A)*P(B). Thus, for example, the chance of getting a 5 and on the next roll again getting a 5 is: 1/6*1/6=1/36. But we are dealing here with events that are dependent on each other (we saw above that the chance of getting the result 5 depends on whether in that same roll the result is odd or greater than 4). In such a case the chance of obtaining both events is:
P(A^B) = P(B/A)*P(A)=P(A/B)*P(B)
When the events are independent, the conditional probabilities equal the absolute probabilities: P(A/B)=P(A) and P(B/A)=P(B), and then you get the earlier results.
The right-hand equality in the last formula gives us Bayes’ formula, and its importance is that if we have one conditional probability, say P(A/B), we can flip its direction and ask what the opposite conditional probability is: P(B/A). Now you can also see that the relation between these probabilities depends on the absolute probabilities of the two events, each separately, exactly as we saw in the examples above. If one of them is large, that does not necessarily mean that the other is large, and vice versa.
Above we presented an atheist argument that is based on soft implication. We are now ready to understand why the MT rule, which is valid for hard implication, does not necessarily apply to soft implication. But before that I will present the flaw in this atheist argument from a different angle.
The atheist’s argument begs the question
If the absolute probabilities P(A) and P(B) are equal to each other, then we can cancel them and the conditional probabilities are equal to each other. In such a situation we may infer that if one is large then the other is large, and therefore if this is plausible then that is plausible as well. In that case we may apply MT also to soft implication. But when these probabilities are not equal, the situation becomes complicated.
To analyze the question in terms of Bayes’ formula, we begin by identifying our variables. The atheist’s assumption is that if God exists (event A) then it is likely that He is involved (event B). That is, he assumes that P(B/A) is high. But now note that his conclusion deals with the reverse direction: he assumes that He is not involved and infers from this that it is likely He does not exist.
Note that in our case P(A/B) is 1, since if He is involved then He certainly exists. Beyond that, the a priori chance that He exists, P(A), is also very high (this is point 2 in my answer above). But if so, Bayes’ formula gives us:
P(B/A) = P(B)/P(A)
The conclusion is that the atheist-frum assumption that P(B/A) is high is simply not correct.
Of course, if you assume a high plausibility that God does not exist—that is, that P(A) is low, i.e., approaching the chance of His involvement P(B) (which is also small),[3]—then you will obtain that the conditional probability is high, but that is begging the question, since you have already assumed that God’s existence is implausible.
What, then, is the conclusion we should draw in such a case? As we have seen, one can assume whatever one wants and get a consistent conclusion. Thus the atheist’s argument, which sought to prove that God’s existence is implausible, fails.
Back to statistical (soft) implication
That was an answer to the substance of the question. Now I will try to take the bull by the horns and show the general conclusion to which this column is devoted: one must not apply the MT rule to soft implication. We are essentially looking for the conditional probability that if He is not involved then He does not exist: P(A/B). The questioner inferred that if P(B/A) is high, then necessarily this chance should also be high.
It is important to understand that this is exactly the application of the MT rule to soft implication. Hard implication says that if A then necessarily B. Soft implication says that if A then it is likely that B. In other words, it says that the conditional probability P(B/A) is high. The question is whether, according to the MT rule, we may infer from this that the opposite soft implication is also correct—that is, that the conditional probability P(A/B) is also high. Let us check whether it is indeed correct to assume this.
First we must express this conditional probability in terms of P(A) and P(B). Here we simply swap the variables in Bayes’ formula:
P(B/A)*P(A)=P(A/B)*P(B)
Of course, the following relations hold:
P(A) = 1 – P(A) ; P(B) = 1 – P(B)
(The probability that God exists + the probability that He does not exist = 1, and likewise for involved and not involved.)
Note that in light of what we saw above, the left-hand chance is very low (there is good evidence that God exists, regardless of His involvement in the world) and the right-hand one is very high (there is no indication that He is involved in the world). In addition, the opposite conditional probability is of course: P(B/A)=1, for if He does not exist He cannot be involved.[4]
If we now substitute these three data into Bayes’ formula, we obtain:
P(A/B) = P(A)/P(B)
which is of course a very small number (a small number divided by a large number).
What have we obtained? Two important conclusions regarding the application of the MT rule to soft implications:
- Even without assuming anything about the absolute probabilities, we can infer that the MT transition in soft implication is not necessary and we must not assume it offhand.
- In light of the considerations I presented (point 2 in my answer to him), it is a priori clear that in our case P(A) is high and P(B) is low, and therefore here it is also false—not merely unnecessary.
The conclusion regarding the atheist’s argument is that even if we adopt the assumption that if God exists then it is likely that He is involved, there is no necessity that the opposite direction obtained by the MT rule—namely, that if He is not involved then it is likely He does not exist—also has a high probability. On the contrary, as we saw from a priori considerations, this probability is very low in our case.
We have shown here that soft statistical implication is not subject to the MT rule, except in very special cases. Now we will point to another implication of this insight, and in fact connect it to previous discussions we held (beyond the columns on Bayes’ formula to which I already referred above).
The Raven Paradox
About two years ago a question about Hempel’s Raven Paradox reached my site. By way of background I will mention that Karl Popper argued that a scientific theory can only be falsified and not proven. For example, the theory that all ravens are black cannot be proven (unless somehow you manage to verify that you observed all ravens—and even then the theory ceases to be a theory and becomes an observational claim). But a single raven that is not black suffices to refute it. Other philosophers of science argued against Popper that even if a scientific theory cannot be proven, it can be confirmed (corroborated). Every raven we observe and find to be black strengthens or confirms our theory (even if it does not prove it).
Carl Hempel attacked the notion of ‘confirmation’ by means of the following paradox. The claim “all ravens are black” is logically equivalent to “everything that is not black is not a raven.” This is an application of our acquaintance, the MT rule of course. If so, according to those who support the possibility of confirming a scientific theory, something odd follows. How do we confirm the theory “everything that is not black is not a raven”? We observe objects that are not black and check whether they are ravens or not. It follows that every pink table we observe confirms the theory that everything that is not black is not a raven. But this claim is equivalent to the claim that all ravens are black. Thus we learn that observing a pink table confirms the claim that all ravens are black. This seems very odd.
I have already explained in the past (see columns 221 and 87) where the flaw in this argument lies. Here I merely wish to point out the connection between that flaw and the discussion in this column. Note that a claim about confirmation deals with plausibility and not with absolute claims. The existence of a single black raven confirms the claim that all ravens are black, i.e., that if we have seen one black raven, then it is (more) plausible that all ravens are black. This is exactly what I defined here as soft implication. If so, we should not be surprised that the MT rule does not necessarily apply to it; that is, that confirmation of the softly equivalent claim does not necessarily confirm this claim.
In probabilistic phrasing, we can express it thus. The claim that if something is a raven then it is likely (but not certain) that it is black is not equivalent to the claim that if something is not black it is likely not a raven. The MT rule does not apply to soft implications. And since, according to Popper and Hempel (and also according to the truth, of course), laws of nature are not necessary claims but plausible ones, it is very important to be careful in applying the MT rule to them. In other words, when speaking about probabilities of claims rather than their truth, then a measure of the plausibility of the equivalent claim (everything that is not black is not a raven) is not necessarily directly proportional to the plausibility of the original claim (every raven is black).[5]
The strength of this connection naturally depends on the probabilities of objects being black or being ravens (the absolute probabilities). The more objects there are that are ravens, or objects that are not black, the weaker the MT connection becomes. And this is precisely the flaw I pointed out in those columns. It is easy to see that the additional examples discussed in those columns are connected to my contention here regarding applying the MT rule to soft implications.
Back to applying MT to soft implication: summary and demonstration
This is an addition after I saw in the comments that there is a lack of understanding regarding my latest claims. I offered here two formulations for rejecting the atheist’s argument. The first argument shows that it begs the question, and the second explains that one must not apply MT to soft implication. I will now clarify and sharpen both again, this time using the Raven Paradox to help me do so.
In the comments they presented the atheist’s argument as follows. I showed in the column that
P(B/A) = P(B)/P(A)
If one assumes (and I agreed to this for the sake of discussion) that P(B/A) is high, and also that P(B) is low (which I also accept), it necessarily follows that P(A) is low. I answered that this is true but worthless, since the atheist is begging the question. I, as a believer, claim that P(A) is high (because there is very good evidence for God’s existence irrespective of the question of His involvement in the world). And hence I am in contradiction in light of the last formula. There are two possibilities: (a) give up the assumption that P(B/A) is high; (b) give up the assumption that P(A) is high. You cannot live with both. The atheist, in his argument, assumes that P(A) is an open quantity and therefore adopts option (b). I think the evidence for His existence is excellent and therefore I adopt option (a) (and thus am, of course, forced to give up the assumption that the conditional probability is high). Therefore I argued that he is begging the question. This is the basis of claim 2 in my answer to him at the beginning of the column.
In the second formulation of my rejection of the atheist’s argument I showed that one must not apply MT to soft implication. I have now thought of a good illustration of this. To that end I will take the Raven Paradox that I analyzed at the end of the column, because it allows me to make an explicit numerical calculation and prove my claim about MT. Suppose, for the sake of discussion, that the chance that if something is a raven then it is black is high. Can we derive from this that the chance that if something is not black it is not a raven is high? Let us assume that there are 1,000 ravens in the world, of which 990 are black. Besides them there are 10 other objects in the world, of which 9 are black. We can now calculate the chance that if X is a raven then X is black. The result is 0.99—very high. And what is the chance that if X is not black then X is not a raven? The result here is 0.1, which is rather low. The explanation for this gap is of course the very high absolute chance of being a raven (almost 1), and the very small absolute chance of being not black (0.01). This is exactly the situation in the theological discussion, except that there it is difficult to show it with numerical calculation.
[1] I very much liked the “With God’s help” at the opening. This of course reminds me of Anselm’s prayer at the beginning of the ontological argument presented in his Proslogion. See on this at the beginning of the first notebook, or at the beginning of the first conversation in my book The First Being.
[2] Admittedly, if the discussion is about the esteem owed to a team for its fighting spirit and resilience, then none of this is relevant. And if in the end they succeed, it will be correct to say that they belong to the five teams that came back from 2:0, did not break, and reversed the result. The esteem they deserve is indeed determined according to the estimate of 4/30.
[3] The chance that He exists is always greater than the chance that He is involved, since for Him to be involved He must, in particular, exist, and even then it is not certain that He intervenes. Therefore the conditional probability here is always less than 1, as we would expect of a probability.
[4] Incidentally, this is itself an application of the MT rule, except that here it is applied to hard implication and therefore it is legitimate. The conditional probability P(A/B) = 1. This means that the implication “if He is involved then He exists” is a hard implication (since we are not speaking about plausibility but about an absolute entailment: its probability is 1). Therefore we may infer from this, according to the MT rule, that if He does not exist then He is not involved; namely, that the conditional probability P(B/A) = 1.
[5] There is a very delicate point here. If there are two claims that are logically equivalent, then the probability of one must be exactly equal to the probability of the other, since we are dealing with the same claim. Therefore it is clear that it must be the case that: P(A -> B) = P(B -> A). My claim above concerns a different relation: the relation between the claim A -> P(B) (if something is a raven it is plausible at level P that it is black) and the claim B -> P(A) (if something is not black it is plausible at level P that it is not a raven). The phrasing above is not precise, but I adopted it for the sake of simplicity and clarity.
Discussion
That may answer the teretz. But the kushya is completely different. 🙂
Divine intervention is before the verdict is issued.
Once the verdict has been decreed (a terminal illness, a crashing airplane, a broken leg, losing a million on the stock market), the possibility of miraculous divine intervention is extremely low, even if people pray.
Just as a judge does not alter a justified sentence even if thousands of advocates speak in favor of a criminal.
Therefore, dear Michi, this is why we do not see differences between groups of terminally ill patients for whom people prayed and unbelieving patients for whom people prayed.
Consider it.
But in terms of the number of sick people among those who pray, the statistical difference is completely significant. Interesting. The wonders of apologetics.
A. This is unclear. I’ll restate your point and then continue to the calculation.
The atheist’s argument assumes that the probability God would intervene, given that He exists, is high; and since it is given a high probability that in fact He does not intervene, the conclusion is that the probability that He exists is low. Completely valid. You say one can assume from the outset that the probability God exists is high, and then, by Bayes, reject the atheist’s assumption that the probability God would intervene, given that He exists, is high.
But that is exactly what we are debating.
A small and agreed-upon introduction: there is a hypothesis h1 to which we assign probability 0.8, and it predicts the data with probability 0.5 (that is, if the hypothesis is true, then the chance we will see the results we got is 0.5). And there is a hypothesis h2 to which we assign probability 0.2, and it predicts the data with probability 1. The question now is: which hypothesis is more probable?
The answer is that h1 has 0.8*0.5=0.4, whereas h2 has 0.2*1=0.2, and therefore h1 has probability 2/3. That is, we still think hypothesis h1 is more probable, but the data caused us to update its probability to 0.66 instead of 0.8.
This is exactly how the argument from hiddenness works. The higher a probability we assign to the claim that God would intervene if He exists (and to the claim that He does not intervene), the more we will have to lower the probability that God exists. Although if we started out with a high probability that He exists, that probability can still remain high—but lower.
Suppose we assign to God’s existence a probability of 0.9. And it predicts the data that He does not intervene for the better with probability 0.2 (that is, the chance that God will not intervene for the better, given that He exists, is 0.2). The probability of God’s non-existence is 0.1, and it predicts the data that He does not intervene for the better with probability 1. In that case, the updated probability that God exists drops to 0.64. Do you agree with this calculation? [If one assigns God’s existence probability 0.99 and the state that He will not intervene for the better probability 0.01, then the probability of God’s existence drops to a little less than half.]
B. There is a God (there are good proofs), and He does not intervene for the better (there are fairly good proofs). The conclusion, of course, is that God is not good. The strange insistence that God is good seems to me utterly childish, and the argument from hiddenness slaps it forcefully across its astonished face. (I read the response on the site based on the continuity of the laws of nature. When the topic comes up on the site sometime, I will defend in detail my pompous certainty. Let the good God kindly replace laws or people, or perform more miracles.) It is natural to assume that in His eyes people are like lice, and talm ve-latzag vekal.
[For the benefit of a casual reader—the matter is called MAP estimation. I haven’t found a good explanation of it in Hebrew online at the moment, so whoever wanders on Google may wander.]
“I should note that added information (the additional datum) always reduces the total number of possibilities, and therefore also increases the probability of the final result.”
If the datum were that the result is even, the probability would drop to 0.
The whole discussion of the probability that something exists or does not exist is mistaken.
If something exists, then it exists, and if not, then not. There is no probability in that.
Exactly. Therefore, even if non-intervention lowers the chance that He exists, the question is what that chance is, not whether the chance went down or not. My claim is that regarding the question what the chance is, the answer is not unequivocal.
As for God’s goodness, that is a different discussion, and you already mentioned my claim about it.
Dear posek, this is how decisions are made in life: suppose you are a doctor and your patient has bleeding in his stool. The chance that he has such a tumor in his intestine when there is bleeding is 0.8. Would you send him for a colonoscopy, or would you say probabilities are irrelevant here—either there is a tumor or there isn’t?
And in the context of the article: if one assigns more predictive value to our theological assumptions about how God is supposed to behave than to the assumptions that ground God’s existence, then S'mindelof is indeed right that the fact that He does not intervene lowers the probability of His existence. But if one understands that these assumptions are ridiculous once one understands that God is not an entity that is supposed to behave according to our rules, then in order to understand whether there is a God one should deal with what decides the question itself.
Actually, in addition to the 3 rebuttals that the mara de-atra wrote, it seems to me that the most ridiculous thing in the argument is its hidden assumption that if there were a God, we would know how to predict how He is supposed to behave.
So each person has to attach the probabilities he thinks appropriate, and then calculate which hypothesis (there is/isn’t God) comes out better in the end, and the atheist’s argument is perfectly fine. When I, for example, attach the probabilities that seem right to me (it’s hard to assign probabilities, but we go with it), it comes out to me that it is more likely that God does not exist. And the only way out is to give up the hypothesis that He is good (especially since even apart from the data I assign it a small prior).
We use that hidden assumption all the time. When God commands you the Torah, you assume that His will is that we keep the Torah and not, on the contrary, that His will is precisely that we should not listen to Him. Therefore, if you pull the rug out from under reasonable assumptions about God, then all observance of the commandments collapses. So clearly everyone makes assumptions about God that seem reasonable to them. The assumption that God is good is also an assumption about God, incidentally.
Avishai. What you are describing is confusion. The following sentence is not correct: “The chance that he has such a tumor in his intestine when there is bleeding is 0.8.”
What is correct is that the chance of finding this among past patients in a similar condition is 0.8. Regarding the new patient, you cannot state the probability, because either he has it or he doesn’t. You can say that if in the future you have many such patients, the chance that you will be mistaken is 0.2, but that is only if there will be many such cases and you make an average calculation over all of them. That is, the probability is not whether he has it or not, but whether you are wrong or not wrong in your guess that he has it.
But what matters is that the probability was given after there were a number of patients in a similar condition.
As for the question whether some specific thing exists or does not exist, one cannot assign a probability, because it is something that happened or did not happen only once. That has nothing to do with probability.
Posek—if I guess correctly, then the tumor necessarily exists, so I do not understand the point of saying that probability is irrelevant with regard to existence (if I am right in my guess, then the probability is 1 that it exists). In any case, even on your view, I only said that this is how decisions are made, so perhaps one only needs to change the wording and speak about the correctness of our hypothesis regarding God’s existence and not about existence itself.
S'mindelof—I also make assumptions about God and do so endlessly, but I understand that the certainty of those assumptions is weak. I have no tools to understand anything with respect to God’s essence, and all the anthropomorphism we engage in regarding Him (assuming He is good, that He likes it when people listen to Him) is not a condition that defines His existence, but rather more our interpretation of what we understand from His revelation in the world.
Someone who defines God as only what is revealed of Him in the world, and says that God has to fit his values or else He does not exist, seems ridiculous to me, even though I too have all kinds of assumptions—because for me, God is not only my assumptions about how God ought to behave.
By the way, in terms of the structure of this objection, the common “Where was God in the Holocaust?” is built the same way.
I didn’t understand the answer. Why is your assumption that God wants us to observe His commandments, and not the opposite—that we should transgress them—an assumption by whose force Jews lived and died for sanctifying His name, any less ridiculous and anthropomorphic than the assumption (which I do indeed make) that if God exists and is able and good, then He would intervene?
This is an even greater kind of confusion, in which many err out of lack of understanding.
A tumor is not a thing that exists. What exists does not change in time. What changes in time does not exist.
Nothing we know exists. These are transient phenomena.
In my opinion, the question “Where was God in the Holocaust?” (and in many other places) is an objection as hard as iron, and one ought to draw conclusions from it.
There is no significance because everyone sins.
If the religious/Haredi people of Israel were completely and unmistakably righteous, then there would be a bit more room for your reasoning about intervention (although even then one could explain it away with reincarnations), because then it really would seem strange that religious people who are truly like angels get sick like everyone else and lose money on the stock market like everyone else.
But look right and left and you too will see: everyone sins—Haredim, religious, secular—
robbery, gossip, reckless driving, lying, adultery.
So does God work for us—that we’ll sin here and there, and in the end pray and expect everything to turn out fine because we were taught that if we pray then everything will change for the better?
Are those your expectations of God?
Hello,
I haven’t yet read the whole trilogy, but I did read the section on revelation,
and if in your view there is no involvement, then how could revelation be possible?
Heaven forbid. Not ridiculous or anthropomorphic—exactly to the same degree.
Both assumptions are excellent.
The opposite assumption is ridiculous: if He does not intervene on the basis of my assumptions, then He does not exist. I am not claiming that since I think that if there were a God He would want Torah observance, then if He does not want people to listen to Him, He does not exist.
I have no problem with one assumption or another; I have a problem with the decision that these assumptions carry weight in the question of God’s existence.
If you are dealing with His goodness—if you assume that good = intervening in the Holocaust, then if He did not intervene, He is not good.
A correct conclusion and reasonable assumptions, but not necessary ones. And they say nothing about existence. (The additional assumption—if not good, then nonexistent—is, in my opinion, the root of the problem, because that one must be multiplied by a very low certainty coefficient—how do you know that His existence depends on your assumptions about what counts as good?)
Posek—as for the existence of things in our changing world, that is beyond me. The first Posek did not decide the matter in Yesodei HaTorah 1:1, so I do not know.
And on the substance of the matter—I understand that only with regard to God there is no probability. And I am willing to accept that if you agree that with regard to all other things there is probability in relation to what the world calls their existence.
In our everyday world of happenings there are no probabilities, there are likelihoods.
And in practice, how does that affect anything, in your opinion?
In that I wrote it in a comment. Other than that, not at all.
In the past there was. Creation of the world is also involvement, no? Every prophecy is involvement, and so is every miracle. My view on this matter is set out in detail in the second book (and also here on the site in several places).
I didn’t understand why MT is not valid for soft implication—in this particular case p(A'|B') is p(a')/p(b'), meaning one minus p(a) divided by one minus p(b); this quantity depends on the size of p(b)/p(a), which equals p(B|A). (I couldn’t write the math in a reasonable way so I used verbal description. I used ' to indicate the complementary case.)
In the general case one can show that p(A'|B') equals 1 minus the probability of A minus the probability of B plus the conditional probability p(B|A)*p(A), all divided by 1 minus p(b). That is, in the case where p(B|A)=1 it is 1 (this is hard implication), and in every case the closer p(B|A) gets to 1, the closer p(A'|B') gets to it.
What you showed in the post is that if we give up the assumption that p(B|A) is high, then p(A'|B') is not high either.
Not true. I showed that even if one conditional probability is high, the other is not necessarily high. It depends on the values of the absolute probabilities. And from that it follows that the MT rule is not valid for soft implication.
And indeed, as I wrote, when the conditional probability is 1, then the opposite one is also 1. That is precisely the MT rule for hard implication.
You are right about the statement regarding approximation, and Sandomilov already noted this above. But you are not right about the absolute value (whether it is high or not). The atheist argument does not claim something about getting closer; it claims something about the absolute values.
At long last the opportunity has presented itself https://mikyab.net/posts/71241#comment-49327 .
Is a Torah scroll holy with the holiness of content (h1) or the holiness of representation (h2)? Let us assume these two hypotheses are a priori equivalent in our eyes. Now suppose it is given that it is holy in any language (A). According to h1, the probability of A is 1, because the representation does not matter. According to h2, it is still possible that all possible representations are holy with the holiness of representation, but it is also possible that a specific representation is required, namely the holy tongue. Therefore according to h2 the probability of A is, let us say, 0.8. Given A, is that evidence in favor of the hypothesis that a Torah scroll is holy with the holiness of content? In a formal calculation it comes out that yes. Because 0.5*1>0.5*0.8. Therefore the probability of h1 has now risen to 5/9 instead of 0.5. This is a move you made in article 381 (and similarly also in article 380).
Is this indeed, in your view, a valid way to draw conceptual/Talmudic conclusions? That is, if law A necessarily follows from hypothesis h1, whereas hypothesis h2 is indifferent to it (that is, orthogonal to it and says nothing about it), then law A confirms hypothesis h1?
I once collected several examples where, in my humble opinion, the Rishonim and Acharonim do not use such a method, and if something works well enough under both hypotheses, then it is not interesting that it is strictly required by one of them. Unfortunately I lost them. But I would be glad if you would confirm that in general you accept and use such a method in halakhic proofs.
Absolutely correct. I used a similar line of reasoning in my comments about Rav Shach in article 87. Of course, such a consideration adds only a bit of probability to one hypothesis, but does not prove it. Therefore one must be careful in using this method. See what I wrote in the aforementioned article.
I didn’t notice this distinction. (In the limiting case, the probability of God’s existence is 1 and the probability of intervention is high, and then p(B|A) is high and also p(A'|B')=0.)
It seems to me that the atheist argument is based on the assumptions:
1. p(B|A) is high.
2. It is plausible that p(B') is high (from the very fact that we do not see intervention), and therefore p(B) is low.
That is, p(B)/p(A) is high and p(B) is low, therefore p(A) is low.
(Suppose for example that x=0.9; then in order for p(A'|B') to be less than 0.7, p(B) has to be greater than 0.73. And if x=0.99, then in order for p(A'|B') to be less than 0.7, p(B) has to be greater than 0.96, and with x=0.999 we get p(B)>0.996.)
There is a subtle point here, because the claim is that since we see there is no intervention, it is plausible that the probability of intervention is low.
In any case, there is no situation in which p(A) is high, p(B) is low, and p(A|B) is high.
1. The argument you presented in the atheist’s name begs the question, as I explained in the article.
2. I completely agree that the probability of involvement is low. I wrote this in the article. There is nothing subtle here. It is trivial. If one does not see intervention, then apparently there isn’t any.
3. P(A/B) is 1 by definition: if God is involved, then He certainly exists. So there definitely is a situation of the kind you ruled out at the end of the message.
1. The argument says that assuming p(B) is low and p(B|A) is high, then p(A) is low; why is that begging the question?
2. It may be that the probability of intervention is high, but in practice it did not happen. The claim from the fact that something did not occur to the probability of its occurrence adds another thing that must be considered. (Actually, this is the conditional probability that p(B) equals x given that in a certain experiment not-B occurred.)
3. Typo—I meant p(B|A).
Thank you.
A. Against this I offer another thesis: in halakhic literature, evidence against hypothesis B is only a datum that is forced/awkward according to that hypothesis. Let us say for the sake of discussion that a forced datum is one that the hypothesis predicts with less than 0.1. But if one hypothesis predicts the datum with probability 1 and the second hypothesis predicts it with 0.8, nobody cares and it is not used as evidence in favor of the first hypothesis. Although mathematically it is clear that it does not work that way, and if we started with two hypotheses that are a priori equivalent, then a situation like this of posterior predictions 0.8–1 would force us to update our picture of the probabilities of the hypotheses. Evidence one way or the other I hope to obtain in the future. If you have evidence otherwise—by all means.
B. Why, in matters of reasoning, is there no issue that one does not follow the majority in monetary law to extract from the one in possession? Suppose an Amora has the reasoning that the presumption that a person does not pay before the due date is effective to extract money. In his opinion, what is the probability that this reasoning is “indeed correct”? Let us say 0.9. Then how can it have the power to extract? One must say that after a line of reasoning is accepted, one forgets its probability. Something like when a prohibition is established by one witness and then the eater is flogged, even though one witness is not believed for flogging in direct testimony. Is that so?
And if one uses a chain of (independent) lines of reasoning to extract money or prohibit, then one should multiply their probabilities, and it is certainly possible that then there is not even a majority. So every Tanna and Amora would have to attach to every opinion the probability he assigns to it, so that we would know in which chains of reasoning it can participate. If so, why is there nothing like this in the literature?
C. The hypothesis that punishment is meant for deterrence predicts the types of punishments—lashes/punishment—with very high probability. These are the most immediate and powerful things one can do to a person. The hypothesis that punishment is meant to repair spiritual damage predicts no specific type of punishment. The idea that specifically lashes repair spiritual damage is no more plausible a priori than the idea that carving zucchinis into beetle shapes repairs spiritual damage. Therefore here the datum that the punishment in the Torah is specifically lashes strongly confirms the hypothesis that punishment is intended (also) for deterrence and does not contain (only) a component of spiritual repair. Have I understood correctly that you agree with this?
C. Following that, of course, https://mikyab.net/posts/71097#comment-49038
Conditional lashes as recompense mean that there is only spiritual repair and not deterrence or revenge.
1. I explained this and will explain again. The atheist assumes that P(B/A) is high, and leaves open the question whether God exists, meaning he assumes nothing about P(A). But because of the evidence for God’s existence in a way unrelated to His involvement, I cannot assume that P(A) is low. For me it comes out high. Now I find myself in a contradiction, and therefore I must choose one of two possibilities: 1. This assumption of mine is incorrect. P(A) really is not high. 2. The assumption about the conditional probability is incorrect. The atheist assumes 1, and I assume 2. Therefore his argument implicitly assumes that the other evidence for God’s existence is not good, and that is begging the question. I, as a believer, think that evidence is very good, and therefore choose option 2.
2. It may be so, but the probability of that is low. See my reply to Sandomilov regarding Rav Shach’s dispute about the Entebbe operation (article 87).
3. One more clarification regarding applying MT to soft implication. I suggested two formulations for rejecting the argument. The first is described in 1. In the second formulation of my rejection of the argument, I showed that one must not apply MT to soft implication. Now I have thought of a good illustration of this. Take the raven paradox that I presented at the end of the article. You claim that if the probability that if something is a raven then it is black is high, then the probability that if something is not black then it is not a raven is also high. Let us think of a numerical example. Suppose there are 1000 ravens in the world, of which 990 are black. Besides them there are 10 other objects in the world, of which 9 are black and one is not.
The probability that if X is a raven then X is black is 0.99.
The probability that if X is not black then it is not a raven is: 0.1.
And the explanation for this gap is, of course, the very high absolute probability of being a raven (almost 1), and the very low probability of being not black (0.01). This is exactly the situation in our case.
A. There is no contradiction to my remarks here. After all, I said that such an argument only increases the probability a bit, and therefore that is not enough to constitute evidence. That is what I wrote regarding Rav Shach’s Entebbe operation, and that is exactly how I answered regarding the raven paradox. Indeed, a pink table slightly increases the probability that all ravens are black, but that is far from being evidence.
B. Several later authorities wondered about this question. In fact, they formulated it this way: what is the difference between majority and presumption? Why does majority not extract money (according to Shmuel, in certain interpretations; this is far from agreed upon), whereas presumption does? The claim is that when a presumption is established, it may begin from majority, but once established it becomes an absolute principle (so long as it has not been refuted with respect to the case at hand, of course). But that is not true of every majority, only of presumptions. And the proof is that majority indeed does not extract money. And if the reasoning were as you said, that a prohibition is established by one witness, then every majority should be able to extract money.
Something like this was also discussed regarding whether majority is a state of doubt, or whether once there is a majority the matter is considered absolutely decided. A practical ramification: doubtful impurity in the public domain in a situation where there is a majority to rule stringently. Is this still a state of doubt, in which one rules leniently, or once there is a majority is it no longer a doubt, and the rule of doubt in the public domain being ruled leniently does not apply?
What is the criterion for turning a majority into a presumption? When does a majority remain a majority and when does it become a presumption? Good question, and I have no general answer to it. See for example here on a presumption that comes by force of majority: https://www.yeshiva.org.il/wiki/index.php/%D7%9E%D7%99%D7%A7%D7%A8%D7%95%D7%A4%D7%93%D7%99%D7%94_%D7%AA%D7%9C%D7%9E%D7%95%D7%93%D7%99%D7%AA:%D7%97%D7%96%D7%A7%D7%94_-_%D7%9E%D7%91%D7%95%D7%90
I do not agree. It does indeed improve the probability, and still does not make it high. In other words, my response here is similar to what I answered the atheist’s argument: since I have independent evidence (from Hazal and from the Rishonim) that punishment is spiritual recompense, therefore this question is not an open question in my eyes. See the explanatory addition in red at the end of the article that I just added, regarding the atheist’s begging the question.
In the background of this, there can be various explanations for the connection between the intensity of the recompense and the spiritual repair. For example, a person’s suffering is connected in different ways to the spiritual repair he undergoes. Alternatively, the Holy One, blessed be He, created us in such a way that what deters us is exactly what repairs us spiritually, and still the essential reason for punishment is the repair, not the deterrence. And so on.
See the addition in red that I just added at the end of the article.
Then the objection is against Hazal and the Rishonim themselves. It not only improves the probability but makes it very low, almost 0, because the probability of lashes given spiritual repair is one out of all the possibilities in the world—that is, practically 0.
A. One pink table adds a tiny bit of confirmation and is not comparable to simple cases of two hypotheses and a prediction that is required by one and open under the other (like the example of the holiness of a Torah scroll in any language). When I find an example I will return forthwith.
B. I am asking (1) why the reasoning itself is not discussed as a majority (for example, the Hazon Ish’s reasoning against the Pnei Yehoshua regarding owners where it is uncertain whether he guarded his ox from causing damage). A practical ramification: whether one extracts based on this reasoning. And (2) whether in a chain of reasonings one multiplies probabilities and sometimes drops below a majority. The answer that the reasoning becomes an absolute principle indeed explains both phenomena, but it itself seems to me completely arbitrary (although it has analogues in halakhah). So I still have not understood *why* in fact we do not calculate every line of reasoning according to the probability that it is correct.
I answered that. They apparently thought there were good independent reasons to say that punishment is not for deterrence (for example, the need for warning and its acceptance, the exemption from punishment when rather draconian conditions are not met, and more), and they also thought there is a connection between the measure of suffering and the repair.
This is not the place, but I also do not agree that punishment is only for repair and not at all for deterrence. I do not think I wrote such a thing, or that it can be inferred from my words.
I didn’t say you wrote such a thing. I said that you proposed situations in which the punishment is only repair—and they are situations of punishment without warning according to the Sefer HaChinuch, where this is conditional recompense, meaning only spiritual repair and not deterrence. Was I mistaken?
I understand. In principle, even in such situations it can be deterrence. For example, there are attempts to deter zealots from acting (zealots attack) even though the act in itself is positive.
In any case, the nature of the punishment in these situations proves nothing. The fact that lashes were chosen is because the general punishment in halakhah is lashes.
I understand.
Excellent and clear post, many thanks!
I only think there is a small pedantic mistake:
“I should note that added information (the additional datum) always reduces the total number of possibilities, and therefore also increases the probability of the final result. From this it follows that conditional probability is always greater than absolute probability.”
In my opinion there are cases where the conditional probability is דווקא smaller than the absolute probability.
The probability of cardiac arrest < the probability of cardiac arrest given that you are under age 30.
The number of possibilities for having a heart attack at any age is greater than the number of possibilities for having one under age 30. Per year it is smaller.
A sharper formulation of section B:
Suppose that in order to rule something prohibited in a certain case, four assumptions must be made. For example: if one pickles bone marrow in sour milk, is it forbidden? So one rules that cooking meat in milk is forbidden, and also that bone marrow counts as meat, and also that sour milk fit only for a dog’s consumption counts as milk, and also that pickling counts as cooking. Now suppose all halakhic decisors in all generations indeed ruled each of these assumptions.
But what is the probability of these assumptions in their opinion? Let us say they all thought each assumption had the respectable probability of 0.8. But 0.8 to the 4th power is already less than half.
Rational conduct according to a monistic approach of discovering halakhic truth should therefore in such a case rule that one who pickles bone marrow in sour milk is permitted to eat it. Of course we have never heard of such a thing anywhere. (R. Ovadia sometimes does make a sefek sefeika even when both doubts go against Maran, and on each doubt by itself he rules like Maran. But an explicit calculation of probabilities, which is entirely necessary according to that monistic approach, we have neither heard nor seen.) If there is a plausible resolution for this ignoring of probabilities, I will rejoice and exult to hear it. I do not even remember having seen anyone address this point, although it is quite possible it appears here in some article I have not read.
It seems to me that at the root of the matter lies the issue of simplicity. Such a doubt usually cannot be quantified for us in percentages. How would you know how many people pay before the due date? In addition, probabilistic calculation is not really familiar to halakhic decisors. Therefore it was decided that one goes by the number of relevant doubts, irrespective of the strength of each one. And then the rules are only about doubt and double doubt. The resolution is not raised.
Beyond that, there is a good deal of sense in treating an accepted ruling as certain. Here it is not only considerations of simplicity, but also strengthening the attitude toward established halakhah.
I hear, but I neither rejoice nor exult.
A clearer counterexample—
What is the probability of rolling a 2 on a die? 1/6.
What is the probability of rolling a 2 on a die given that an odd number came up? 0.
Therefore, to the best of my understanding, although conditioning reduces the total number of possibilities, the conditional probability is not necessarily greater than the absolute probability.
I agree, but note that the number of possibilities here too is smaller. It is just that the result is not one of them. What I wrote, that the number of possibilities is always smaller, is correct. But the statement that conditional probability is always greater is correct unless it is 0 (that is, if it is not one of those possibilities).
Is it correct that without monism the matter is very clear? If the Holy One, blessed be He, expects us to do whatever is ruled halakhically in each matter on its own (even if in truth that is not “the truth”—there are problems here that I am ignoring for the moment), then it is understandable that in a chain of things one need not multiply probabilities smaller than 1. But if so, what will the approach answer that makes a sefek sefeika from certain rejected views? Such a sefek sefeika indicates, for example, that the ruling in each of the doubts is 0.6, so the sefek sefeika gives 0.4 plus another 0.24 toward leniency, namely 0.64 toward leniency. I know you are very monistic and will not be moved by such arguments, but does this matter, like ravens, reduce the probability of monism?
If there is no truth, then all of halakhah becomes a kind of game. In such a situation there is no meaning at all to probability. Probability of what? Of your being right? By definition you are right. Therefore the whole discussion becomes unnecessary. And if you are speaking about a factual question (about which even pluralists would admit there is truth), then we are back to monism.
As an aside: if you know that one of the doubts is 0.6, then it is not a doubt but a majority. A doubt is when you do not have a clear datum and so you assume it is fifty-fifty.
I suggested the 0.6 for a situation where one rules each of the doubts like Maran, and only makes a sefek sefeika with them. If it were 0.5, then one would treat even one doubt as a doubt. One only needs to say that the plausibility of a line of reasoning’s correctness is not judged exactly like a majority.
Following Sandomilov’s remarks, perhaps it is also worth mentioning the position of Fisher, the father of modern statistics, who was very strongly opposed (to the point of boycott and excommunication) to the use of Bayes’ law in statistics.
The reason emerges very clearly here: the need to bring in a “prior”—a prior assumption brought from home regarding the assessment of probabilities—pulls the rug out from under the objectivity of the argument. You have to assume what prior probability you assign to things, and that differs from person to person, so one cannot build an objective argument on it.
Even conceptually he did not agree to speak of the “probability” of something when no sample space is defined. When there are different possibilities that can occur (a die landing on one of its six faces), I can count points in the sample space and say what the probability of a certain event is. But what does it mean to ask for the probability of God’s existence? Are there worlds in which God exists and worlds in which He does not, and we count what percentage of the worlds are ones in which He exists?
Rabbi Michi also emphasized many times that in such cases one can speak of plausibility but not of probability.
In our case one can raise 3 hypotheses:
1. God exists and is characterized by non-intervention
2. God exists and is characterized by intervention
3. God does not exist
From the discussion it emerges that the probability of getting a world like ours if one assumes (2) is low. We are therefore left with two hypotheses; the way to decide between them does not stem from what we have seen in the world, but from our prior views or from other arguments.
As opposed to Fisher, Bayesian statisticians really do define probability differently—a degree of belief in something—and then all the calculations raised above can be made.
A few comments—
1. There were various methods here (hypothesis testing, MAP estimation); perhaps we should try MAXIMUM LIKELIHOOD?
2. Instead of writing it this way
P(B/A) = P(B)/P(A)
it would be better to write it this way
P(A) = P(B)/P(B/A)
and then P(A) is the result from the observations and assumptions. P(A) is the result we want to calculate. Dealing with the complementary events only complicates things in my opinion.
3. The atheist begs the question, and it is possible that he will say about you that you beg the question too…
4. You write, “I, as a believer, claim that P(A) is high (because there is very good evidence for God’s existence unrelated to the question of His involvement in the world)”—but this is what we want to calculate according to the atheist’s argument (or the neutral attempt—aspiring to be objective).
There would be agreement (if there would be) on observations and assumptions, and then we would infer P(A).
That is, let’s generalize the method to add more arguments for and against, and in this method try to present the “very good evidence for existence” in a formal probabilistic way (even if approximate)—and try to find some kind of framework that will yield the result (not necessarily 42).
Maybe NAIVE BAYES, or just a weighted linear sum (Kahneman)—and make a table in Excel.
5. If only there were such a framework—for political, philosophical, evaluative, professional discussions, etc.
6. The apparent understanding that P(B/A) is high—comes from the conception in Tanakh and Hazal that educates us toward it—that there is involvement, whether overt or hidden. (Hagar, the sun at Gibeon, Nineveh, the ten plagues, the donkey, etc.)
The understanding that P(B) is low—comes from our observations (some of us) in today’s world, in everyday life, and from understanding alternatives to miracles, and from preferring rational explanations. So there is a gap here, which in my opinion is hard to dismiss with “hiding of the face.”
I was not able to derive the Bayes of the negative from the Bayes of the positive. It seems these are different probabilities that are not inferred from one another. (But maybe I am mistaken.)
P.S. Maybe it should be possible to upload images in the TALLK BACKS—then it would make the mathematical notation easier for us.
[1. The three methods you mentioned are one and the same. Maximum likelihood is MAP when the priors are equal. What is hypothesis testing if not MAP.]
[P.S. Until it becomes possible to upload images in the talkbacks, one can use a site like this https://imgbb.com/%5D
True, I am seized with terror at the prospect of the ban of our master Rabbi Fisher, head of the academy of ad ha-aminam and the surrounding districts; nevertheless I shall permit myself to open my mouth in supplication—perhaps he will have pity on the poor and needy, perhaps he will have mercy.
Let me begin by saying that your words express a fundamental misunderstanding of the discussion. The atheist came to prove that there is no God, but I came to show that he begs the question, not to prove anything. He did not accomplish his task, precisely because of what I showed. But I certainly did accomplish mine. And the use of Bayes’ formula for this matter is entirely fine.
As for your substantive point: if this is a question for which there is no sample space (and indeed, as I wrote, one can speak here of plausibility and not probability), there are two possibilities: not to discuss it, or to discuss it as I did (a Bayesian discussion). The relations among plausibilities are like those among probabilities (as I showed in the example of the ravens in the red section at the end of the article). Whoever does not wish to discuss it—good health to him. But let him not exploit his statistical status in order to support his personal philosophical approach (which is not logical at all). The fact that this calculation depends on assumptions is known and obvious. That is the nature of every logical argument. And what was here was not a probabilistic calculation but a logical argument (soft implication). Regarding that, Fisher’s words have no relevance.
What I meant to say is two things: 1. These assumptions are plausible in my eyes (for philosophical reasons, not statistical ones, and therefore Fisher’s words are irrelevant). 2. This is the conclusion that follows from them. I showed that the view that there is a high chance of God’s existence stands in tension with the view that there is a high chance that He intervenes. That is what the Bayesian calculation shows, with or without the support of Fisher, may he live long. From here on, each person can decide whatever he wants, again with or without the support of the aforementioned.
Therefore I do not see why it was relevant to bring his remarks.
1. I see no difference at all.
2. I left the transposition of the terms to the intelligent reader. If you prefer, there are several other ways to present the same formula.
3. This is simply a complete misunderstanding. He came to prove that there is no God, and I showed that he begs the question. I did not prove from this that there is a God, and therefore I did not beg the question.
4. Again the same misunderstanding. I showed that the assumption that His existence is plausible contradicts the assumption that the plausibility of His involvement is high. One may now choose either of the two possibilities.
5. This framework exists for every discussion; one just has to be careful to use it correctly and not draw from it what cannot be drawn.
6. What emerges from the Tanakh is that the Holy One, blessed be He, is involved. That does not mean that the probability of His involvement is high. My claim is that this has gone with the wind over the course of history. What does the Tanakh teach about that? Looking at the world shows that in all likelihood He does not intervene. But as stated, the assumption that He does not intervene is shared by both sides in this dispute, and therefore the discussion of it is irrelevant.
You wrote here in one of the comments that conditional probability is always greater unless it is 0. In my opinion it can also be smaller or equal:
The probability of rolling an even number on a die given that the number is less than 4.
The probability of rolling an even number given that the number is divisible by 3.
It is true that there are always fewer possibilities, but it may be that the possibilities that were eliminated are the ones that satisfy it. (Only in the special case where there is only one situation that satisfies the probability are you right.) Am I missing something?
I am trying to generalize—if we are already trying to calculate P(A), then let’s try based on a number of observations and assumptions, and not just one observation. Isn’t the whole trilogy aimed at showing that P(A) is high?
Forget it.
Indeed.
I completely agree that the number of possibilities (the denominator in the probability calculation) will always be smaller under conditioning. I do not agree that this entails that the conditional probability will always be greater than the absolute probability or 0.
A counterexample—
The probability of rolling 5 or more on a die = 1/3
The probability of rolling 5 or more on a die given that 6 did not come up = 1/5.
I repeat and emphasize that I greatly enjoyed the whole article (especially the handling of Hempel’s claim about ravens); this is simply a very small pedantic point of mine.
This was already discussed and I agreed:
https://mikyab.net/posts/72555#comment-53359
Rabbi, I was the one who asked the question, and I had actually wanted to ask you to elaborate about conditional probability, and in the end you wrote a whole article—so thank you very much 🙂 The reason I didn’t manage to respond is that I was terribly busy during that period.
I also wanted to add that in my opinion this is among the finer articles, and the mathematical concepts are also explained very nicely in it. (But I’m biased)…
There was just one point that I may very well not have understood properly, although it is mentioned in the article, and that is: what is the optimal way to refute a theory?
Because it seems a bit from what is said here that one can mostly confirm theories, but refute them less so, especially with such “indirect~” evidence.
I’ll give a somewhat different example to illustrate the questioner’s point:
Suppose you are participating in a TV game show in which they present you with a closed wooden box and ask you whether there is a dog inside or not. When you knock on the box, you do not hear barking. Since it is plausible that if there were a dog there, it would bark because of the knocks, you infer from this that there is no dog there. What is wrong with that way of inference?
That if you have a sufficiently good reason to think there is a dog there, you will prefer to squeeze into the conclusion that for some reason the dog that is there didn’t bark. Just as if the box were transparent and you saw a dog there, and you knocked and it didn’t bark, you would not conclude that there is no dog. Why? Because the probability that you were mistaken in the assumption (or in interpreting what you saw) that there is a dog there is still stronger than the probability that a not-all-that-likely event would fail to happen—namely, that the dog would hear knocking and not bark. Woof woof!
Nice answer.
The meaning of the besad in the question is exactly like the meaning of the besad you put at the beginning of your articles…