On Statistical and Deterministic Implication (Column 402)
With God’s help
Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.
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.
The meaning of the base in the question is exactly the same as the meaning of the base in your opening articles.
That may be an excuse. But the issue is completely different. 🙂
Divine intervention is before the judgment is given.
When the judgment is given (a terminal illness, a plane crash, a broken leg, a loss of a million on the stock market) the possibility of miraculous divine intervention is extremely low even if one prays.
Just as a judge does not change a justified sentence even if thousands of witnesses bring charges against a criminal.
Therefore, my dear, this is why we do not see differences between groups of terminally ill patients who were prayed for, and infidel patients who were prayed for.
Consider this
But in terms of the number of sick people among the worshippers, the statistical difference is quite significant. Interesting. The wonders of the ways of apologetics.
There is no significance because everyone is a sinner.
If the religious/ultra-Orthodox people were completely righteous, then there would be a little more room for your opinion about the intervention (although even then it would be possible to excuse the matter with reincarnations), because then it would really seem strange that religious people are just like angels, get sick like everyone else and lose money on the stock market like everyone else.
But look left and right and you will see too, everyone is a sinner: ultra-Orthodox, religious, secular –
Theft, slander, reckless driving on the road, lying, adultery.
So does God work with us – we will sin here and there, eventually pray and expect it to be okay because we were taught that if we pray then everything will change for the better?
Are these your expectations of God?
A. Not understood. I will repeat what you said and then continue with the calculation
The atheist's argument assumes that the chance that God will intervene, given that he exists, is a high chance, and since there is a high chance that he actually does not intervene, then the conclusion is that the chance that he exists is low. Completely valid. You say that it is possible to assume in advance that the chance that God exists is high and then, according to Bayes, to reject the atheist's assumption that the chance that God will intervene, given that he exists, is a high chance.
But that is exactly what we are discussing.
A small and agreed-upon introduction: There is a hypothesis h1 to which we attribute a probability of 0.8 and it predicts the data with a probability of 0.5 (meaning that if the hypothesis is correct, then the chance that we will see the results we obtained is 0.5). There is a hypothesis h2 to which we attribute a probability of 0.2 and it predicts the data with a probability of 1. Now the question is which hypothesis is more likely?
The answer is that h1 has 0.8*0.5=0.4 while h2 has 0.2*1=0.2, so h1 has a 2/3 chance. In other words, we still believe that hypothesis h1 is more likely, but the data has caused us to update its probability to 0.66 instead of 0.8.
This is exactly how the argument from concealment works. The higher the probability we attribute to the probability that God will intervene if he exists (and to the claim that he does not intervene), the lower the probability that God exists. Even if we started with a high probability that he exists, then this probability can still remain high, but less.
Suppose we attribute a probability of 0.9 to the existence of God. And it predicts the data that he does not intervene favorably with a probability of 0.2 (i.e. the chance that God will not intervene favorably 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 good with probability 1. In such a situation, the current probability that God exists drops to 0.64. Do you agree with this calculation? [If we give God's existence a probability of 0.99 and the situation that he will not intervene for good a probability of 0.01, then the probability of God's existence drops to a little less than half].
B. There is a God (there is good evidence) and he does not intervene for good (there is quite good evidence). The conclusion, of course, is that God is not good. The puzzling insistence that God is good seems completely childish to me, and the argument from concealment slaps its bewildered face vigorously. (I read the answer on the website from the continuity of natural laws. When the topic comes up on the website sometime, I will stand in detail behind my pompous determination. May the good God deign to change laws or people or perform more miracles). It is suggested that people in his eyes are squeamish, and sly, and sly, and sly.
[For the benefit of the casual reader, this is called MAP estimation. I haven't found a good Hebrew explanation of the matter online, so Google will have to search for it]
Exactly. Therefore, even if non-intervention reduces the chance that it exists, the question is what the chance is, not whether the chance has decreased or not. My argument is that regarding the question of what the chance is, the answer is not unequivocal.
Regarding the goodness of God, that is a different discussion, and you have already mentioned my argument regarding it.
So everyone has to add up the probabilities they think of and then calculate which hypothesis (there is/is not a God) comes out better in the end, and the atheist's argument is perfectly fine. For example, when I add up the probabilities that seem to me (it's hard to add up probabilities, but they flow), it turns out that it's more likely that God doesn't exist. And the only way out is to give up on the hypothesis that is good (especially since I also attach a small prior to it regardless of the data).
“I would point out that additional information (the additional data) always reduces the overall number of possibilities, and therefore also increases the probability of the final outcome.”
If it were given that the outcome was even. The probability would drop to 0.
The whole discussion about probability of whether something exists or not is wrong.
If something exists then it exists and if it doesn't then it doesn't. There is no probability in that.
Dear Posk, this is how decisions are made in life: Let's say you are a doctor and your patient has bleeding in the 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 that there are no chances here - either there is a tumor or not?
And in the context of the column: If we attribute more predictive value to our theological assumptions about how God is supposed to behave than to the assumptions that establish the existence of God, Smindeloff is indeed right that He does not intervene, which lowers the chance of His existence. But if we understand that these assumptions are ridiculous when we understand that God is not a being that is supposed to behave according to our rules, then in order to understand whether there is a God, we need to deal with what is crucial about the question itself.
In fact, in addition to the 3 rejections that Mara Datra wrote, it seems to me that the most ridiculous thing in the argument is his hidden assumption that if there was a God, we would know how to predict how He is supposed to behave.
We use this hidden assumption all the time. When God commands you Torah, you assume that it is His will that we keep the Torah, not that it is His will that we not listen to Him. Therefore, if we drop the ground beneath reasonable assumptions about God, then the entire observance of the commandments collapses. Of course, everyone makes assumptions about God that seem reasonable to them. Even the assumption that God is good is an assumption about God, by the way.
Avishai. What you are describing is confusion. The following sentence is incorrect: “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 a similar condition in patients from the past is 0.8, regarding the new patient you cannot say what the probability is because he either has it or does not have it. You can say that he will have many such patients in the future, so your chance of being wrong is 0.2, but that is only if there are many of them and you make an average calculation on all of them. In other words, the probability is not whether he has it or not, but whether you are wrong or not in your guess if he has it.
But what is important is that the probability was given after there were a number of patients in a similar condition.
Regarding the question of whether something specific exists or does not exist, you cannot give a probability because it is something that happened or did not happen only once. It does not belong to probability.
The Posk - If I guess correctly then the growth necessarily exists therefore I don't understand the point of saying that there is no probability regarding existence (if I am right in guessing probability 1 that there is). In any case also according to your method, I only said that this is how decisions are made, so maybe we just need to change the wording, and talk about the correctness of our hypothesis regarding the existence of God and not about the existence itself
Smindeloff - I also make assumptions about God and do it endlessly, but I understand that the certainty of these assumptions is weak. I don't have the tools to understand anything regarding the essence of God and all the anthropomorphization we make about Him (we assume He is good, that He loves to be listened to) is not a condition that defines His existence but rather our interpretation of what we understand from His revelation in the world.
Anyone who defines that God is only what is revealed from Him in the world and that God must conform to His values otherwise He does not exist is ridiculous in my eyes even though I too have all kinds of assumptions - because for me God is not just my assumptions about how God should behave.
By the way, in terms of the structure of this question, the common question "Where was God in the Holocaust?" is also structured like this:
I didn't understand the answer. Why is your assumption that God wants us to keep His commandments and not, on the contrary, to break them, an assumption by virtue of which Jews lived and died for the sanctification of God, less ridiculous and dehumanizing than the assumption (which I do assume) that if God exists, is powerful and good, then He would intervene.
This is an even greater kind of confusion that many people make mistakes in due to lack of understanding.
Growth is not something that exists. What exists does not change over time. What changes over time does not exist.
Nothing that we know of exists. These are transient phenomena.
The question of where God was during the Holocaust (and in many other places) is, in my opinion, a difficult question, and it is appropriate to draw conclusions from it.
Hello,
I haven't read the entire trilogy yet, but I have read the part about revelation,
and if in your opinion there is no involvement, then how is revelation possible?
In the past there was. The creation of the world is also a mix, isn't it? Every prophecy is a mix, and so is every miracle. My commentary on the matter is detailed in the second book (and also here on the site in several places).
God forbid. Not ridiculous, just as humane.
Both assumptions are excellent
The opposite assumption is ridiculous: If he doesn't intervene based on my assumptions, he doesn't exist. I'm not claiming that because I think if there was a God, he would want Torah observance, then if he doesn't want to be listened to, he doesn't exist.
I don't have a problem with one assumption or another, I have a problem with deciding that these assumptions have weight in the question of God's existence.
If you're concerned with his goodness - if you assume that goodness is equal to intervening in the Holocaust, then if he didn't intervene, he's not good.
A correct conclusion and reasonable assumptions, but not necessary. And they say nothing about existence. (The additional assumption - not good does not exist is in my opinion the root of the problem - because it needs to be multiplied by a very low certainty factor - you're right that its existence depends on your assumptions about what is good)
The decision - regarding the existence of things in our changing world, is greater than mine. The first posek did not rule on the matter in the fundamentals of the Torah, so I do not know.
And to the point - I understand that only with regard to God is there 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 events, there are no probabilities, there are likelihoods.
And what do you think this actually affects?
Because I wrote this in response. Other than that, nothing.
I didn't understand why MT is not true for soft drag – in this particular case p(A’|B’) is p(a’)/p(b’) i.e. one less p(a) divided by one less p(b) This size depends on the size of p(b)/p(a) which is equal to p(B|A) (I couldn't write the math reasonably so I used a verbal description. I used ‘ to indicate the complementary case)
In the general case it can be shown that p(A’|B’) is equal to 1 minus the probability that A minus the probability that B plus the conditional probability p(B|A)*p(A) all of this divided by 1 minus p(b). That is, in the case that p(B|A)=1 it is 1 (this is a hard drag) and in any case, as p(B|A) approaches 1, p(A’|B’) approaches 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. Hence, the MT rule is not true in soft drag.
Indeed, as I wrote, when the conditional probability is 1, the opposite is also 1. This is exactly the MT rule for hard drag.
You are right about the approximation, and Sandomilov has already commented on it above. But you are wrong about the absolute value (whether it is high or not). The atheistic argument does not claim anything about approximation, but about absolute values.
I didn't notice this difference. (In the case where the extreme probability of God's existence is 1 and the probability of intervention is high, then p(B|A) is high and p(A’|B’)=0).
It seems to me that the atheistic argument is based on the assumptions
1. p(B|A) is high.
2. It is likely that p(B’) is high (since we don't see intervention) and therefore p(B) is low.
That is, p(B)/p(A) is high and p(B) is low, which means p(A) is low.
(For example, suppose that x=0.9 then in order for p(A’|B’) to be less than 0.7 p(B) must 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) must be greater than 0.96 and at x=0.999 we arrive at p(B)>0.996).
There is a subtle point here because the argument is that since we see that there is no interference, it is reasonable that the probability of interference 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 on behalf of the atheist assumes the desired, as I explained in the column.
2. I completely agree that the probability of involvement is low. I wrote this in the column. There is nothing subtle here. It is trivial. If you don't see any involvement, there probably isn't.
3. P(A/B) is 1 by definition: if God is involved, he must exist. Therefore, there is definitely 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 this a desired assumption?
2. It is possible that the probability of intervention is high but in fact it did not happen. The argument from the fact that nothing happened to the probability of its occurrence adds another thing to consider. (In fact, this is the conditional probability that p(B) equals x given that in a particular experiment B is not).
3. Typo, I meant p(B|A).
Thanks.
1. I have explained and will explain again. The atheist assumes that P(B/A) is high, and leaves the question of whether God exists open, that is, assumes nothing about P(A). But because of the evidence for God's existence that is unrelated to His involvement, I cannot assume that P(A) is low. It turns out that it is high. Now I am in a contradiction and therefore I must choose one of two options: 1. This assumption is incorrect. P(A) really is not high. 2. The assumption regarding 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 this is the desired assumption. As a believer, I think they are very good, and therefore choose option 2.
2. It is possible, but the probability of this is low. See my response to Sandomilov regarding Rabbi Schach's argument about Operation Entebbe (column 87).
3. Further clarification regarding the application of MT to soft drag. I proposed two formulations to reject the argument. The first is described in 1. In the second formulation of my rejection of the argument, I showed that MT should not be applied to soft drag. Now I thought of a good illustration of this. Take the crow paradox that I showed at the end of the column. You claim that if the chance that if something is a crow then it is black is high, then the chance that if something is not black then it is not a crow – is also high. Let's think of a numerical example. Suppose there are 1000 crows 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 chance that if X is a crow then X is black is 0.99.
The chance that if X is not black then it is not a crow is: 0.1.
And the explanation for this gap is, of course, the very high absolute chance of being a raven (almost 1), and the very small chance of not being black (0.01). This is exactly the case with us.
See the addition in red that I have now added at the end of the column.
At a good time, the opportunity arose https://mikyab.net/posts/71241#comment-49327 .
Is a Torah scroll holy in the holiness of its content (h1) or the holiness of its representation (h2), let's assume that these two hypotheses are a priori equivalent in our eyes. Now, it is given that it is holy in every 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 in the holiness of its representation, but it is also possible that a specific representation is required, that is, in the sacred language. Therefore, according to h2, the probability of A is, let's say, 0.8. When we are given A, is this evidence in favor of the hypothesis that a Torah scroll is holy in the holiness of its content? In a formal calculation, it turns out that it is. Because 0.5*1>0.5*0.8. Therefore, now the probability of h1 has increased to 5/9 instead of 0.5. This is a move you made in column 381 (and similarly in column 380).
Is this indeed a valid way in your opinion to draw scholarly conclusions? That is, if law A necessarily requires hypothesis h1 while hypothesis h2 is indifferent to it (i.e., it is orthogonal to it and says nothing about it), then law A is a confirmation of hypothesis h1?
I once collected several examples in which the first and last answers do not use such a method, and if something works out-sufficiently according to two hypotheses then it is not interesting that it is actually necessary from one hypothesis. Unfortunately, they are lost. But I would be happy if you would confirm that in general you accept and use such a method in halakhic proofs.
Absolutely true. I used a similar method in my comments about Rabbi Shach in column 87. Of course, such a consideration only adds a little probability to the one hypothesis but does not prove it. Therefore, one should be careful in using this method. See my comments in the above column.
A. Against this I put forward another thesis: Evidence in the halakhic literature against hypothesis B is only a figure that is squeezed according to this hypothesis. Let's assume for the sake of the matter that a squeezed figure is one for which the hypothesis predicts less than 0.1. But if one hypothesis predicts a probability of 1 for a given figure and a second hypothesis predicts 0.8 for it, then this does not move anyone and does not serve as evidence in favor of the first hypothesis. Although it is clear that in mathematical calculation this does not work that way, and if we started with two hypotheses that are a priori equivalent, then such a situation of a posteriori predictions of 0.8-1 will force us to update the picture of the probabilities for the hypotheses. Evidence here and there I hope to obtain in the future. If you have evidence that is not so, then on the contrary.
B. Why is it not a matter of not following the majority financially, except for the one that is held. Let's assume that the Amora has a reason that the assumption that no one breaks the law during its useful time is to exclude. What is the probability in his opinion that this reason is “indeed correct”? Let's say 0.9. So how can it exclude? We must say that after deciding on a reason, we forget about its probability. It's as if the prohibition in favor of one is upheld and then the food is spoiled even though one witness is not faithful to the whips by direct testimony. Is that true?
And if a chain of (independent) reasons is used to exclude money or prohibit, then their probability must be doubled and it is certainly possible that then there is no longer even a majority. So every Tanna and Amora must attach to each opinion the probability that he attributes to it so that we know in which chains of reasons it can participate. So why is there no such thing in the literature.
A. There is no contradiction in my words. After all, I said that such an argument only slightly increases the chance and therefore is not enough to constitute evidence. That is how I wrote about Rabbi Schach's Entebbe operation, and that is exactly how I answered about the crow paradox. Indeed, a pink table slightly increases the chance that all crows are black, but this is far from being evidence.
B. Several recent scholars have wondered about this question. They actually formulated it this way: What is the difference between a majority and a presumption? Why does a majority not spend money (according to Shmuel, in certain interpretations. This is far from being agreed upon) and a presumption does. The argument is that when a presumption is established, it may start with a majority, but after it has been established, it becomes an absolute principle (as long as it has not been refuted in the case at hand, of course). But this is not true for every majority, but only for presumptions. And the evidence is that a majority really does not spend money. And if the reasoning were as you say, that a prohibition would be upheld in favor of one, every majority would have to spend money.
And so we discussed the question of whether a majority is a state of doubt or that after there is a majority, the situation is considered to be absolutely decided. It is forbidden to provide impurity in a state of being See for example here about a presumption that comes from a 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
A. One pink table adds a tiny confirmation and is not similar to simple cases of two hypotheses and a prediction that is necessary from one and open from the other (like the example of the sanctity of the Holy Scriptures in every language). When I find an example, I will come back now alive.
B. I ask (1) why the explanation itself is not discussed as a cabbage (for example, the explanation of the predicted against the internal in the owner who doubted that he kept his ox from harm). We ask if we deduce according to this explanation, and whether (2) in a chain of explanations, we sometimes fall below the majority. The answer that the explanation becomes an absolute principle does explain both phenomena, but it is itself completely arbitrary (although it has analogues in halakha). So I still don't understand *why* we really don't calculate every explanation according to the probability of its correctness.
A sharper formulation of section b:
Suppose that in order to rule on a prohibition in a particular case, four premises must be assumed. For example, is it forbidden to boil bone marrow in sour milk? Then they rule that cooking meat in milk is forbidden, and that boiling bone marrow is also considered meat, and that sour milk, which is only suitable for dog food, is considered milk, and that boiling is considered cooking. Now let's assume that all the poskim of all generations have indeed ruled on each of these premises.
But what is the probability of these premises according to their opinion? Suppose that everyone together thought that each of the premises had a respectable probability of 0.8. But 0.8 to the power of 4 is already less than half.
Rational conduct according to a monistic approach to discovering halachic truth should therefore in such a case rule that boiling bone marrow in sour milk is permissible to eat. Of course, we have not heard anything like this anywhere (Rabbi Ovadia does sometimes make a doubt even when both doubts are against a Maran, and in case of any doubt on its own merits, he rules as a Maran. But we have not heard or seen a detailed calculation of probabilities, which is absolutely necessary according to the aforementioned monistic approach). If there is a plausible solution to this disregard of probabilities, Ashish and Agil are the ones who listen to it. I don't even remember seeing anyone addressing this matter, although it is very possible that it appears here in some column that I have not read.
It seems to me that the basis of things is the issue of simplicity. We usually have no way to quantify such a doubt in percentages. How do you know how many people break the law over time? In addition, probabilistic calculations are not really clear to the poskim. Therefore, it was decided to follow the number of relevant spikahs, regardless of the strength of each one. Then the rules are only about doubt and spikah doubt. No resolution is raised.
Beyond that, there is a lot of logic in treating a ruling that has been accepted as certain. Here it is not only considerations of simplicity, but also strengthening the attitude towards settled law.
Hear, but not excited.
Is it true that without monism the matter is most clear. If God expects us to do what has been decided by the halakha in each matter on its merits (even if in reality this is not the "truth". There are problems here that I am ignoring for now) then of course in the chain of events one should not multiply probabilities smaller than 1. But if so, what would the method that the SS does with certain rejected methods answer? Such an SS indicates, for example, that the decision in each of the doubts is 0.6, and then the SS gives 0.4 and another 0.24 for the remainder, i.e. 0.64 for the remainder. I know that you are a very monist and do not waver from such claims, but does this matter of crows reduce the chance of monism?
If there is no truth, then all of Halacha becomes a kind of game. In such a situation, probability has no meaning at all. Probability for what? That you are right? By definition, you are right. Therefore, the whole discussion is redundant. And if you are talking about a factual question (for which even pluralists will admit that there is truth), then we have returned to monism.
On the sidelines: If you know that one of the spiky is 0.6, then it is not doubt but a majority. Doubt is when you do not have a clear figure and then you assume that it is half-and-half.
I suggested 0.6 for a situation where each of the doubts is ruled out as a single one and only a single question is made with them. If 0.5, then one doubts even one doubt. It should only be said that the probability of the correctness of a belief is not discussed precisely because of a majority.
C. The hypothesis that punishment is intended to deter predicts the types of punishments, flogging/punishment, with a very high probability. These are the most immediate and powerful things that can be done to a person. The hypothesis that punishment is intended to repair spiritual damage does not predict any specific type of punishment. The idea that flogging repairs spiritual damage is no more plausible a priori than the idea that carving zucchini into the shape of beetles repairs spiritual damage. Therefore, here the fact that the punishment in the Torah is actually flogging confirms the hypothesis that punishment is (also) intended to deter and does not (only) have an element of spiritual correction. Did I understand correctly that you agree with this?
C. Further to this of course https://mikyab.net/posts/71097#comment-49038
Flogging as a conditional reward means that there is only spiritual correction and not deterrence or revenge.
I disagree. It does improve the chances and still does not make them high. In other words, my answer to this is similar to what I answered to the atheist's argument: Since I have independent evidence (from the Hebrew and from the Rishonim) that punishment is spiritual retribution, this question is not an open question in my opinion. See the additional explanation in red at the end of the column that I have now added, regarding the atheist's requested assumption.
In the background to this, there can be different explanations for the connection between the strength of retribution and spiritual correction. For example, a person's suffering is related in various ways to the spiritual correction he undergoes. Alternatively, God created us in such a way that what deters us is precisely what spiritually corrects us, and yet the essential reason for punishment is the correction and not deterrence. And so on.
So the question is about Hazal and the first ones themselves. This not only improves but becomes very low, almost 0, because the probability of whipping given a spiritual correction is one part of all the possibilities in the world, meaning practically 0.
I answered that. They probably believed that there were good independent reasons to say that punishment is not a deterrent (for example, the need for warning and its acceptance, the exemption from punishment when fairly draconian conditions are not met, etc.), and they also believed that there was a connection between the degree of suffering and correction.
This is not the place, but I also don't agree that punishment is only for correction and generally for deterrence. I don't think I wrote anything like that, or that it could be inferred from my words.
I didn't say you wrote anything like that. I said you suggested situations in which the punishment is only correction - and they are situations of punishment without warning, according to the education that this is a conditional reward, meaning only spiritual correction and not deterrence. Was I wrong?
I understand. In principle, even in such situations this can be a deterrent. For example, there are attempts to deter zealots from acting (jealous people harm) even though the act itself is positive.
In any case, the nature of the punishment in these situations does not prove anything. The fact that they chose flogging is because the general punishment in Halacha is flogging.
I understand.
Excellent and clear post, thank you very much!
I just think there was a small pedantic mistake:
“I would point out that additional information (the additional data) always reduces the total number of possibilities, and therefore also increases the probability of the final outcome. Hence, conditional probability is always greater than absolute probability.”
In my opinion, there are cases in which the conditional probability is actually less than absolute probability.
The chance of cardiac arrest < the chance of cardiac arrest given that you are under 30.
The number of chances of having a cardiac arrest at any age is greater than the number of chances of having one under the age of 30. Per year, it is smaller.
A clearer counterexample –
What is the probability of getting a 2 on a die? 1/6.
What is the probability of getting a 2 on a die given that an odd number comes up? 0.
So as far as I understand, even though conditioning reduces the overall number of possibilities, the conditional probability is not necessarily greater than the absolute probability.
I agree, but note that the number of possibilities here is also smaller. Except that the outcome is not one of them. What I wrote that the number of possibilities is always small is true. But the statement that the conditional probability is always greater is true unless it is 0 (that is, if it is not one of these possibilities).
I completely agree that the number of possibilities (the denominator in the probability calculation) will always be smaller in the conditional. I do not agree that this implies that the conditional probability will always be greater than the absolute probability or 0.
For example, against –
The probability of getting a 5 or higher on the die = 1/3
The probability of getting a 5 or higher on the die given that it did not come out a 6 = 1/5.
I reiterate that I really enjoyed the entire column (especially in dealing with the argument of the waterfall over the crows), just a very small pedantic point of mine.
Already discussed and I agreed:
https://mikyab.net/posts/72555#comment-53359
Following Sandomilov's words, it might be worth noting the position of Fisher, the father of modern statistics, who strongly opposed (to the point of boycott and ostracism) the use of Bayes's law in statistics.
The reason here is very clear: the need to bring a “prior” – a priori assumption about estimating probabilities undermines the objectivity of the argument. You have to assume what the a priori probability is that you give things, and it differs from person to person, and it is impossible to build an objective argument on it.
Even conceptually, he did not agree to talk about a “probability” of something when a sample space is not defined. When there are different possibilities that can happen (the die falling on one of its six sides), I can count points in the sample space and say what the probability of a certain event is, but what does that mean? What is the probability of God's existence? Are there worlds in which God exists and worlds in which He does not exist, and we count the percentage of worlds in which He exists?
Rabbi Michi also emphasized many times that in such cases one can talk about likelihood but not probability.
In our case, 3 hypotheses can be raised:
1. God exists and is characterized by non-intervention
2. God exists and is characterized by intervention
3. God does not exist
The discussion shows that the chance of accepting a world like ours if we assume (2) is low. We are therefore left with two hypotheses. The decision between them does not stem from what we have seen in the world, but from our preconceptions or other arguments.
In contrast to Fisher, Bayesian statisticians really define probability differently – A level of belief in something and then you can do all the calculations mentioned above.
I am indeed terrified of the Shemta Demohar”r Fisher Reish Matyvta Dad HaEminem and the Galileans, but I will still allow myself to make up for it with pleas, perhaps he will have mercy on the poor and needy, perhaps he will have mercy.
I will start 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 assumes what is wanted and not to prove anything. He did not do his part, check it out because of what I showed. But I certainly did do mine. And the use of Bayes' formula in this matter is completely fine.
As for your words, if it is a question for which there is no sample space (and indeed, as I wrote, one can talk about likelihood here and not probability), there are two options: not to discuss it, or to discuss it as I did (a Bayesian discussion). The relationships between likelihoods are like those between probabilities (as I showed in the example of the crows in the section on the subject at the end of the column). Whoever does not want to discuss, may his health be with him. But let him not exploit his statistical status to support his personal (and completely illogical) philosophical approach. The fact that this calculation depends on assumptions is well known and clear. This is the nature of any logical argument. And what was here is not a probabilistic calculation but a logical argument (soft deduction). In this regard, Fisher's words have no relevance.
What I came to say are two things: 1. These assumptions are reasonable in my opinion (from a philosophical and not statistical point of view, and therefore Fisher's words are irrelevant). 2. This is the conclusion that arises from them. I showed that the concept that there is a high chance of God's existence stands in contrast to the concept that there is a high chance that He intervenes. This is what the Bayesian calculation shows, with or without the support of Fisher Shalit”a. From now on, anyone can decide what they want, and that too with or without the support of the aforementioned.
Therefore, I do not see why it was relevant to bring his words.
A few comments -
1. There were different methods here (hypothesis testing, MAP estimation) Maybe we should try MAXIMUM LIKLIHOOD?
2. Instead of writing like this
P(B/A) = P(B)/P(A)
It is better to write like this
P(A) = P(B)/P(B/A)
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 assumes what is wanted, and he may say about you that you assume what is wanted…
4. You write “As a believer, I claim that P(A) is high (because there is very good evidence for the existence of God regardless of the question of his involvement in the world) – But this is what we want to calculate according to the atheistic argument (or neutral experience - with an objective aspiration)
On observations and assumptions there will be agreement (if there is) and then we will deduce P(A)
That is, let's generalize the method and add more arguments for and against and with this method try to bring the ”very good evidence for its existence” in a formal probabilistic form (even if approximate) - and try to find some kind of Framework that will give the result (and not necessarily 42)
Maybe NAIVE BASE, or just a linear sum with weights (Kahneman) – and make a table in Excel.
5. I wish there was a Framework – like this for political, philosophical, professional value discussions, etc.’
6. The theoretical understanding that P(B/A) is high – It stems from the perception in the Bible and Chazal that we are educated to – that there is a mix between the overt and the covert. (Hagar, Sun in Gibeon, Nineveh, 10 plagues, Aton, etc.)
The understanding that P(B) is low – is from our observations (some of us) in this world we live in everyday life, and understanding the alternatives to miracles, and preferring rational explanations. – So there is a gap here, which I think is difficult to dismiss as ”hiding face”
[1. The three methods you mentioned are one. Maximum likelihood is MAP when the priors are equal. What is hypothesis testing if not MAP.]
1. I don't see any difference.
2. I left the shifting of the wings to the intelligent reader. If you prefer, there are several other options for presenting the same formula.
3. This is already a complete misunderstanding. He came to prove that there is no God and I showed that he assumes the desired. I did not prove from this that there is a God, and therefore I did not assume the desired.
4. The same misunderstanding again. I showed that the assumption that his existence is probable contradicts the assumption that the probability of his involvement is high. Now you can choose either of the two options.
5. This framework exists for every discussion, you just have to be careful to work with it correctly and not conclude from it what cannot be concluded.
6. What emerges from the Bible is that God is involved. This does not mean that the probability of his involvement is high. My argument is that it has passed with the wind throughout history. What does the Bible teach about this? A look at the world shows that he is highly likely not to intervene. But as mentioned, the assumption that he is not to intervene is shared by both sides in this debate, and therefore discussing it is irrelevant.
I'm trying to generalize - if we're going to try to calculate P(A), then let's try based on multiple observations and assumptions, not just one observation. Isn't the whole trilogy aimed at showing that P(A) is high?
abandoned
I was unable to develop the basis of the negative from the basis of the positive. It seems that these are different probabilities that do not follow from each other. (But maybe I am wrong)
P.S. Maybe it will be possible to upload images in TALK BACKS – then it will make the mathematical notation easier for us
[P.S. Until it is possible to upload photos to Talkbacks, you can use a site like this https://imgbb.com/%5D
You wrote here in one of the comments that conditional probability is always greater than or equal to 0. In my opinion, it can also be less than or equal to:
The probability of rolling an even die, given that a number less than 4 is rolled.
The probability of rolling an even die, given that a number is divisible by 3.
It is true that there will always be possibilities, but it is possible that the possibilities that have been rolled are the ones that are true. (Only in a special case where there is only one situation that fulfills the probability, you are right). Am I missing something?
indeed
Rabbi, I was the one who asked the question and I specifically wanted to ask you to elaborate on conditional probability and finally you wrote a whole column so thank you very much 🙂 The reason I didn't have time to respond was because I was very busy during this period.
I also wanted to say and add that in my opinion this is among the more beautiful columns and the mathematical concepts are also explained well in them. (But I'm biased)…
There was just a point that I may not have understood properly even though it is mentioned in the column, and it is what is the optimal way to refute a theory?
Because we see some of the things here that can mainly be used to confirm theories but less to refute them, especially with such “indirect~” evidence..
I will give a slightly different example to illustrate the questioner's argument:
Suppose you are on a television game show and are shown a closed wooden box and asked whether or not there is a dog inside. When you tap the box, you hear no barking. Since it is likely that if there were a dog there, it would bark because of the tapping, you conclude that there is no dog there. What is wrong with this way of inference?
That if you have a good enough reason to think that there is a dog there, then you would rather be forced to conclude that for some reason the dog there did not bark. Just as if the box is transparent and you see a dog there and you knock and it does not bark, you would not conclude that there is no dog. Why? Because the probability that you were wrong in assuming (or interpreting the vision) that there is a dog there is still stronger than the probability that the not-so-likely event will not happen, that the dog will hear the knocks and not bark. Woof woof!
You answered well.