Since when are we obligated by Immanuel Kant's imperatives? Did the Jewish people accept these imperatives upon themselves? Asking seriously.

You are not obligated to Kant but to morality and reason. The fact that Kant formulated it does not invalidate its force. Why would I need a verse for something that reason dictates?!

I heard the claim that the very low chance of being hit by missiles in Israel is miraculous (and they bring examples of terrorists elsewhere in the world who fired similar improvised rockets at areas less populated than Israel and the results were far more deadly). I know I'm in territory here that is not friendly to miracle theories. But just in case there are miracles, maybe one could say that it is forbidden to rely on a miracle and include it in the statistics, and therefore it would be permitted to desecrate the Sabbath.
(By the way, the Iron Dome statistics are complete nonsense for our purposes. They calculate all the rockets fired into Israeli territory, not only the ones that Iron Dome decides to intercept—the calculation is apparently relevant if they also want to know about Iron Dome's ballistic calculation abilities, but for us all the missiles they don't even try to intercept are irrelevant. And by the way, I'm sure Rafael benefits from the public misunderstanding of these statistics. Just to give some perspective, the numbers were something like 6,000 rockets and 800 interception attempts, and among those there were really not a few impacts. That gives you a sense of how inflated the success percentages are.)

That is nonsense. There is no miracle in not being hit. The miracle would be to be hit. You do not need any interception statistics for this (which are also very impressive, contrary to what you wrote). Just calculate the threatened area against the number of missiles and you will see that a specific person has essentially no chance of being hit.

You're right, as usual 🙂
As for the statistics, I did not mean that they are not impressive, especially since they are much better than similar systems made by other companies around the world (S-300, Patriot, and the like).
Rather, the protection from Iron Dome is not as dramatic as one might think from numbers like 96% success and so on that we got in one of the previous rounds.

What about interrupting the Amidah prayer in order to reach shelter?

Apparently the same thing. Since this is a rare case, it is hard to see this as justification. Of course, interrupting the Amidah is different in severity from desecrating the Sabbath, and I can understand someone who would permit it.

Hello, honorable Rabbi,
According to your approach, you claim that since the chance of personally being hit by a missile is very low, this is not considered a real case of "danger to life" that justifies desecrating the Sabbath. In my opinion, this is a problematic approach.

First, desecrating the Sabbath is permitted and even required when there is even a doubt of danger to life, not only when there is absolute certainty.

Second, even if the personal chance is low, on the general level this is a real threat to human life. It is not correct to ignore that just because the statistical chance for each individual is low.

Third, in a real emergency such as missile fire, it is very hard to estimate the exact level of risk,
and difficult to calculate the personal chance of being hit, because it depends on many factors that are unknown to us, such as the number of missiles fired, the effectiveness of the Iron Dome system, and more.
Therefore it is preferable to take maximum precautions.
Also, one has to add to the equation the interception shrapnel from Iron Dome, which is no less dangerous than the missiles themselves in the event of a hit.

Therefore, in my opinion, in such a case one should permit desecrating the Sabbath on the grounds of a doubtful danger to life, where Jewish law is lenient.

Strange arguments.
Who spoke about certainty? I wrote that the chance is low, not that there is certainty. According to your approach, it would be forbidden to travel by car because there is a chance of being hurt there too (and there the chance is much higher than with missiles).
The second argument was answered in my response.
The third argument is incorrect. There is no problem estimating the risk.
So I did not see any argument here that would change my mind.

A. So on a weekday too there is no obligation to run to a shelter under the rule of "guard yourselves carefully"?
B. Are you aware of sources in Jewish law that refer to the categorical imperative?

A. I explicitly wrote that there is a full obligation to run to a protected space both on weekdays and on the Sabbath. The categorical imperative is a collective law of "guard yourselves carefully."
B. Not explicitly.

A video of Rabbi Asher Weiss on the subject.
Interesting that he formulates a kind of halakhic categorical imperative.

Wow! Fascinating! He really does make this distinction, but notice that he does not arrive at my distinction. He sees this as an instruction to the public and therefore permits transgressing a prohibition. I wrote that in my opinion, if this is not a prohibition that applies essentially to everyone, but only a situation that arises for isolated individuals, then in such a case there is no permission to commit a transgression.

However, one can discuss the case he deals with, which is a siren during prayer on the Sabbath. That is different from the case I was asked about here. There it is a matter of several synagogues in the area, each with many worshippers inside (not like one individual locked out of his home), and apparently that is already a matter involving the public.
I am undecided about this, because it still is not a general law (most people are not in synagogue). Beyond that, all these people are in one place (or three), and not spread uniformly across the entire threatened area. In such a situation the chance that something will happen is still small, because the missile has to hit precisely there. But it is true that if this exceptional case does happen, many people and not just a few would be hurt, and therefore the expected risk may still justify permitting an interruption and running to a shelter.
Since in matters of danger to life we are stringent in cases of doubt, in practice I would rule like him.
Thanks for the source. You made me very happy.

Glad to have made you happy.
Here is another similar consideration from the same lesson (formulated less clearly), regarding carrying a weapon on the Sabbath:

By the way, another possible source for a kind of categorical imperative in Jewish law is the Shulchan Arukh's statement regarding learning during the cantor's repetition: "Each person should view himself as though there are not nine others besides him, and should pay attention to the cantor's blessing."

Shouldn't one take into account the mental state of a person who cannot get into a shelter during a siren? It can create quite serious panic, and I seem to remember that halakhic decisors permit breaking down a door for a minor stuck in a room just because of the fear, which can be dangerous for him.

That should be taken into account if one sees that this is indeed the situation. But there is no reason to assume that. People who are out on the roads during a siren also do not enter a safe room.