It suddenly occurs to me that the first woman is the right choice for raising the child, regardless of who the birth mother is.
I wonder if Solomon had plans in mind if both women had said the same thing.
I wonder if Solomon had plans in mind if both women had said the same thing.
That's what the next pair of claimants did, after learning about the case. That time Solomon's decision was not wise enough to be included in the sacred texts: he sold the baby into slavery and then promptly executed both claimants. Not surprisingly, no further cases like this were brought before the king.
I wonder if Solomon had plans in mind if both women had said the same thing.
Parchment, shears, rock.
This is an excellent point I should've noticed myself (though it's been long and long since I encountered the parable). Who says you own a baby just by being its genetic mother?
Albeit sufficiently young babies are plausibly not sentient.
They left out the part where the "good mother" rolled over her own baby two weeks later and everyone just threw up their hands and declared "What can we do, these things just happen, ya' know?"
Co-sleeping is controversial, not one-sided. It seems that co-sleeping increases the risk of smothering but decreases the risk of SIDS, leading to a net decrease in infant mortality. Always be wary of The Seen and The Unseen.
Such are your desires, then, at the object level. But do you also desire that they be your desires? Are you satisfied with being the sort of person who cares more about avoiding guilt and personal responsibility than about the actual survival and well-being of his/her child? Or would you change your preferences, if you could?
The Solomon story has always bugged me as being the sort of thing a not-wise person would come up with as an example of wisdom. There are too many ways it could have gone wrong.
I have my own preferred take on the story, and what else that sort of solution might imply. In that version, it ends with
And because he was the king, beheld by his subjects with awe and terror, the women did not protest his judgment.
And nobody ever bothered the king with domestic disputes again.
I often notice how people use arguments that fail to distinguish the hypotheses under discussion. For example, someone gives an argument that favors their hypothesis, but it also happens to favor the opponent's hypothesis to about the same degree. Interpreting arguments in terms of the likelihood ratio they provide seems like an easy-to-use heuristic that fixes such errors.
Since the elements of the empty set satisfy arbitrary properties, all the examples you provided are technically evidence in favor for your observation. Also, against it.
The analogy seems a bit tortuous... Bayes wasn't needed to understand the story, and seeing the story in the light of Bayes doesn't seem to add any new understanding - at least, in my opinion.
What other stories do you know that show this sort of qualitative Bayesian thinking?
I strongly suspect there are other stories where this sort of interpretation seems natural, but as the memory of this story and its interpretation floated into my memory unbidden, I am not sure where to look for others.
Solomon wasn't actually using Bayes here.
The prior here (A has stolen B's baby) is actually quite low. It just doesn't happen very often. Of course, Solomon actually has to consider some extra evidence (B has accused A of stealing her baby). Solomon (by your account) doesn't consider these things at all.
Solomon's analysis only considered the likelihood given a single test.
The prior here (A has stolen B's baby) is actually quite low.
Irrelevant, because it is certain that one of them attempted to steal the other's baby: the question is whether it was by a midnight baby-swap, or by bearing false witness. What's your prior for the likelihood of attempting by each method conditioned on that an attempt was made? (Note that it could even be a conjunction- when the baby-swap fails, rush to the court and claim that she attempted a baby-swap!)
Why is Bayes' Rule useful? Most explanations of Bayes explain the how of Bayes: they take a well-posed mathematical problem and convert given numbers to desired numbers. While Bayes is useful for calculating hard-to-estimate numbers from easy-to-estimate numbers, the quantitative use of Bayes requires the qualitative use of Bayes, which is noticing that such a problem exists. When you have a hard-to-estimate number that you could figure out from easy-to-estimate numbers, then you want to use Bayes. This mental process of testing beliefs and searching for easy experiments is the heart of practical Bayesian thinking. As an example, let us examine 1 Kings 3:16-28:
Now two prostitutes came to the king and stood before him. One of them said, “Pardon me, my lord. This woman and I live in the same house, and I had a baby while she was there with me. The third day after my child was born, this woman also had a baby. We were alone; there was no one in the house but the two of us.
“During the night this woman’s son died because she lay on him. So she got up in the middle of the night and took my son from my side while I your servant was asleep. She put him by her breast and put her dead son by my breast. The next morning, I got up to nurse my son—and he was dead! But when I looked at him closely in the morning light, I saw that it wasn’t the son I had borne."
The other woman said, “No! The living one is my son; the dead one is yours.”
But the first one insisted, “No! The dead one is yours; the living one is mine.” And so they argued before the king.
The king said, “This one says, ‘My son is alive and your son is dead,’ while that one says, ‘No! Your son is dead and mine is alive.’”
Notice that Solomon explicitly identified competing hypotheses, raising them to the level of conscious attention. When each hypothesis has a personal advocate, this is easy, but it is no less important when considering other uncertainties. Often, a problem looks clearer when you branch an uncertain variable on its possible values, even if it is as simple as saying "This is either true or not true."
Then the king said, “Bring me a sword.” So they brought a sword for the king. He then gave an order: “Cut the living child in two and give half to one and half to the other.”
The woman whose son was alive was deeply moved out of love for her son and said to the king, “Please, my lord, give her the living baby! Don’t kill him!”
But the other said, “Neither I nor you shall have him. Cut him in two!”
Then the king gave his ruling: “Give the living baby to the first woman. Do not kill him; she is his mother.”
Solomon considers the empirical consequences of the competing hypotheses, searching for a test which will favor one hypothesis over another. When considering one hypothesis alone, it is easy to find tests which are likely if that hypothesis is true. The true mother is likely to say the child is hers; the true mother is likely to be passionate about the issue. But that's not enough; we need to also estimate how likely those results are if the hypothesis is false. The false mother is equally likely to say the child is hers, and could generate equal passion. We need a test whose results significantly depend on which hypothesis is actually true.
Witnesses or DNA tests would be more likely to support the true mother than the false mother, but they aren't available. Solomon realizes that the claimant's motivations are different, and thus putting the child in danger may cause the true mother and false mother to act differently. The test works, generates a large likelihood ratio, and now his posterior firmly favors the first claimant as the true mother.
When all Israel heard the verdict the king had given, they held the king in awe, because they saw that he had wisdom from God to administer justice.