Why Bayes? A Wise Ruling

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.