What Evidence Filtered Evidence?

I dis­cussed the dilemma of the clever ar­guer, hired to sell you a box that may or may not con­tain a di­a­mond. The clever ar­guer points out to you that the box has a blue stamp, and it is a valid known fact that di­a­mond-con­tain­ing boxes are more likely than empty boxes to bear a blue stamp. What hap­pens at this point, from a Bayesian per­spec­tive? Must you hel­plessly up­date your prob­a­bil­ities, as the clever ar­guer wishes?

If you can look at the box your­self, you can add up all the signs your­self. What if you can’t look? What if the only ev­i­dence you have is the word of the clever ar­guer, who is legally con­strained to make only true state­ments, but does not tell you ev­ery­thing they know? Each state­ment that the clever ar­guer makes is valid ev­i­dence—how could you not up­date your prob­a­bil­ities? Has it ceased to be true that, in such-and-such a pro­por­tion of Everett branches or Teg­mark du­pli­cates in which box B has a blue stamp, box B con­tains a di­a­mond? Ac­cord­ing to Jaynes, a Bayesian must always con­di­tion on all known ev­i­dence, on pain of para­dox. But then the clever ar­guer can make you be­lieve any­thing they choose, if there is a suffi­cient va­ri­ety of signs to se­lec­tively re­port. That doesn’t sound right.

Con­sider a sim­pler case, a bi­ased coin, which may be bi­ased to come up 23 heads and 13 tails, or 13 heads and 23 tails, both cases be­ing equally likely a pri­ori. Each H ob­served is 1 bit of ev­i­dence for an H-bi­ased coin; each T ob­served is 1 bit of ev­i­dence for a T-bi­ased coin.1 I flip the coin ten times, and then I tell you, “The 4th flip, 6th flip, and 9th flip came up heads.” What is your pos­te­rior prob­a­bil­ity that the coin is H-bi­ased?

And the an­swer is that it could be al­most any­thing, de­pend­ing on what chain of cause and effect lay be­hind my ut­ter­ance of those words—my se­lec­tion of which flips to re­port.

  • I might be fol­low­ing the al­gorithm of re­port­ing the re­sult of the 4th, 6th, and 9th flips, re­gard­less of the re­sult of those and all other flips. If you know that I used this al­gorithm, the pos­te­rior odds are 8:1 in fa­vor of an H-bi­ased coin.

  • I could be re­port­ing on all flips, and only flips, that came up heads. In this case, you know that all 7 other flips came up tails, and the pos­te­rior odds are 1:16 against the coin be­ing H-bi­ased.

  • I could have de­cided in ad­vance to say the re­sult of the 4th, 6th, and 9th flips only if the prob­a­bil­ity of the coin be­ing H-bi­ased ex­ceeds 98%. And so on.

Or con­sider the Monty Hall prob­lem:

On a game show, you are given the choice of three doors lead­ing to three rooms. You know that in one room is $100,000, and the other two are empty. The host asks you to pick a door, and you pick door #1. Then the host opens door #2, re­veal­ing an empty room. Do you want to switch to door #3, or stick with door #1?

The an­swer de­pends on the host’s al­gorithm. If the host always opens a door and always picks a door lead­ing to an empty room, then you should switch to door #3. If the host always opens door #2 re­gard­less of what is be­hind it, #1 and #3 both have 50% prob­a­bil­ities of con­tain­ing the money. If the host only opens a door, at all, if you ini­tially pick the door with the money, then you should definitely stick with #1.

You shouldn’t just con­di­tion on #2 be­ing empty, but this fact plus the fact of the host choos­ing to open door #2. Many peo­ple are con­fused by the stan­dard Monty Hall prob­lem be­cause they up­date only on #2 be­ing empty, in which case #1 and #3 have equal prob­a­bil­ities of con­tain­ing the money. This is why Bayesi­ans are com­manded to con­di­tion on all of their knowl­edge, on pain of para­dox.

When some­one says, “The 4th coin­flip came up heads,” we are not con­di­tion­ing on the 4th coin­flip hav­ing come up heads—we are not tak­ing the sub­set of all pos­si­ble wor­lds where the 4th coin­flip came up heads—but rather are con­di­tion­ing on the sub­set of all pos­si­ble wor­lds where a speaker fol­low­ing some par­tic­u­lar al­gorithm said, “The 4th coin­flip came up heads.” The spo­ken sen­tence is not the fact it­self; don’t be led astray by the mere mean­ings of words.

Most le­gal pro­cesses work on the the­ory that ev­ery case has ex­actly two op­posed sides and that it is eas­ier to find two bi­ased hu­mans than one un­bi­ased one. Between the pros­e­cu­tion and the defense, some­one has a mo­tive to pre­sent any given piece of ev­i­dence, so the court will see all the ev­i­dence; that is the the­ory. If there are two clever ar­guers in the box dilemma, it is not quite as good as one cu­ri­ous in­quirer, but it is al­most as good. But that is with two boxes. Real­ity of­ten has many-sided prob­lems, and deep prob­lems, and nonob­vi­ous an­swers, which are not read­ily found by Blues and Greens shout­ing at each other.

Be­ware lest you abuse the no­tion of ev­i­dence-fil­ter­ing as a Fully Gen­eral Coun­ter­ar­gu­ment to ex­clude all ev­i­dence you don’t like: “That ar­gu­ment was filtered, there­fore I can ig­nore it.” If you’re ticked off by a con­trary ar­gu­ment, then you are fa­mil­iar with the case, and care enough to take sides. You prob­a­bly already know your own side’s strongest ar­gu­ments. You have no rea­son to in­fer, from a con­trary ar­gu­ment, the ex­is­tence of new fa­vor­able signs and por­tents which you have not yet seen. So you are left with the un­com­fortable facts them­selves; a blue stamp on box B is still ev­i­dence.

But if you are hear­ing an ar­gu­ment for the first time, and you are only hear­ing one side of the ar­gu­ment, then in­deed you should be­ware! In a way, no one can re­ally trust the the­ory of nat­u­ral se­lec­tion un­til af­ter they have listened to cre­ation­ists for five min­utes; and then they know it’s solid.

1“Bits” in this con­text are a mea­sure of how much ev­i­dence some­thing pro­vides—they’re the log­a­r­ithms of prob­a­bil­ities, base 12.

Sup­pose a ques­tion has ex­actly two pos­si­ble (mu­tu­ally ex­clu­sive) an­swers, and you ini­tially as­sign 50% prob­a­bil­ity to each an­swer. If I then tell you that the first an­swer is cor­rect (and you have com­plete faith in my claim), then you have ac­quired one bit of ev­i­dence. If there are four equally likely op­tions, and I tell you the first one is cor­rect, then I have given you two bits; if there are eight and I tell you the right one, then I have given you three bits; and so on. This is dis­cussed fur­ther in “How Much Ev­i­dence Does It Take?” (in Map and Ter­ri­tory).