Suspiciously balanced evidence

What prob­a­bil­ity do you as­sign to the fol­low­ing propo­si­tions?

  • “Hu­man ac­tivity has caused sub­stan­tial in­creases in global mean sur­face tem­per­a­ture over the last 50 years and bar­ring ma­jor policy changes will con­tinue to do so over at least the next 50 years—say, at least half a kelvin in each case.”

  • “On av­er­age, Chris­ti­ans in Western Europe donated a larger frac­tion of their in­come last year to non-re­li­gious char­i­ta­ble causes than athe­ists.”

  • “Over the next 10 years, a typ­i­cal in­dex-fund-like bas­ket of US stocks will in­crease by more than 5% per an­num above in­fla­tion.”

  • “In 2040, most global elec­tric­ity gen­er­a­tion will be from re­new­able sources.”

Th­ese are con­tro­ver­sial and/​or difficult ques­tions. There are surely a lot of peo­ple who think they know the an­swers and will con­fi­dently pro­claim that these propo­si­tions are just true or, as the case may be, false. But you are a so­phis­ti­cated rea­soner, able to think in prob­a­bil­is­tic terms, and I ex­pect your an­swers to ques­tions like these mostly lie be­tween p=0.1 and p=0.9 or there­abouts, just like mine. No fool­ish black-and-white think­ing for the likes of us!

(I con­fess that my es­ti­mate for the first of those propo­si­tions is above . But it’s not above .)

… But isn’t it odd that the ev­i­dence should be so evenly bal­anced? No more than 3 bits or so ei­ther way from perfect equal­ity? Shouldn’t we ex­pect that the to­tal­ity of available ev­i­dence, if we could eval­u­ate it prop­erly, would make for a much larger im­bal­ance? If we en­counter only a small frac­tion of it (which we’d need to, to ex­plain such evenly bal­anced re­sults), shouldn’t we ex­pect that ran­dom­ness in the sub­set we hap­pen to en­counter will in some cases make us see a large im­bal­ance even if any given sin­gle piece of ev­i­dence is about as likely to go one way as the other? What’s go­ing on here?

Let me add a bit of fur­ther fuel to the fire. I could have tweaked all those propo­si­tions some­what—more than 3%, or more than 7%, above in­fla­tion; more than 40%, or more than 60%, or 2050 in­stead of 2040. Surely that ought to change the prob­a­bil­ities quite a bit. But the an­swers I’d have given to the ques­tions would still have prob­a­bil­ities be­tween 0.1 and 0.9, and I bet oth­ers’ an­swers would have too. Can things re­ally be so finely enough bal­anced to jus­tify this?

I can think of two “good” ex­pla­na­tions (mean­ing ones that don’t re­quire us to be think­ing badly) and one not-so-good one.

Good ex­pla­na­tion #1: I chose propo­si­tions that I know are open to some de­gree of doubt or con­tro­versy. When I referred to “ques­tions like these”, you surely un­der­stood me to mean ones open to doubt or con­tro­versy. So ques­tions where the ev­i­dence is, or seems to us to be, much more one-sided were filtered out. (For in­stance, I didn’t ask about young-earth cre­ation­ism, be­cause I think it’s al­most cer­tainly wrong and ex­pect most read­ers here to feel the same way.) … But isn’t it strange that there are so many ques­tions for which the ev­i­dence we have is so very bal­anced?

Good ex­pla­na­tion #2: When as­sess­ing a ques­tion that we know is con­tro­ver­sial but that seems one-sided to us, we tend to ad­just our prob­a­bil­ities “in­ward” to­wards 1:1 as a sort of a nod to the “out­side view”. I think this, or some­thing like it, is prob­a­bly a very sen­si­ble idea. … But I think it un­likely that many of us do it in a prin­ci­pled way, not least be­cause it’s not ob­vi­ous how to.

Not-so-good ex­pla­na­tion: We have grown used to see­ing prob­a­bil­ity es­ti­mates as a sign of clear thought and so­phis­ti­ca­tion, and ev­ery time we ac­com­pany some opinion with a lit­tle an­no­ta­tion “” we get a lit­tle twinge of pride at how we quan­tify our opinions, avoid black-and-white think­ing, etc. And so it be­comes a habit, and we trans­late an in­ter­nal feel­ing of con­fi­dence-but-not-cer­tainty into some­thing like “” even when we haven’t done the sort of ev­i­dence-weigh­ing that might pro­duce an ac­tual nu­mer­i­cal re­sult.

Now, I’m not sure which of two quite differ­ent con­clu­sions I ac­tu­ally want to en­dorse.

  • “The temp­ta­tion to push all prob­a­bil­ities for not-crazy-sound­ing things into the mid­dle of the pos­si­ble range is dan­ger­ous. We are apt to treat things as sub­stan­tially-open ques­tions that re­ally aren’t; to be timid where we should be bold. Let’s over­come our cow­ardice and be­come more will­ing to ad­mit when the ev­i­dence sub­stan­tially favours one po­si­tion over an­other.”

  • “Our prac­tice is bet­ter than our prin­ci­ples. Em­piri­cally, we make lots of mis­takes even in cases where nu­mer­i­cal ev­i­dence-weigh­ing would lead us to prob­a­bil­ities close to 0 or 1. So we should con­tinue to push our prob­a­bil­ity es­ti­mates in­ward. The challenge is to figure out a more prin­ci­pled way to do it.”

Here is a pos­si­ble ap­proach that tries to com­bine the virtues of both:

  • Allow ac­cu­mu­lat­ing ev­i­dence to push your prob­a­bil­ity es­ti­mates to­wards 0 or 1; be unashamed by these ex­treme-sound­ing prob­a­bil­ities. BUT

  • Keep a sep­a­rate es­ti­mate of how con­fi­dent you are that your ap­proach is cor­rect; that your ac­cu­mu­la­tion of ev­i­dence is ac­tu­ally con­verg­ing on some­thing like the right an­swer. THEN,

  • When you ac­tu­ally need a prob­a­bil­ity es­ti­mate, bring these to­gether.

Sup­pose your “in­ter­nal” prob­a­bil­ity es­ti­mate is , your prob­a­bil­ity that your ap­proach is cor­rect is , and your prob­a­bil­ity con­di­tional on your ap­proach be­ing wrong is . Then your over­all prob­a­bil­ity es­ti­mate is and (hold­ing con­stant) in effect your in­ter­nal prob­a­bil­ity es­ti­mates are lin­early squashed into the in­ter­val from to . So, for in­stance, if you’re 90% sure your ap­proach is right and your best guess if your ap­proach is all wrong is that the thing’s 75% likely to be true, then your es­ti­mates are squashed into the range [7.5%,97.5%].

Cau­tion­ary note: There’s an im­por­tant er­ror I’ve seen peo­ple make when try­ing to do this sort of thing (or en­courage oth­ers to do it), which is to con­fuse the propo­si­tions “I’m think­ing about this all wrong” and “the con­clu­sion I’m fairly sure of is ac­tu­ally in­cor­rect”. Un­less the con­clu­sion in ques­tion is a very spe­cific one, that’s likely a mis­take; the prob­a­bil­ity I’ve called above mat­ters and surely shouldn’t be ei­ther 0 or 1.