Lorec's Shortform

While writing a recent post [LW · GW], I had to decide whether to mention that Nicolaus Bernoulli had written his letter posing the St. Petersburg problem specifically to Pierre Raymond de Montmort, given that my audience and I probably have no other shared semantic anchor for Pierre's existence, and he doesn't visibly appear elsewhere in the story.

I decided Yes. I think the idea of awarding credit to otherwise-silent muses in general is interesting.

Footnote to my impending post about the history of value and utility:

After Pascal's and Fermat's work on the problem of points, and Huygens's work on expected value, the next major work on probability was Jakob Bernoulli's Ars conjectandi, written between 1684 and 1689 and published posthumously by his nephew Nicolaus Bernoulli in 1713. Ars conjectandi had 3 important contributions to probability theory:

[1] The concept that expected experience is conserved, or that probabilities must sum to 1.

Bernoulli generalized Huygens's principle of expected value in a random event as

$E=\frac{p_1\ast a_1+p_2\ast a_2+p_3\ast a_3}{p_1+p_2+p_3}$

[ where $p_i$ is the probability of the $i$th outcome, and $a_i$ is the payout from the $i$th outcome ]

and said that, in every case, the denominator - i.e. the probabilities of all possible events - must sum to 1, because only one thing can happen to you

[ making the expected value formula just

$E=p_1\ast a_1+p_2\ast a_2+p_3\ast a_3$

with normalized probabilities! ]

[2] The explicit application of strategies starting with the binomial theorem [ known to ancient mathematicians as the triangle pattern studied by Pascal

and first successfully analyzed algebraically by Newton ] to combinatorics in random games [which could be biased] - resulting in e.g. [ the formula for the number of ways to choose k items of equivalent type, from a lineup of n [unique-identity] items ] [useful for calculating the expected distribution of outcomes in many-turn fair random games, or random games where all more-probable outcomes are modeled as being exactly twice, three times, etc. as probable as some other outcome],

written as $(\frac{n}{k})$:

$(\frac{n}{k})=\frac{n!}{k!(n-k)!}$

[ A series of random events [a "stochastic process"] can be viewed as a zig-zaggy path moving down the triangle, with the tiers as events, [whether we just moved LEFT or RIGHT] as the discrete outcome of an event, and the numbers as the relative probability density of our current score, or count of preferred events.

When we calculate $(\frac{n}{k})$, we're calculating one of those relative probability densities. We're thinking of $n$ as our total long-run number of events, and $k$ as our target score, or count of preferred events.

We calculate $(\frac{n}{k})$ by first "weighting in" all possible orderings of $n$, by taking $(n\ast (n-1)...)$, and then by "factoring out" all possible orderings $k!$ of $k$ ways to achieve our chosen W condition [since we always take the same count $k$ of W-type outcomes as interchangeable], and "factoring out" all possible orderings $(n-k)!$ of our chosen L condition [since we're indifferent between those too].

[My explanation here has no particular relation to how Bernoulli reasoned through this.] ]

Bernoulli did not stop with $(\frac{n}{k})$ and discrete probability analysis, however; he went on to analyze probabilities [in games with discrete outcomes] as real-valued, resulting in the Bernoulli probability distribution.

[3] The empirical "Law of Large Numbers", which says that, after you repeat a random game for many turns and add up all the outcomes, the total final outcome will approach the number of turns, times the expected distribution of outcomes in a single turn. E.g. if a die is biased to roll

a 6   40% of the time
a 5   25% of the time
a 4   20% of the time
a 3   8% of the time
a 2   4% of the time, and
a 1   3% of the time

then after 1,000 rolls, your counts should be "close" to

6:   .4*1,000 = 400
5:   .25*1,000 = 250
4:   .2*1,000 = 200
3:   .08*1,000 = 80
2:   .04*1,000 = 40
1:   .03*1,000 = 30

and even "closer" to these ideal ratios after 1,000,000 rolls

- which Bernoulli brought up in the fourth and final section of the book, in the context of analyzing sociological data and policymaking.

One source: "Do Dice Play God?" by Ian Stewart

[ Please DM me if you would like the author of this post to explain this stuff better. I don't have much idea how clear I am being to a LessWrong audience! ]

I think, in retrospect, the view that abstract statements about shared non-reductionist reality can be objectively sound-valid-and-therefore true [? · GW] follows pretty naturally from combining the common-on-LessWrong view that logical or abstract physical theories [LW · GW] can make sound-valid-and-therefore-true abstract conclusions about Reality, with the view, also common on LessWrong, that we make a lot of decisions by modeling other people as copies of ourselves, instead of as entities primarily obeying reductionist physics.

It's just that, despite the fact that all the pieces are there, it goes on being a not-obvious way to think, if for years and years you've heard about how we can only have objective theories if we can do experiments that are "in the territory" in the sense that they are outside of anyone's map. [ Contrast with celebrity examples of "shared thought experiments" from which many people drew similar conclusions because they took place in a shared map - Singer's Drowning Child, the Trolley Problem, Rawls's Veil of Ignorance, Zeno's story about Achilles and the Tortoise, Pascal's Wager, Newcomb's Problem, Parfit's Hitchhiker, the St. Petersburg paradox, etc. ]

Recently, Raginrayguns [LW · GW] and Philosophy Bear both [presumably] read "Cargo Cult Science" [not necessarily for the first time] on /r/slatestarcodex. I follow both of them, so I looked into it. And TIL that's where "cargo-culting" comes from. He doesn't say why it's wrong, he just waves his hands and says it doesn't work and it's silly. Well, now I feel silly. I've been cargo-culting "cargo-culting". I'm a logical decision theorist. Cargo cults work. If they work unreliably, so do reductionistic methods.