(Strong approval for this post. Figuring out how to deal with filtered evidence is close to my heart.)
Suppose that the facts relevant to making optimal decisions about an Issue are represented by nine rolls of the Reality die, and that the quality (utility) of Society’s decision is proportional to the (base-two logarithm) entropy of the distribution of what facts get heard and discussed.
Sorry—what distribution are we measuring the entropy of? When I hear “entropy of a distribution,” I think (∑x−P(x)logP(x)) -- but it’s not clear to me how to get from there to log29, log26, and 67log27+17log221.
You have three green slots, three gray slots, and three blue slots. You put three counters each on each of the green and gray slots, and one counter each on each of the blue slots. The frequencies of counters per slot is [3, 3, 3, 3, 3, 3, 1, 1, 1]. The total number of counters you put down is 3*6 + 3 = 18 + 3 = 21. To turn the frequencies into a probability distribution, you divide everything by 21, to get [1/7, 1⁄7, 1⁄7, 1⁄7, 1⁄7, 1⁄7, 1⁄21, 1⁄21, 1⁄21]. Then the entropy is 6⋅−17log217+3⋅−121log2121, which is 67log27+321log221. Right? (Thanks for checking—it would be really embarrassing if I got this wrong. I might edit the post later to include more steps.)
(Strong approval for this post. Figuring out how to deal with filtered evidence is close to my heart.)
Sorry—what distribution are we measuring the entropy of? When I hear “entropy of a distribution,” I think (∑x−P(x)logP(x)) -- but it’s not clear to me how to get from there to log29, log26, and 67log27+17log221.
You have three green slots, three gray slots, and three blue slots. You put three counters each on each of the green and gray slots, and one counter each on each of the blue slots. The frequencies of counters per slot is [3, 3, 3, 3, 3, 3, 1, 1, 1]. The total number of counters you put down is 3*6 + 3 = 18 + 3 = 21. To turn the frequencies into a probability distribution, you divide everything by 21, to get [1/7, 1⁄7, 1⁄7, 1⁄7, 1⁄7, 1⁄7, 1⁄21, 1⁄21, 1⁄21]. Then the entropy is 6⋅−17log217+3⋅−121log2121, which is 67log27+321log221. Right? (Thanks for checking—it would be really embarrassing if I got this wrong. I might edit the post later to include more steps.)
Ahhh! Yes, that helps a great deal. Thank you!