ʻO ka Simpson paradox kekahi o ka maʻalahi e hoʻomaopopo a ma ka manawa like pūʻiwa nā phenomena i nā helu helu. Kū i kēlā me kēia manawa ke hōʻike nā hui o ka ʻikepili i kekahi ʻano, akā huli ʻia kēlā ʻano ke hoʻohui ʻia nā hui. Me ke kōkua o kahi laʻana maʻalahi, hiki ke hoʻomaopopo koke ʻia ka paradox.
Nānā mākou i nā ʻāpana hoʻokaʻawale ʻelua \(\#1\) a me \(\#2\) a me \(G = \#1 \cup \#2\) a hoʻāʻo i ka helu holomua o \(A\) a ma loko o kēia mau set \(B\):
| \(A\) | \(B\) | \(win\) | |
| \(\#1\) | \(\frac{1}{1}=100\%\) | \(\frac{3}{4}=75\%\) | \(A\) |
| \(\#2\) | \(\frac{2}{5}=40\%\) | \(\frac{1}{3}=33\%\) | \(A\) |
| \(\#1 \cup \#2\) | \(\frac{3}{6}=50\%\) | \(\frac{4}{7}=57\%\) | \(B\) |
ʻIke ʻia ʻo \(A\) ʻoi aku ka holomua ma mua o \(B\) i loko o \(\#1\) a ʻo \(\#2\) \(B\) , akā ʻo ka mea kupanaha ma \(G\) \(B\) ʻoi aku ka holomua ma mua o \(A\) . ʻO kēia laʻana kekahi o nā mea me ka hoʻonohonoho liʻiliʻi \(G\) me \(|G|=13\) . ʻAʻohe \(G\) me \(|G|<13\) (hōʻoia e ka ikaika ikaika).
Hoʻokaʻawale mākou i ka hoʻonohonoho \(G\) ma kahi o \(2\) i loko o \(3\) pūʻulu \(\#1, \, \#2, \, \#3\) me \(\#1 \cup \#2 \cup \#3 = G\) . A laila kūkulu mākou i ka hihia \(e_k \neq \emptyset\) no kēlā me kēia mehana \(e_k \neq \emptyset\) ka mana hoʻonohonoho \(P(G)\) o \(G\) e pili ana kēia: $$\forall e_1, e_2 \in P(G): |e_1| \neq |e_2| \Rightarrow win(e_1) \neq win(e_2) \land |e_1| = |e_2| \Rightarrow win(e_1) = win(e_2)$$ $$\forall e_1, e_2 \in P(G): |e_1| \neq |e_2| \Rightarrow win(e_1) \neq win(e_2) \land |e_1| = |e_2| \Rightarrow win(e_1) = win(e_2)$$
Ma hope o kekahi mau hola o ka ikaika ikaika ma ka Core i7 maʻamau, hiki ke loaʻa i kahi hiʻohiʻona aʻe:
| \(A\) | \(B\) | \(C\) | \(win\) | |
| \(\#1\) | \(\frac{6}{7}=85,71\%\) | \(\frac{12}{15}=80,00\%\) | \(\frac{22}{37}=59,46\%\) | \(A\) |
| \(\#2\) | \(\frac{95}{167}=56,89\%\) | \(\frac{48}{88}=54,55\%\) | \(\frac{38}{67}=56,72\%\) | \(A\) |
| \(\#3\) | \(\frac{48}{144}=33,33\%\) | \(\frac{16}{50}=32,00\%\) | \(\frac{2}{20}=10,00\%\) | \(A\) |
| \(\#1 \cup \#2\) | \(\frac{101}{174}=58,05\%\) | \(\frac{60}{103}=58,25\%\) | \(\frac{60}{104}=57,69\%\) | \(B\) |
| \(\#1 \cup \#3\) | \(\frac{54}{151}=35,76\%\) | \(\frac{28}{65}=43,08\%\) | \(\frac{24}{57}=42,11\%\) | \(B\) |
| \(\#2 \cup \#3\) | \(\frac{143}{311}=45,98\%\) | \(\frac{64}{138}=46,38\%\) | \(\frac{40}{87}=45,98\%\) | \(B\) |
| \(\#1 \cup \#2\cup \#3\) | \(\frac{149}{318}=46,86\%\) | \(\frac{76}{153}=49,67\%\) | \(\frac{62}{124}=50,00\%\) | \(C\) |
Ma laila (e noʻonoʻo ana i ka manawa loiloi lōʻihi ʻana) e ʻike ʻia nā laʻana o \(n\) me nā ʻano like. Ke kū maoli kēia mau hihia, ʻo nā hopena i hoʻokumu ʻia ma ka ʻōlelo aʻoaʻo no ka kūleʻa ʻana o kahi hui a maʻalahi ʻole hoʻi.
I kēia manawa, paipai mākou i ka heluhelu hoihoi Causality: Models, Reasoning and Inferensi na ʻOria Pearl .