Learning Finitely Additive Probabilities: an Impossibility Theorem
نویسنده
چکیده
If (X,X ) is a measure space and F◦ ⊂ X is a field generated either by a countable class of sets or by a Vapnik-Červonenkis class, then if μ is purely finitely additive, there exist uncountably many μ′ agreeing with μ on F◦ and having |μ(A) − μ′(A)| = 1 for uncountably many A. If μ is also non-atomic, then for any r ∈ (0, 1], |μ(Ar)− μ(Ar)| = r for uncountably many Ar. Al-Najjar’s [1] unlearnability result is: if (X,X ) belongs to a class of measure spaces not supporting countably additive nonatomic distributions and C is a Vapnik-Červonenkis class, then there exist purely finitely additive nonatomic probabilities, μ and μ′, agreeing on C and having |μ(Aα)− μ(Aα)| = α for any α ∈ (0, 12 ] for uncountably many Aα. Alice laughed. “There’s no use trying,” she said: “one ca’n’t believe impossible things.” “I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for halfan-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.” (Lewis Carroll, Through the Looking Glass)
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