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By Fearn T., Brown P.J., Besbeas P.

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Iii) In the experiment of flipping two coins, take Ω := {HH, HT, T H, T T }, put H1 := {HH, HT }, T1 := {T H, T T }, S1 := {HH, T T }, S2 := {HT, T H} , and let G := σ (H1 , T1 ) (the field generated by H1 and T1 ) and H := σ (S1 , S2 ) . Is G ∪ H a field? As the result of the first part of the exercise implies, we can work in perfect generality with fields alone whenever sample spaces are finite. The broader concept of σ field is needed only when the nature of the experiment and our interests require the specification of infinitely many outcomes.

Ex ante (before the experiment has been performed), we know only that some member of the class will be drawn and that no one not in the class can be drawn— the trivial information just described. Knowing only this, an event A ∈ F would be assigned probability P(A) = N (A)/N (Ω). Although not usually done except for emphasis, this could also be represented as P (A | Ω) or as P (A | F0 ) where the mark “|” indicates “conditioning”. These symbols would be rendered in words as “the probability of A given that the outcome is in Ω” or as “the probability of A given trivial information”.

The resulting set is measurable, being constructed by countably many differencing operations, and indeed has Lebesgue measure zero. Nevertheless, the set contains uncountably many points. June 5, 2013 10:59 20 BC: 8831 - Probability and Statistical Theory PST˙ws Probability and Statistical Theory for Applied Researchers a. For open intervals (a, b) use the fact that ∞ a+ (a, b) = n=1 n−1 n (b − a), a + (b − a) n n+1 (a union of disjoint sets) plus countable additivity: N λ λ ((a, b)) = lim N →∞ a+ n=1 N = (b−a) lim N →∞ n=1 n−1 n (b−a), a+ (b−a) n n+1 n n−1 − n+1 n N = (b−a) lim N →∞ n=1 = (b−a) lim N →∞ = (b−a) lim N →∞ 1 1 − n n+1 1 1 1 1 1 + + ··· + − − 2 2 3 N N +1 1 1− N +1 1− = b−a.

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A Bayesian decision theory approach to variable selection for discrimination by Fearn T., Brown P.J., Besbeas P.


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