LOST CAUSES IN STATISTICS I: Finite Additivity

Well said!

Normal Deviate

LOST CAUSES IN STATISTICS I: Finite Additivity

I decided that I’ll write an occasional post about lost causes in statistics. (The title is motivated by Streater (2007).) Today’s post is about finitely additive probability (FAP).

Recall how we usually define probability. We start with a sample space $latex {S}&fg=000000$ and a $latex {sigma}&fg=000000$-algebra of events $latex {{cal A}}&fg=000000$. A real-valued function $latex {P}&fg=000000$ on $latex {{cal A}}&fg=000000$ is a probability measure if it satisfies three axioms:

(A1) $latex {P(A) geq 0}&fg=000000$ for each $latex {Ain {cal A}}&fg=000000$.

(A2) $latex {P(S)=1}&fg=000000$.

(A3) If $latex {A_1,A_2,ldots}&fg=000000$ is a sequence of disjoint events then

$latex displaystyle PBigl(A_1 bigcup A_2 bigcup cdots Bigr) = sum_{i=1}^infty P(A_i). &fg=000000$

The third axiom, countable additivity, is rejected by some extremists. In particular, Bruno de Finetti was a vocal opponent of (A3). He insisted that probability should only be required to satisfy the additivity rule for finite unions…

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