Independent Bernoulli arms

Abstract

Many of the examples in the first three chapters assume independent arms: G = F ₁ × ... × F _k. In this chapter we consider independent Bernoulli arms. As has been our convention in the Bernoulli case, we regard F _i as a distribution on the Bernoulli parameter θ _i ∈ [0, 1] rather than on Q _i ∈ D; and consistent with an earlier modification of notation, we write the conditional distribution of (θ ₁, θ ₂,...,θ _k ) given success on arm 1, say, as

$$ {\sigma _1}\left( {{F_1},{F_2},...,{F_k}} \right) = \left( {\sigma {F_1},{F_2},...,{F_k}} \right), $$

and given a failure on arm 1 and as

$$ {\varphi _1}\left( {{F_1},{F_2},...,{F_k}} \right) = \left( {\varphi {F_1},{F_2},...,{F_k}} \right). $$