Proof that prior increases posterior probability of including mini-

imal confounder set

Here we outline why our prior will be useful in controlling for confounding. Let all dis- tributions with a subscript 1 be distributions associated with using our conditional prior specification, and all distributions with a subscript 2 be associated with a flat prior on the

model space similar to BMA. Distributions without a subscript are those that are unaf- fected by the choice of prior. Then without loss of generality if we assume that the first m covariates are the ones necessary for controlling for confounding then the following holds: P1(αy ∈ αy∗|D) = P1(α y 1 = 1, α y 2 = 1, ..., αym = 1|D) = P1(αy1 = 1|D) P1(α2y = 1|α y 1 = 1, D)...P1(αym = 1|α y 1 = 1, ..., α y m−1 = 1, D) = P1(αy1 = 1|D) ∗ C1 = P (D|α y 1 = 1)P1(α y 1 = 1) P (D) ∗ C1 =X αx 1 P (D|αy₁ = 1)P1(αy1 = 1, αx1) P (D) ∗ C1 = P (D|α y 1 = 1)3ω+12ω P (D) ∗ C1 ≥ P (D|α y 1 = 1)12 P (D) ∗ C1 =X αx 1 P (D|αy₁ = 1)P2(αy1 = 1, αx1) P (D) ∗ C1 = P2(αy1 = 1|D) ∗ C1 = P2(αy1 = 1|D)P1(αy2 = 1|α y 1 = 1, D)...P1(αym = 1|α y 1 = 1, ..., α y m−1 = 1, D) = P1(αy2 = 1|α y 1 = 1, D) ∗ C2 = P (D|α y 2 = 1, α y 1 = 1)P1(α y 2 = 1|α y 1 = 1) P (D|αy₁ = 1) ∗ C2 = P (D|α y 2 = 1, α y 1 = 1)P1(αy2 = 1) P (D|αy₁ = 1) ∗ C2 =X αx 2 P (D|αy₂ = 1, αy₁ = 1)P1(αy2 = 1, αx2) P (D|α₁y = 1) ∗ C2 = P (D|α y 2 = 1, α y 1 = 1) 2ω 3ω+1 P (D|αy₁ = 1) ∗ C2 ≥ P (D|α y 2 = 1, α y 1 = 1) 1 2 P (D|αy₁ = 1) ∗ C2 =X αx 2 P (D|αy₂ = 1, αy₁ = 1)P2(α y 2 = 1, αx2) P (D|α₁y = 1) ∗ C2

= P2(αy2 = 1|α y 1 = 1, D) ∗ C2 = P2(α y 1 = 1|D)P2(α y 2 = 1|α y 1 = 1, D) × P1(αy3 = 1|α y 1 = 1, α y 2 = 1, D)...P1(αym = 1|α y 1 = 1, ..., α y m−1 = 1, D)

And the proof concludes by performing analogous algebraic operations for each of the remaining conditional posteriors from j=3,...,m. We have now shown how our condi- tional prior specification assigns more posterior mass to models that contain the true con- founder set.

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