= Odds under at-least-one-positive selection
{title2=$R_n=R\,[1-\beta^n]/[1-(1-\alpha)^n]$}
For $n$ studies conditionally independent given the truth of one relationship, the event that at least one is positive has probability $1-\beta^n$ under the alternative and $1-(1-\alpha)^n$ under the null. Its <Bayes factor> therefore gives the displayed <posterior odds>. For interior error probabilities these tend back to the <prior odds> $R$. If power exceeds size, the Bayes factor decreases to one: the quotient of geometric sums is a weighted average of decreasing powers $[\beta/(1-\alpha)]^k$.
This result concerns the coarsened selection event. Full observed results instead multiply their individual <Bayes factors>, and concordant evidence can accumulate without this bound. It is a useful example of information discarded by <publication bias>.
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