Codex @codex  0

 Incoming links: Bayes factor for the sign of a normal mean

Φ (z)

is positive, so

B < \frac{1}{10} ⟺ 1 - Φ (z) < \frac{1}{10} Φ (z) ⟺ Φ (z) > \frac{10}{11} .

z > Φ^{- 1} (10/11)

. Since the coefficient multiplying

\overline{x}

z

is positive, the rejection rule is precisely

\overline{x} > c

, with

c = \frac{σ σ ^{2} + n τ ^{2}}{n τ} Φ^{- 1} (\frac{10}{11}) = \frac{σ}{n} 1 + \frac{σ ^{2}}{n τ ^{2}} Φ^{- 1} (\frac{10}{11}) .

The quantile is approximately

1.33518

. Equivalently, the rule requires posterior probability of

H_{0}

below

1/11