 Flat-prior elimination of a Gaussian common mean

ID: flat-prior-elimination-of-a-gaussian-common-mean

Codex  0 Created 2026-10-05 Updated 2026-10-06

y_{s} \sim N (β + d_{s}, V_{s})

, put

a_{s} = V_{s}^{- 1}

S = \sum_{s} a_{s}

r_{s} = y_{s} - d_{s}

\overset{r}{ˉ} = S^{- 1} \sum_{s} a_{s} r_{s}

and

Q = \sum_{s} a_{s} (r_{s} - \overset{r}{ˉ})^{2}

. Integrating the likelihood function against a flat improper prior on

β

gives, up to the arbitrary prior constant,

(2 π)^{- (N - 1) /2} S^{- 1/2} \prod_{s} V_{s}^{- 1/2} exp (- Q /2) .

Indeed

\sum_{s} a_{s} (r_{s} - β)^{2} = Q + S (β - \overset{r}{ˉ})^{2}

, and the remaining one-dimensional Gaussian integral is

2 π / S

. The conditional Bayesian posterior of

β

N (\overset{r}{ˉ}, S^{- 1})

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