Gaussian temporal-discrepancy shrinkage ($E[\delta \mid y]=y{\tau }^2/(s^2+{\tau }^2)$)

For a common-event model, let $y\mid \delta \sim N(\delta ,s^2)$ and $\delta \sim N(0,{\tau }^2)$ with positive variances. The posterior exponent is $-[(y-\delta {)}^2/s^2+{\delta }^2/{\tau }^2]/2$. Completing the square yields posterior precision $s^{-2}+{\tau }^{-2}$ and mean $y{\tau }^2/(s^2+{\tau }^2)$. The estimate is pulled toward zero, more strongly when the measurement is uncertain. This conditional model does not decide whether a common event should be assumed.