[jampekka]

At least in the mathematically simpler scenario of a gaussian prior and gaussian observations, the posterior mean is computed by weighing by the the inverses of variances (aka precisions) just like this.

https://en.wikipedia.org/wiki/Conjugate_prior

[ogogmad]

To add, for anyone who's followed the link - that's entries numbers 1 and 2, or "Normal with known variance σ²" and Normal with known precision τ", under "When likelihood function is a continuous distribution".

Also, note that the "precision" τ is defined as 1/σ².