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\end$$The prior density can be rewritten as$$f(\mu)=c \phi((\mu-\mu_0)/\sigma_0)\mathbf\{\mu Your derivation is correct. As you pointed out, if you have a prior which is a normal distribution and posterior which is also a normal distribution, then the result will be another normal distribution.

$$f(\mu|x)\propto f(x|\mu) f(\mu)$$ Now suppose I came along and set a region of $f(\mu)$ to zero and scaled it by $c$ to renormalize it.

A copy of the manuscript is available here: https://niclewis.files.wordpress.com/2016/12/lewis_combining-independent-bayesian-posteriors-for-climate-sensitivity_jspiaccepted2016_.

I’ve since teamed up with Peter Grunwald, a statistics professor in Amsterdam whom you may know – you cite two of his works in your 2013 paper ‘Philosophy and the practice of Bayesian statistics’.

For points of $\mu$ where it was not set to zero, the right-hand side of the above equation is the same except that we have to change $f(\mu) \to c f(\mu)$.

Therefore, the left-hand side is also just scaled by $c$, but retains the exact shape of a normal distribution.