Bayesian updating normal distribution
This seems a not uncommon situation in science, and it is a good approximation to that which exists when estimating climate sensitivity.
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So we end up with a scaled normal distribution, except of course for points where $f(\mu)$ is zero and the left hand side is also zero.
It won't cause problems since it's correct, although it might not be a nice function to work with if you're trying to derive something analytically.
It turns out that my proposed method for combining evidence agrees with that implied by the Minimum Description Length principle, which he has been closely involved in developing.
We have a joint paper under review by a leading climate science journal, which applies the method developed in my JSPI paper.
A copy of the manuscript is available here: https://niclewis.files.wordpress.com/2016/12/lewis_combining-independent-bayesian-posteriors-for-climate-sensitivity_jspiaccepted2016_.