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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.

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We're going to first start by reviewing some simple terminology and definitions regarding Bayesian methods to make the discussion later a bit easier to follow.I think the reason Bayesian updating can give poor probability matching, even when the original posterior used as the prior gives exact probability matching in relation to the original data, is that conditionality is not always applicable to posterior distributions.Conditional probability was developed rigorously by Kolmogorov in the conext of random variables.Frequentist coverage is almost exact using my analytical solution, based on combining Jeffreys’ priors in quadrature, whereas Bayesian updating produces far poorer probability matching.I also show that a simple likelihood ratio method gives almost identical inference to my Bayesian combination method.

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