# Bayesian updating normal distribution

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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’.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.Frequentist coverage is almost exact using my analytical solution, based on combining Jeffreys’ priors in quadrature, whereas Bayesian updating produces far poorer probability matching.

Not quite the same problem but it’s in the same general class of questions.\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.Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Suppose further that the prior distribution is given by truncated normal distribution $\mathcal(\mu_0,\sigma^2_0,t)$, i.e., density $f(\mu)=c/\sigma \phi((\mu-\mu_0)/\sigma_0)$ if $\mu\mu$ and $c$ is a normalizing constant.Visit Stack Exchange Suppose we observe one draw from the random variable $X$, which is distributed with normal distribution $\mathcal(\mu,\sigma^2)$. (Interpretation: we get noisy signals about $\mu$, which are known to be normally distributed with known variance---this is the draw of $X$.