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I am with Elements of Statistical Learning 8.4 Relationship between the bootstrap and bayesian inference.

We observe a single observation $z$ from a normal distribution $z \sim N(\theta,1)$

We carry out a bayesian analysis for $\theta$ and need to specify a prior. We choose $\theta \sim N(0, \tau)$.

This is said to give the posterior probability $\theta|z \sim N(\frac{z}{1+1/\tau},\frac{1}{1+1/\tau})$.

How do we derive this posterior probability ? This is using Bayes' rule, posterior probability distribution equals likelihood times prior.

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    $\begingroup$ $p(\theta| z) \propto p(z|\theta) p(\theta)$. Note that you know the densities for $z|\theta$ and $\theta$. Try to simplify right side and show that it's basically an unscaled Gaussian density with the desired parameters. $\endgroup$ – James Yang May 27 at 12:25
  • $\begingroup$ @JamesYang would you like to elaborate an answer with the rigourous notation and derivation ? $\endgroup$ – kiriloff May 28 at 6:08

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