# Gaussian product - posterior probability distribution

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.

• $p(\theta| z) \propto p(z|\theta) p(\theta)$. 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. – James Yang May 27 at 12:25
• @JamesYang would you like to elaborate an answer with the rigourous notation and derivation ? – kiriloff May 28 at 6:08