Generating data from the posterior distribution

Question

Let $$p(D \mid \mu,\sigma^2) \sim \mathcal{N}(\mu,\sigma^2)$$ where $D=(x_1\ldots x_n)$ is my data. I imposed a normal prior on the mean as $$\pi(\mu) \sim \mathcal{N}(\mu_0,\sigma_0^2)$$ Using Bayes, I know that the posterior is $$p(\mu \mid D) \propto \mathcal{N}(\mu_n,\sigma_n^2)$$ where $\mu_n$ is a convex combination between the ML estimate of $\mu$ and the initial belief $\mu_0$, that is $$\mu_n = w\mu_{ML} + (1-w)\mu_0 \tag{1}$$ $$\sigma_n^{-2} = \frac{n}{\sigma^2} + \frac{1}{\sigma_0^2}\tag{2}$$ Using any random normal generator (R,MATLAB for ex), I could generate $D$ using $\mu,\sigma^2$.

Question: If possible, how do I generate $\mu \mid D$ without $\mu_n,\sigma_n$, i.e. only using the evidence $D$? I just want to compare the empirical posterior moments, $\hat{\mu}_n,\hat{\sigma}_n^2$, to the true ones in equations $(1,2)$.

Example: In this question, there is a posterior histogram (evaluated empirically). How do you do that ?

Answer 1

It appears that $\sigma$ is known in your case. Then, just use the R command rnorm to generate a random sample from a normal distribution with the corresponding parameters

rnorm(n, mu = mun, sigma = sigman)

where mun and sigman are defined by your equations.

https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Normal.html

If you don't know what the posterior parameters are, just check the Conjugate Prior wiki:

https://en.wikipedia.org/wiki/Conjugate_prior#When_likelihood_function_is_a_continuous_distribution