Although I am completely new to Bayesian Analysis I struggle sometimes when trying to investigate some intersections between Bayesian and Frequentist analysis. I would like to discuss the different implications based on the following simple example:
Let $y=(y_1,\ldots,y_N)$ were $y_i|\mu \sim N(\mu,1)$, $\mu\in\mathbb{R}$. Hereby $\mu$ is not known to us. A Frequentist would compute $\hat{\mu}=\frac{1}{N}\sum_{i=1}^{N}y_i$. For the estimator $\hat{\mu}$ it holds that $\hat{\mu} \sim N(\mu,\frac{1}{N})$.
In a Bayesian approach we could assign a normal prior for $\mu$ with infinite variance $\mu\sim N(\mu_0,\lambda^{-1}), \lambda\rightarrow \infty$ and obtain a normal distributed posterior density $\mu|y\sim N(\hat{\mu},\frac{1}{N})$.
I am wondering how $\hat{\mu} \sim N(\mu,\frac{1}{N})$ and $\mu|y\sim N(\hat{\mu},\frac{1}{N})$ can be interpreted and what different implications you can draw out of those formulas. Is it a correct Bayesian interpretation to state that after observing our data $y$ the variance term $\frac{1}{N}$ reflects the 'uncertainty' that is incorporated in choosing $\hat{\mu}$ as our decision rule? On the other hand side what does the Frequentist approach tells us regarding the 'true' parameter? Each and every comment is welcome!