# Better understanding of Posterior Hyper Parameters for Normal Likelihood with known Variance

I must be missing something in terms of the way the posterior distribution is parameterized when a conjugate (normal) prior is applied to normal data.

From https://en.wikipedia.org/wiki/Conjugate_prior#cite_note-ppredNt-8 I gather that with for normally distributed data with a known variance, the mean is: \begin{align} ((\mu_0 / \sigma^2_0) + (\sum_{i=1}^{n}(x_i) / \sigma^2)) / ((1 / \sigma^2_0) + (n / \sigma^2)) \end{align} and the variance is: \begin{align} ((1 / \sigma^2_0) + (n / \sigma^2)) ^{-1} \end{align} I think I am missing something fundamental because I would expect that the posterior mean and variance could not be smaller than both the prior variance and the variance of the data, but the most simple example suggests that this is not the case:

For the variance, if all values are 1:

(1/1 + 1/1)^-1 = 0.5


Intuitively, as the sample size increases, the posterior variance should approach the known variance sigma squared. Obviously I am misinterpreting something. Help is appreciated!

EDIT: I think it makes sense for the posterior mean, But I am still lost regarding the variance.