Scale of Normal posterior stays constant despite increasing the number of observed data points

Misc.
3

Thank you for the answer @yilmazcanozyurt! I understand that the data has unit variance, but I don’t see why the posterior of the mean should also have unit variance. Concretely, in Pattern Recognition and Machine Learning (section 2.3.6), the author computes analytically the posterior and its variance is inversely proportional to the number of observed points N:
image
The author also mentions:

[I]f the number of data points N → ∞ , the variance σ²_N goes to zero and the posterior distribution becomes infinitely peak around the maximum likelihood solution.

Edit: Maybe the inconsistency appears because I’m iterating through the points one by one, instead of using batches of N points?