Adapted from the two books: Pattern Recognition and Machine Learning Christopher M. Bishop and A Probabilistic Perspective Murphy 2012.
Given a data set:
which are i.i.d and drawn from a 1d Gaussian which likelihood function is given by:
Where the conjugate prior distributions for 
is given by:
With the mean field method we factorize the approximate distribution such that it is given by:
The following video sequence was generated with 15 data points and the following parameters:
And the following shows the exact posterior with the approximate (20 data points):
Where the exact posterior is given by:
Murphy, K. P. (2007). Conjugate Bayesian analysis of the Gaussian distribution. def, 1(2σ2), 16.