The

full Bayesian inference adds up all the posterior distribution output and then

takes the average as its output. The maximum a posteriori probability (MAP) picks

a weight vector which maximises the posterior distribution. If the prior and additive noise model are

Gaussian then the Posterior is also a Gaussian. In linear Gaussian, the

convergence of the posterior mean and the MAP estimator coincide. The sample mean of random linear function is equivalent

to the MAP mean of Gaussian for any value of standard deviation.

Since

we have prior knowledge about pervious posterior distribution, we can use it to

enhance the MAP hence we can get the complete posterior distribution – as shown

above in part ‘e’ the Gaussian have a mean of in high dimension which is more accurate than sample mean

from the linear function.