At our statistics course we had multivariate normal distribution (MVN) and Bayesian linear model based on it. I understand what's going on here:
So I have a posterior x|y, where from prior knowledge x and observation y I estimate e. g. location of point from radar echo (posterior).
We also used sequential update, i. e. using posterior after first estimation as a prior for the next one, and so on for all observations.
I have to get an equation for batch update and prove it, i. e. use all observations at once. I've only managed to get recursive formula, which I think is still sequential:
Sigma_0 = Sigma_x
u_0 = u_x
Sigma_(x|y)_n = [Sigma_(x|y)_(n-1) + (Sigma_y)^(-1)]^(-1)
u_(x|y)_n = Sigma_(x|y)_n * [(Sigma_y)^(-1) * y + Sigma_(x|y)_(n-1) * u_x]
Can you give me equation or some hints for the batch form? I know I'm missing some small detail here. I want to prove it through induction.