I am studying in preparation for an exam and got stuck with the following question. The only thing I found out is that maybe I should use a Kalman filter, but no idea how.
Given a model $ x = \theta + \epsilon $ where $x$ is an observation, $\theta$ an unknown constant and $\epsilon$ additive white Gaussian noise with zero mean and variance $\sigma^2$.
(1) Give the recursive representation of the posterior distribution of $\theta$, $p(\theta|x_1, x_2,...,x_k) $ with respect to k when the observations $ {x_1, x_2, ..., x_n} $ are obtained. Provided that $ p(\theta|x_1, x_2,...,x_k) $ is a Gaussian distribution from the given conditions, let the mean and variance be $\theta_k$ and $\sigma^2_k$
(2) Based on the results of (1) find the MAP estimator of $\theta$ as a recursive formula using the known $\sigma^2$, which indicates an update for each observation.
(3) The estimator of (2) requires the prior probability of $\theta$ before the observation. What probability should be selected as the prior probability to perform maximum likelihood estimation.
I have absolutely no idea since this is the first time I try this kind of problem. Any help is greatly appreciated!
