[1] (##) Given the Markov property \begin{equation*} p(x_n|x_{n-1},x_{n-2},\ldots,x_1) = p(x_n|x_{n-1}) \tag{A1} \end{equation*} proof that, for any $n$, \begin{align*} p(x_n,x_{n-1},&\ldots,x_{k+1},x_{k-1},\ldots,x_1|x_k) = \\ &p(x_n,x_{n-1},\ldots,x_{k+1}|x_k) \cdot p(x_{k-1},x_{k-2},\ldots,x_1|x_k) \tag{A2}\,. \end{align*} In other words, proof that, if the Markov property A1 holds, then, given the "present" ($x_k$), the "future" $(x_n,x_{n-1},\ldots,x_{k+1})$ is independent of the "past" $(x_{k-1},x_{k-2},\ldots,x_1)$.

[2] (#)
(a) What's the difference between a hidden Markov model and a linear Dynamical system?

(b) For the same number of state variables, which of these two models has a larger memory capacity, and why?
[3] (#) (a) What is the 1st-order Markov assumption?
(b) Derive the joint probability distribution $p(x_{1:T},z_{0:T})$ (where $x_t$ and $z_t$ are observed and latent variables respectively) for the state-space model with transition and observation models $p(z_t|z_{t-1})$ and $p(x_t|z_t)$.
(c) What is a Hidden Markov Model (HMM)?
(d) What is a Linear Dynamical System (LDS)?
(e) What is a Kalman Filter?
(f) How does the Kalman Filter relate to the LDS?
(g) Explain the popularity of Kalman filtering and HMMs?
(h) How relates a HMM to a GMM?