**[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?

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