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"# Dynamic Models \n",
"\n",
"- **[1]** (##) Given the Markov property\n",
"\\begin{equation*}\n",
"p(x_n|x_{n-1},x_{n-2},\\ldots,x_1) = p(x_n|x_{n-1}) \\tag{A1}\n",
"\\end{equation*}\n",
"proof that, for any $n$,\n",
"\\begin{align*}\n",
"p(x_n,x_{n-1},&\\ldots,x_{k+1},x_{k-1},\\ldots,x_1|x_k) = \\\\\n",
"&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}\\,.\n",
"\\end{align*}\n",
"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\" $(x_{k-1},x_{k-2},\\ldots,x_1)$.\n",
"\n",
"\n",
"- **[2]** (#) \n",
" (a) What's the difference between a hidden Markov model and a linear Dynamical system? \n",
" \n",
" (b) For the same number of state variables, which of these two models has a larger memory capacity, and why? \n",
" \n",
"- **[3]** (#) \n",
"(a) What is the 1st-order Markov assumption? \n",
"(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 $p(x_t|z_t)$. \n",
"(c) What is a Hidden Markov Model (HMM)? \n",
"(d) What is a Linear Dynamical System (LDS)? \n",
"(e) What is a Kalman Filter? \n",
"(f) How does the Kalman Filter relate to the LDS? \n",
"(g) Explain the popularity of Kalman filtering and HMMs? \n",
"(h) How relates a HMM to a GMM? \n",
"\n"
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