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    "# Factor Graphs \n",
    "\n",
    "- **[1]** Consider the following state-space model:\n",
    "$$\\begin{align*}\n",
    "z_k &= A z_{k-1} + w_k \\\\\n",
    "x_k &= C z_k + v_k \n",
    "\\end{align*}$$\n",
    "where $k=1,2,\\ldots,n$ is the time step counter; $z_k$ is  an *unobserved* state sequence; $x_k$ is an *observed* sequence; $w_k \\sim \\mathcal{N}(0,\\Sigma_w)$ and $v_k \\sim \\mathcal{N}(0,\\Sigma_v)$ are (unobserved) state and observation noise sequences respectively; $z_0 \\sim \\mathcal{N}(0,\\Sigma_0)$ is the initial state and $A$, $C$, $\\Sigma_v$,$\\Sigma_w$ and $\\Sigma_0$ are known parameters. The Forney-style factor graph (FFG) for one time step is depicted here:     \n",
    "<img src=\"./i/ffg-5SSB0-exam-Kalman-filter.png\" style=\"width:500px;\">     \n",
    "  (a) Rewrite the state-space equations as a set of conditional probability distributions.                 \n",
    "  $$\\begin{align*}\n",
    " p(z_k|z_{k-1},A,\\Sigma_w) &= \\ldots \\\\\n",
    " p(x_k|z_k,C,\\Sigma_v) &= \\ldots \\\\\n",
    " p(z_0|\\Sigma_0) &= \\ldots\n",
    "\\end{align*}$$      \n",
    "  (b) Define $z^n \\triangleq (z_0,z_1,\\ldots,z_n)$, $x^n \\triangleq (x_1,\\ldots,x_n)$ and $\\theta=\\{A,C,\\Sigma_w,\\Sigma_v\\}$. Now write out the generative model $p(x^n,z^n|\\theta)$ as a product of factors.     \n",
    "  (c) We are interested in estimating $z_k$ from a given estimate for $z_{k-1}$ and the current observation $x_k$, i.e., we are interested in computing $p(z_k|z_{k-1},x_k,\\theta)$. Can $p(z_k|z_{k-1},x_k,\\theta)$ be expressed as a Gaussian distribution? Explain why or why not in one sentence.       \n",
    "  (d) Copy the graph onto your exam paper and draw the message passing schedule for computing $p(z_k|z_{k-1},x_k,\\theta)$ by drawing arrows in the factor graph. Indicate the order of the messages by assigning numbers to the arrows.      \n",
    "  (e) Now assume that our belief about parameter $\\Sigma_v$ is instead given by a distribution $p(\\Sigma_v)$ (rather than a known value). Adapt the factor graph drawing of the previous answer to reflects our belief about $\\Sigma_v$.      \n",
    "\n",
    "\n"
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