pgm_tutorial.ipynb

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    "# Introduction to probabilistic graphical models\n",
    "\n",
    "**Sources:** \n",
    "1. Christopher Bishop's book [Pattern recognition and machine learning](https://www.microsoft.com/en-us/research/uploads/prod/2006/01/Bishop-Pattern-Recognition-and-Machine-Learning-2006.pdf)\n",
    "2. Philipp Hennig's [course on Probabilistic ML](https://uni-tuebingen.de/en/180804)\n",
    "3. The [course on probabilistic graphical models](https://ermongroup.github.io/cs228-notes/) by Stefano Ermon's group\n",
    "3. The python library [`pgmpy`](https://pgmpy.org)"
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    "## Bayesian inference\n",
    "\n",
    "1. Construct a joint distribution $p(X,Y)$ over the data $(X,Y)$.\n",
    "2. Condition on the observed variable $X$ to compute the posterior distribution\n",
    "\n",
    "$$p(Y|X)= \\frac{p(X,Y)}{p(X)}= \\frac{p(X|Y)p(Y)}{ \\int p(X|y)p(y) \\mathrm{d}y}.$$\n",
    "\n",
    "3. Query the posterior distribution $p(Y|x)$ to predict the distribution of values $Y$ for a new data point $x$."
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    "**Problem:** Naive Bayesian inference is often intractable if the range of $(X,Y)$ is \"too large\".\n",
    "\n",
    "$\\Rightarrow$ *(Probabilistic) graphical models (PGMs)* leverage conditional independencies to compactly represent joint distributions and efficiently perform inference.\n",
    "\n",
    "We will look at the following models:\n",
    "1. *Bayes(ian)/belief models/networks* (directed graphs)\n",
    "2. *Markov models/networks / Markov random fields* (undirected graphs)"
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    "## Bayesian models\n",
    "\n",
    "* Use a directed acyclic graph (DAG) to represent a set of random variables and their conditional dependencies.\n",
    "* A node $N$ in the graph represents the conditional distribution $p(N | \\mathrm{Pa}(N))$, where $\\mathrm{Pa}(N)$ is the set of parents of $N$.\n",
    "* Joint distribution is given by:\n",
    "\n",
    "$$p(N_1,\\dots,N_K) = \\prod_{i=1}^K p(N_i | \\mathrm{Pa}(N_i))$$\n",
    "\n",
    "$\\Rightarrow$ Assuming each variable $N_i$ has $n$ outcomes and $k \\ll K$ parents, this reduces the number of values from $n^K$ to $K n^{k+1}$."
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    "### Example: Student model\n",
    "\n",
    "Describes the intelligence of a student (average/high), his score in the SAT exam (bad/good), the grade he gets in the class (A,B,C), the difficulty of the class (easy/hard), and the quality of the recommendation letter he gets from the professor after completing the course (bad/good).\n",
    "\n",
    "Notation:\n",
    "1. Gray nodes are called *observed variables*, which we expect to obtain for new data points.\n",
    "2. White nodes are *hidden variables / latent variables*, some of which we are interested in estimating."
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\n",
      "text/plain": [
       "<Figure size 217.21x224.174 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import utils\n",
    "\n",
    "model = utils.student_model()\n",
    "utils.render(model)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7999a215",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Let us look at the conditional distributions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "799bd304",
   "metadata": {
    "scrolled": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+------------+------------------+------------------+------------------+------------------+\n",
      "| Intel      | Intel(Average)   | Intel(Average)   | Intel(High)      | Intel(High)      |\n",
      "+------------+------------------+------------------+------------------+------------------+\n",
      "| Difficulty | Difficulty(Easy) | Difficulty(Hard) | Difficulty(Easy) | Difficulty(Hard) |\n",
      "+------------+------------------+------------------+------------------+------------------+\n",
      "| Grade(A)   | 0.3              | 0.05             | 0.9              | 0.5              |\n",
      "+------------+------------------+------------------+------------------+------------------+\n",
      "| Grade(B)   | 0.4              | 0.25             | 0.08             | 0.3              |\n",
      "+------------+------------------+------------------+------------------+------------------+\n",
      "| Grade(C)   | 0.3              | 0.7              | 0.02             | 0.2              |\n",
      "+------------+------------------+------------------+------------------+------------------+\n"
     ]
    }
   ],
   "source": [
    "node = \"Grade\" # try \"Intel\"\n",
    "utils.print_cpd(model.get_cpds(node))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "209fd529",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Joint distribution\n",
    "\n",
    "The structure leads to a reduced representation of the joint distribution (see [chain rule of probability](https://en.wikipedia.org/wiki/Chain_rule_(probability)))\n",
    "\n",
    "$$ P(L,G,D,I,S) = \\underbrace{P(L|G,D,I,S)}_{P(L|G)} \\ \\underbrace{P(G|D,I,S)}_{P(G|D,I)} \\ \\underbrace{P(S|D,I)}_{P(S|I)} \\ \\underbrace{P(D|I)\\ }_{P(D)} P(I), $$\n",
    "\n",
    "which reduces the number of values from $48$ to $26$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ae72415c",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P(Difficulty), 2 values\n",
      "P(Intel), 2 values\n",
      "P(Grade|Intel, Difficulty), 12 values\n",
      "P(Letter|Grade), 6 values\n",
      "P(SAT|Intel), 4 values\n",
      "---\n",
      "Joint: 48 values\n",
      "PGM: 26 values\n"
     ]
    }
   ],
   "source": [
    "from itertools import product\n",
    "\n",
    "utils.print_cpds(model)\n",
    "print(\"---\")\n",
    "print(f\"Joint: {len(list(product(*model.states.values())))} values\")\n",
    "print(f\"PGM: {sum(cpd.values.size for cpd in model.cpds)} values\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f57b8bf",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Inference\n",
    "\n",
    "The structure helps in marginalization, as we can break the computation into smaller parts (*variable eliminiation*). For instance:\n",
    "\n",
    "\\begin{align*}\n",
    "P(G) = \\sum_{L,D,I,S} P(L,G,D,I,S) &= \\sum_{L,D,I,S} P(L|G) P(G|D,I) P(S|I) P(D) P(I) \\\\&= \\sum_{D} P(D) \\sum_{I} P(G|D,I)P(I) \\sum_{S}   P(S|I) \\sum_{L} P(L|G).\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "f554e6d2",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+----------+--------------+\n",
      "| Grade    |   phi(Grade) |\n",
      "+==========+==============+\n",
      "| Grade(A) |       0.3620 |\n",
      "+----------+--------------+\n",
      "| Grade(B) |       0.2884 |\n",
      "+----------+--------------+\n",
      "| Grade(C) |       0.3496 |\n",
      "+----------+--------------+\n"
     ]
    }
   ],
   "source": [
    "from pgmpy.inference import VariableElimination\n",
    "\n",
    "node = \"Grade\"\n",
    "\n",
    "infer = VariableElimination(model)\n",
    "print(infer.query([node]))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de48982b",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "This also allows us to compute conditional distributions and *maximum a posteriori* (MAP) estimates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "948400ff",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+----------+--------------+\n",
      "| Grade    |   phi(Grade) |\n",
      "+==========+==============+\n",
      "| Grade(A) |       0.7400 |\n",
      "+----------+--------------+\n",
      "| Grade(B) |       0.1680 |\n",
      "+----------+--------------+\n",
      "| Grade(C) |       0.0920 |\n",
      "+----------+--------------+\n",
      "MAP estimate: {'Grade': 'A'}\n"
     ]
    }
   ],
   "source": [
    "node = \"Grade\"\n",
    "evidence = {\"SAT\": \"Good\", \"Intel\": \"High\"}\n",
    "\n",
    "map_est = infer.map_query([node], evidence=evidence, show_progress=False)\n",
    "print(infer.query([node], evidence=evidence))\n",
    "print(f\"MAP estimate: {map_est}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9af63c24",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Local independence\n",
    "\n",
    "Let us look more closely at the independencies given by our model. \n",
    "\n",
    "First, we see that \n",
    "\n",
    "$$N \\perp \\mathrm{NonDe}(N) | \\operatorname{Pa}(N),$$ \n",
    "\n",
    "where $\\mathrm{NonDe}(N)$ is the set of nodes which are not descendents of the node $N$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "c158edc8",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Grade ⟂ SAT | Intel, Difficulty)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 217.21x224.174 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nodes = [\"Grade\"]\n",
    "\n",
    "utils.render(model)\n",
    "model.local_independencies(nodes)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3354a50e",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Independence (3 nodes)\n",
    "\n",
    "**Causal / evidential / cascade / head-to-tail:** $ D \\rightarrow G \\rightarrow L$ implies $D \\perp L | G$, but, in general, $ D \\not\\perp L$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "164a9139",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Letter ⟂ Difficulty | Grade)\n",
       "(Difficulty ⟂ Letter | Grade)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 194.094x211.453 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cascade = model.copy()\n",
    "cascade.remove_nodes_from([\"Intel\", \"SAT\"])\n",
    "cascade.latents = {\"Difficulty\", \"Letter\"}\n",
    "utils.render(cascade)\n",
    "cascade.get_independencies(include_latents=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "674f66b2",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Independence (3 nodes)\n",
    "\n",
    "**Common cause / fork / fan-out / tail-to-tail:** $ S \\leftarrow I \\rightarrow G$ implies $S \\perp G | I$, but, in general, $ S \\not\\perp G$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "134618a1",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(SAT ⟂ Grade | Intel)\n",
       "(Grade ⟂ SAT | Intel)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 125.591x136.554 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fork = model.copy()\n",
    "fork.remove_nodes_from([\"Letter\", \"Difficulty\"])\n",
    "fork.latents = {\"Grade\", \"SAT\"}\n",
    "utils.render(fork)\n",
    "fork.get_independencies(include_latents=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b23a170",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Independence (3 nodes)\n",
    "\n",
    "**Common evidence / collider / V-structure / fan-in / head-to-head:** $ I \\rightarrow G \\leftarrow D$ implies $I \\perp D$ but, in general, $ I \\not\\perp D | G$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "42ac20de",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Intel ⟂ Difficulty)\n",
       "(Difficulty ⟂ Intel)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 194.094x211.453 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "collider = model.copy()\n",
    "collider.remove_nodes_from([\"Letter\", \"SAT\"])\n",
    "collider.latents = {\"Intel\", \"Difficulty\"}\n",
    "utils.render(collider)\n",
    "collider.get_independencies(include_latents=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "567c9d73",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Example: [Monty Hall problem](https://en.wikipedia.org/wiki/Monty_Hall_problem)\n",
    "\n",
    "Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 0, and the host, who knows what's behind the doors, opens another door, say No. 2, which has a goat. He then says to you, \"Do you want to pick door No. 1?\" Is it to your advantage to switch your choice?\n",
    "\n",
    "**Assumption:** The host must always open a door which has not been picked by the contestant and which reveals a goat."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "8d3adf03",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+---------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+\n",
      "| Choice  | Choice(0) | Choice(0) | Choice(0) | Choice(1) | Choice(1) | Choice(1) | Choice(2) | Choice(2) | Choice(2) |\n",
      "+---------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+\n",
      "| Prize   | Prize(0)  | Prize(1)  | Prize(2)  | Prize(0)  | Prize(1)  | Prize(2)  | Prize(0)  | Prize(1)  | Prize(2)  |\n",
      "+---------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+\n",
      "| Host(0) | 0.0       | 0.0       | 0.0       | 0.0       | 0.5       | 1.0       | 0.0       | 1.0       | 0.5       |\n",
      "+---------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+\n",
      "| Host(1) | 0.5       | 0.0       | 1.0       | 0.0       | 0.0       | 0.0       | 1.0       | 0.0       | 0.5       |\n",
      "+---------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+\n",
      "| Host(2) | 0.5       | 1.0       | 0.0       | 1.0       | 0.5       | 0.0       | 0.0       | 0.0       | 0.0       |\n",
      "+---------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+-----------+\n"
     ]
    }
   ],
   "source": [
    "node = \"Host\" # try \"Prize\"\n",
    "\n",
    "mh_model = utils.monty_hall_model()\n",
    "utils.print_cpd(mh_model.get_cpds(node))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "adc1fdda",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The Bayesian model has a collider structure and, as excepted, it holds that $C \\perp P$. However, in general, the door opened by the host can provide information on the location of the car, i.e.,\n",
    "$$C \\not\\perp P | H.$$\n",
    "We can infer that the posterior probability of the car being behind door 1 is the largest and thus the contestant should switch the door."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "33a01d01",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+----------+--------------+\n",
      "| Prize    |   phi(Prize) |\n",
      "+==========+==============+\n",
      "| Prize(0) |       0.3333 |\n",
      "+----------+--------------+\n",
      "| Prize(1) |       0.6667 |\n",
      "+----------+--------------+\n",
      "| Prize(2) |       0.0000 |\n",
      "+----------+--------------+\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 139.291x151.533 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mh_infer = VariableElimination(mh_model)\n",
    "print(mh_infer.query([\"Prize\"], evidence={\"Choice\": 0, \"Host\": 2}))\n",
    "utils.render(mh_model)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c5a20d65",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Independence (d-separation)\n",
    "\n",
    "The local independencies motivate the following general theorem on conditional independencies in Bayesian models.\n",
    "\n",
    "Let A, B, C be disjoint subsets of nodes of a DAG and consider all paths from any node in A to any node in B (treating edges as undirected). We call a path *blocked* if it includes a node $N$ such that the nodes at $N$ meet\n",
    "* head-to-tail or tail-to-tail, and $N\\in C$, or\n",
    "* head-to-head, and $(\\{N\\} \\cup \\mathrm{De}(N)) \\cap C = \\emptyset $ (neither $N$ nor its descendants are in $C$).\n",
    "\n",
    "If all paths are blocked, then A is said to be *d-separated* from B by C, and \n",
    "\n",
    "$$A \\perp B | C.$$\n",
    "\n",
    "A path which is not blocked is also called *active*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "626c123f",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "d-separated: True\n",
      "{'Difficulty'} ⟂ {'SAT'} | {'Intel'}\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 217.21x224.174 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = {\"Difficulty\"}\n",
    "B = {\"SAT\"}\n",
    "C = {\"Intel\"} # try {\"Intel\", \"Letter\"}, {\"Letter\"}, and set()\n",
    "\n",
    "model.latents = model.nodes - C\n",
    "active_trail_nodes = model.active_trail_nodes(list(A), observed=C, include_latents=True)\n",
    "active_trail_nodes = set.union(*active_trail_nodes.values())\n",
    "dseparated = not bool(active_trail_nodes.intersection(B))\n",
    "print(f\"d-separated: {dseparated}\") \n",
    "if dseparated:\n",
    "    print(f\"{A} ⟂ {B} | {C}\")\n",
    "utils.render(model)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b61b915b",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Markov blanket\n",
    "\n",
    "We call a subset of nodes $S$ a *Markov blanket* of node $N$, if $N$ is independent of the rest of the graph conditioned on $S$.\n",
    "\n",
    "In a general Bayesian model, the minimal Markov Blanket (*Markov boundary*) contains all parents $\\mathrm{Pa}(N)$, children $\\mathrm{Ch}(N)$, and co-parents $\\mathrm{CoPa}(N)$ (parents of children other than $N$), i.e.,\n",
    "\n",
    "$$ \\mathrm{Pa}(N) \\cup \\mathrm{Ch}(N) \\cup \\mathrm{CoPa}(N).$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "2cc4d808",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Markov blanket of 'Intel': {'SAT', 'Edu', 'Difficulty', 'Grade'}\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 370.079x370.079 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "node = \"Intel\" \n",
    "\n",
    "ext_model = model.copy()\n",
    "for tail, head in zip([\"Edu\", \"Country\", \"Subject\"], [\"Intel\", \"Edu\", \"Difficulty\"]):\n",
    "    ext_model.add_edge(tail, head)\n",
    "blanket = set(ext_model.get_markov_blanket(node))\n",
    "print(f\"Markov blanket of '{node}': {blanket}\")\n",
    "ext_model.latents = ext_model.nodes - blanket\n",
    "utils.render(ext_model, grid_unit=3.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2c900e64",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Imperfect map\n",
    "\n",
    "Let $C_1,C_2$ describe two independent fair coin tosses and let $B$ describe a bell which is activated if the coins show the same side. \n",
    "We observe that $C_1 \\perp C_2$, $C_1 \\perp B$, and $C_2 \\perp B$.\n",
    "This gives rise to three factorizations of the joint (and thus three Bayesian models), however none represents every (conditional) independence property of the distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "65220f3a",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 84.4882x91.6142 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 84.4882x91.6142 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 84.4882x91.6142 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pgmpy.models import BayesianNetwork\n",
    "\n",
    "nodes = {\"B\", \"C1\", \"C2\"}\n",
    "for head in nodes:\n",
    "    utils.render(BayesianNetwork([(n, head) for n in nodes - {head}], latents=nodes))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "46305345",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Markov random fields\n",
    "\n",
    "Markov random fields are graphical models based on undirected graphs. They represent conditional independence more conveniently by graph separation (*strong Markov property*):\n",
    "\n",
    "For any disjoint subsets $X,Y,S$ of nodes, where $S$ is a *separating set* (every path from $X$ to $Y$ passes through $S$), it holds that $X \\perp Y |S$.\n",
    "\n",
    "$\\Rightarrow$ The Markov boundary of a node $N$ is just given by the set of neighbors of $N$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "1daecedb",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Markov blanket of 'A': {'C', 'D'}\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 83.3858x83.3858 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "node = \"A\"\n",
    "markov_model = utils.voting_model()\n",
    "blanket = set(markov_model.markov_blanket(node))\n",
    "print(f\"Markov blanket of '{node}': {blanket}\")\n",
    "markov_model.latents = markov_model.nodes - blanket\n",
    "utils.render(markov_model)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb99697b",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Note that no directed graph over four variables implies the set of conditional independence properties given by this undirected graph, i.e., \n",
    "\n",
    "$$A\\not\\perp B, \\quad C \\perp D | \\{A , B\\}, \\quad A \\perp B | \\{C,D\\}.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "004eab48",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 83.3858x83.3858 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 83.3858x83.3858 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 83.3858x83.3858 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "markov_model.latents = markov_model.nodes\n",
    "utils.render(markov_model)\n",
    "markov_model.latents = {\"C\", \"D\"}\n",
    "utils.render(markov_model)\n",
    "markov_model.latents = {\"A\", \"B\"}\n",
    "utils.render(markov_model)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a28ee776",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Joint distribution\n",
    "\n",
    "We cannot specify the conditional distributions as in Bayesian models. Instead we identify dependent variables and define the strength of their interactions.\n",
    "\n",
    "We can represent such interactions by *potential functions / factors* $\\phi_c$ defined on *cliques* $(N_i)_{i\\in c}$ (connected subgraphs). The joint distribution is then given by\n",
    "\n",
    "$$\n",
    "p(N_1,\\dots,N_K) = \\frac{1}{Z} \\tilde{p}(N_1,\\dots,N_K) = \\frac{1}{Z}\\prod_{c \\in \\mathrm{cliques}} \\phi_c\\big((N_i)_{i\\in c}\\big),\n",
    "$$\n",
    "\n",
    "where the *partition function* (normalization constant) is given by\n",
    "\n",
    "$$Z = \\sum_{N_1,\\dots,N_K} \\tilde{p}(N_1,\\dots,N_K).$$  \n",
    "\n",
    "We use cliques, as our model imposes that nodes not sharing an edge are independent conditioned on the rest of the graph, i.e., they should not appear in the same factor."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "47b2572f",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "For the sake of illustration, let us interpret $A,B,C,D$ in our previous example as persons. Let's assume that $\\{A,C\\}$, $\\{C,B\\}$, $\\{B,D\\}$, and $\\{D,A\\}$ are pairs of friends, which tend to have similar voting preferences. For simplicity, we assume that the preferences are the same across all pairs  and that $\\phi_c =1$ for all other cliques."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "0aafd982",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+------+------+------------+\n",
      "| A    | C    |   phi(A,C) |\n",
      "+======+======+============+\n",
      "| A(0) | C(0) |     5.0000 |\n",
      "+------+------+------------+\n",
      "| A(0) | C(1) |     1.0000 |\n",
      "+------+------+------------+\n",
      "| A(1) | C(0) |     1.0000 |\n",
      "+------+------+------------+\n",
      "| A(1) | C(1) |    10.0000 |\n",
      "+------+------+------------+\n",
      "11327.0\n"
     ]
    }
   ],
   "source": [
    "clique_index = 0\n",
    "print(markov_model.factors[clique_index])\n",
    "print(markov_model.get_partition_function())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff2bb694",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "For illustration, let us perform (approximate) inference on our Markov random field using [*Belief propagation / sum–product message passing*](https://en.wikipedia.org/wiki/Belief_propagation)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "8fab0103",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+------+----------+\n",
      "| A    |   phi(A) |\n",
      "+======+==========+\n",
      "| A(0) |   0.0216 |\n",
      "+------+----------+\n",
      "| A(1) |   0.9784 |\n",
      "+------+----------+\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 83.3858x83.3858 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pgmpy.inference import BeliefPropagation\n",
    "\n",
    "evidence = {\"B\": 1} # try {\"D\": 1} and {\"D\": 1, \"C\": 0}\n",
    "\n",
    "markov_model.latents = markov_model.nodes - set(evidence.keys())\n",
    "voting_infer = BeliefPropagation(markov_model)\n",
    "print(voting_infer.query([\"A\"], evidence=evidence))\n",
    "utils.render(markov_model)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8c7811cf",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Imperfect map\n",
    "\n",
    "Finally, note that also Markov random fields do not always offer perfect maps, where every conditional independence property of a distribution\n",
    "is reflected in the graph. For instance, the distribution given by a collider, $A\\perp B$ and $A\\not\\perp B |C$, has no corresponding undirected graph (over the same variables) that is a perfect map."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "249a341e",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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x8OBBbN++HW+88YZN+0E0Gg2io6Od8oIsV8N6ejd79mxUVFRYXQSj1WpBRMO1taCgIISHh2P37t3/+B6l+yEiqNVqJCQksA3RPWE78bbkIZyxPuN+ysvLnZpPcTV25zpmzpzp1IpzWloaamtrUV9f/7+Z6yD6/3UazrraLPmU/Px8p/j/N7BrQyfDMFAqlbh+/Tpqamo43Tfd1dWFadOmQSqVoqys7Ml5n7S9Z6i5uZkEAgGlpKRwtp/PZDLRggULSCAQkF6v58Snu+BQzbCsrIy8vLwoJSXF4UpLV1cXLViwgLy9vZ+YG+D9OFwFLysrI4FAQJMnT7ZboLKyMpJIJCQQCJ5IkYk4WtfR3NxMc+fOJQC0YsUKUqvVNq3rqKiooLS0NAJA8+fPf+KGi/vh5J8pAPdukAUFBcjNzUVjYyNkMhkUCgXkcjnCw8Ph7e2N/v5+XL16FRqNBmfPnkVDQwMiIiLw7rvvIj09/cm58T0CzoS2wDAMTp8+jeLiYly4cAF1dXUjXtXj4eGB6OhoyOVyrF69Gkql8smYJ1uBc6EfpK+vD62trejv74e3tzcmTpwIX19fZ3bpljhd6FHu8eQOim7GqNAuYlRoFzEqtIsYFdpFjArtIkaFdhGjQruIUaFdxKjQLuL/AM7gms6eLpQzAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 70.7874x76.6344 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "utils.render(BayesianNetwork([(\"A\", \"C\"), (\"B\", \"C\")], latents={\"A\", \"B\"}))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e842f77",
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    "slideshow": {
     "slide_type": "slide"
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   },
   "source": [
    "### Pros & Cons\n",
    "\n",
    "**Undirected Graphical Models / Markov Random Fields:**\n",
    "\n",
    "$+$ can be applied to problems without directionality of the associated variable dependencies.\n",
    "\n",
    "$+$ directly encode conditional independence structure.\n",
    "\n",
    "$-$ expressing the joint requires to find cliques and compute the normalization constant $Z$ (often requires approximations).\n",
    "\n",
    "$-$ interpreting undirected models and generating data is typically harder than for Bayesian models.\n",
    "\n",
    "**Directed Graphical Models / Bayesian Networks:**\n",
    "\n",
    "$+$ directly encode a facorization of the joint.\n",
    "\n",
    "$-$ analyzing conditional independence requires to consider\n",
    "d-separation.\n"
   ]
  }
 ],
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