Show casing a sum-to-one constraint node using a discrete Bayesian network with pyAgrum

sum_to_one_node.ipynb

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     ]
    }
   ],
   "source": [
    "!pip install watermark"
   ]
  },
  {
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   "execution_count": 2,
   "id": "1cb69552-2ee8-4182-89af-b81bdacf1bac",
   "metadata": {
    "tags": []
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   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%reload_ext autoreload\n",
    "%autoreload 2\n",
    "%load_ext watermark"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "644a235d-281b-486d-90cb-406c62f45f3d",
   "metadata": {},
   "source": [
    "* this notebook was run in the Binder image provide by pyAgrum https://agrum.gitlab.io/extra/agrum_at_binder.html, click the Binder icon found on the bottom left corner of the agrum project home page\n",
    "* the link for the binder image used for this notebook is https://agrum.gitlab.io/extra/agrum_at_binder.html\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "617e0b4b-0957-4de8-959a-2ed8dc3958ce",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sat Jul  1 22:34:42 UTC 2023\n"
     ]
    }
   ],
   "source": [
    "!date"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "979b7cb8-76f2-4e93-a01e-232c2b6cc2f9",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "import pyAgrum as gum\n",
    "import pyAgrum.lib.notebook as gnb\n",
    "import numpy as np\n",
    "import itertools"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "4bf91d0c-fb92-473a-b376-151123af54a8",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python implementation: CPython\n",
      "Python version       : 3.9.16\n",
      "IPython version      : 8.14.0\n",
      "\n",
      "numpy  : 1.25.0\n",
      "pyAgrum: 1.8.3\n",
      "\n",
      "Compiler    : GCC 11.3.0\n",
      "OS          : Linux\n",
      "Release     : 5.15.0-69-generic\n",
      "Machine     : x86_64\n",
      "Processor   : x86_64\n",
      "CPU cores   : 72\n",
      "Architecture: 64bit\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%watermark -v -m -p numpy,pyAgrum"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fcd592bf-3c58-496b-862f-d85fe08c13c7",
   "metadata": {},
   "source": [
    "# Purpose\n",
    "* want to add a *sum-to-one* constraint such that among all $P_k$ nodes there is implicit $\\sum_k P_k = 1$\n",
    "* There will be a node, $sum$, who is a shared child of each $P_k$ that models the normalization constraint for all the probabilities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "8140a966-9cc0-4944-a416-c90389975779",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def sum_to_one_network(*, r: int, prob_bins: int):\n",
    "    bn = gum.BayesNet(\"sum-to-one network\")\n",
    "\n",
    "    # Range and number of bins for each variable\n",
    "    min_val, max_val, num_bins = 0, 100, 10\n",
    "    min_prob, max_prob= 0, 1\n",
    "\n",
    "    # Add nodes P_k and N_k for each state\n",
    "    P = [bn.add(gum.NumericalDiscreteVariable(f'P{k}', f'probability parameter for state {k}', min_prob, max_prob, prob_bins)) for k in range(1, r+1)]\n",
    "\n",
    "    sum_node = bn.add(gum.NumericalDiscreteVariable('sum', 'sum of probabilities', [0.,1.]))\n",
    "\n",
    "    # Add arcs (edges) to the BN\n",
    "    for k in range(r):\n",
    "        bn.addArc(P[k], sum_node)\n",
    "\n",
    "\n",
    "    # Fill nodes\n",
    "    for p in P:\n",
    "        bn.cpt(p).fillWith(np.ones(prob_bins) / prob_bins)\n",
    "\n",
    "    # Define the domain of the P_k nodes\n",
    "    domain = np.linspace(min_prob, max_prob, prob_bins)\n",
    "\n",
    "    # Get the Cartesian product of the domain with itself r times\n",
    "    cartesian_product = list(itertools.product(domain, repeat=r))\n",
    "\n",
    "    bn.cpt(sum_node).fillWith(np.array([[0.,1.] if sum(combo) == 1 else [1.,0.] for combo in cartesian_product]).flatten())\n",
    "    return bn"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "15e0f146-2fea-4bd1-a392-b010b937f58c",
   "metadata": {},
   "source": [
    "* the *sum-to-one* constraint node is a deterministic node\n",
    "* map all the combinations from the cartesian product of all the domains of each $P_k$ probability node with the following:\n",
    "\n",
    "    $$ sum(P_1,P_2,...,P_k)=   \\left\\{\n",
    "\\begin{array}{ll}\n",
    "      1 & \\text{if} & \\sum_k P_k = 1 \\\\\n",
    "      0 & \\text{otherwise} \\\\\n",
    "\\end{array} \n",
    "\\right.  $$\n",
    "\n",
    "see the example table below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "3079d406-b826-400e-86b2-86d454ac5fa1",
   "metadata": {
    "tags": []
   },
   "outputs": [
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       "<tr><th colspan='2'></th>\n",
       "      <th colspan='2' style='border:1px solid black;color:black;background-color:#808080;'><center>sum</center>\n",
       "      </th></tr>\n",
       "<tr><th style='border:1px solid black;color:black;background-color:#808080'><center>P2</center></th><th style='border:1px solid black;color:black;background-color:#808080'><center>P1</center></th><th style='border:1px solid black;border-bottom-style: double;color:black;background-color:#BBBBBB'>\n",
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       "(pyAgrum.Potential<double>@0x561b97187690) \n",
       "             ||  sum              |\n",
       "P1    |P2    ||0        |1        |\n",
       "------|------||---------|---------|\n",
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       "0.5   |0     || 1.0000  | 0.0000  |\n",
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       "0.5   |1     || 1.0000  | 0.0000  |\n",
       "1     |1     || 1.0000  | 0.0000  |"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bn = sum_to_one_network(r=2, prob_bins=3)\n",
    "bn.cpt(\"sum\")"
   ]
  },
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   "id": "e1b4e23d-562d-4629-8585-f78e40768b5c",
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   "source": [
    "bn = sum_to_one_network(r=4, prob_bins=5)\n",
    "gnb.showInference(bn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "a94adf85-15d1-4076-9bc7-14c88f4fb2aa",
   "metadata": {
    "tags": []
   },
   "outputs": [
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   "source": [
    "gnb.showInference(bn,evs={\"sum\": 1})"
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   "execution_count": 10,
   "id": "cf81e421-475a-4fdb-8d19-67a91551bb0f",
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   "source": [
    "gnb.showInference(bn,evs={\"P1\":2,\"P2\":1, \"sum\": 1})"
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  {
   "cell_type": "code",
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   "id": "f0317aa4-0ff6-4f2f-a38b-484d1165c44b",
   "metadata": {
    "tags": []
   },
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   "source": [
    "gnb.showInference(bn,evs={\"P1\":0,\"sum\":1})"
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   "id": "e212d11b-a4d1-4c94-bc87-5a345623d82d",
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    "tags": []
   },
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    "gnb.showInference(bn,evs={\"P1\":1,\"sum\":1})"
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   "id": "ae3ac362-9e33-4bc4-acd7-4bc5eb26c46b",
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    "tags": []
   },
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   ],
   "source": [
    "gnb.showInference(bn,evs={\"P1\":2,\"sum\":1})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "79e01948-3ebd-414f-bad2-03f1657104b1",
   "metadata": {
    "tags": []
   },
   "outputs": [
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   "source": [
    "gnb.showInference(bn,evs={\"P1\":3,\"sum\":1})"
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   "execution_count": 15,
   "id": "90b2bf1c-e29b-4418-b539-cd5678a1f652",
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    "tags": []
   },
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       " <defs>\n",
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       "   <rect x=\"32.865625\" y=\"32.1555\" width=\"111.6\" height=\"69.3\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n",
       "\n",
       "</g>\n",
       "<!-- P4&#45;&gt;sum -->\n",
       "<g id=\"edge3\" class=\"edge\">\n",
       "<title>P4-&gt;sum</title>\n",
       "<g id=\"a_edge3\"><a xlink:title=\"3 → 4\">\n",
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       "</svg>"
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       "<IPython.core.display.SVG object>"
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     },
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    }
   ],
   "source": [
    "gnb.showInference(bn,evs={\"P1\":2,\"P2\":1,\"sum\":1})"
   ]
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