long due comeback

Francisco Coelho
1 parent d080b400
Showing 9 changed files with 1003 additions and 148 deletions Show diff stats
students/amartins/tarefas/Untitled1.ipynb
students/amartins/tarefas/bninput.py
students/amartins/tarefas/tarefa2.py
text/paper_01/pre-paper.pdf
text/paper_01/pre-paper.tex
text/paper_01/temp.fc/LecBayesNetsAndInference.pdf
text/paper_01/temp.fc/hj18.pdf
text/paper_01/temp.fc/lec15.pdf
text/paper_01/zugzwang.bib
@@ -0,0 +1,424 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "a1e66bad-47aa-49b5-ac59-a2183646e74e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "0dbacf2a-1b98-4a20-89f7-1b46ac725a13",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "x = np.linspace(0,1)\n",
+    "y = x*(1-x)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "178862b2-2fc8-42ec-b8e9-ceb8b78ea7a9",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[<matplotlib.lines.Line2D at 0x7f7473269e80>]"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    },
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<Figure size 640x480 with 1 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plt.plot(x,y)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 94,
+   "id": "26e02fe3-a079-4b64-b986-56f37f967e85",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympy import pprint, symbols, expand, factor, simplify, collect, cancel, apart, Matrix, solve, Poly, roots, degree\n",
+    "from sympy.printing.latex import latex\n",
+    "from sympy.plotting import plot"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 57,
+   "id": "baf8a3e6-0a39-410a-85c8-8c536b7603ae",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "t,x = sympy.symbols(\"t x\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "5d570fe9-37cf-4680-a608-99e015d9b977",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "Z = 8.8 + t*(1-t)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "de141bca-0e2a-42c7-86a3-355a1eda8287",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "pa = 0.7/Z"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "id": "0194f6b5-c206-47d7-bc0d-8b1617186d46",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "T = 1 - t"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 58,
+   "id": "d1b858fc-2432-42e7-ba79-e012e76a6a99",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "s = Matrix([\n",
+    "    37,\n",
+    "    9,\n",
+    "    9,\n",
+    "    3,\n",
+    "    3,\n",
+    "    0,\n",
+    "    0,\n",
+    "    2,\n",
+    "    1])\n",
+    "m = Matrix([\n",
+    "    0.0,\n",
+    "    0.0,\n",
+    "    0.7,\n",
+    "    0.3 * t,\n",
+    "    0.3 * T,\n",
+    "    0.7 + 0.3 * t,\n",
+    "    0.7 + 0.3 * T,\n",
+    "    0.3 * t * T,\n",
+    "    0.7 + 0.3 * t * T\n",
+    "])\n",
+    "z = (s.T * m)[0,0]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 50,
+   "id": "2b461810-5bbe-4e4c-a938-b3c916dd95ee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "z = (s.T * m)[0,0]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 51,
+   "id": "3946e0dc-fc81-4030-9cac-ccdccf05b9e1",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle - 0.9 t \\left(t - 1\\right) + 7.9$"
+      ],
+      "text/plain": [
+       "-0.9*t*(t - 1) + 7.9"
+      ]
+     },
+     "execution_count": 51,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "simplify(z)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 68,
+   "id": "a34c1fc2-3810-4054-bcee-b3d8db7b852f",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle - 7.9 \\cdot \\left(0.113924050632911 t^{2} - 0.113924050632911 t - 1.0\\right)$"
+      ],
+      "text/plain": [
+       "-7.9*(0.113924050632911*t**2 - 0.113924050632911*t - 1.0)"
+      ]
+     },
+     "execution_count": 68,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "factor(z)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 53,
+   "id": "89ac28fa-0c47-4a27-a0cc-f7fe0ddf63bd",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle - 0.9 t^{2} + 0.9 t + 7.9$"
+      ],
+      "text/plain": [
+       "-0.9*t**2 + 0.9*t + 7.9"
+      ]
+     },
+     "execution_count": 53,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "expand(z)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 47,
+   "id": "f575ba56-647a-4e01-8c0f-39499abf6300",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "'- 1.3 t^{2} + 1.3 t + 7.5'"
+      ]
+     },
+     "execution_count": 47,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "latex(expand(z))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 61,
+   "id": "a4c62a99-9ad7-4b79-9dda-a28092383a07",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "zx = z.replace(t,x)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 70,
+   "id": "9d67f9d2-48d9-454e-bbfa-4d185eb86200",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<Figure size 640x480 with 1 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "p1 = plot(zx, xlim=(0.0, 1.0), ylim=(0,10), show=False, legend=True)\n",
+    "p2 = plot( (t*T).replace(t, x), xlim=(0.0,1.0), show=False)\n",
+    "p1.append(p2[0])\n",
+    "p1.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 78,
+   "id": "27da5481-766b-4062-a947-aa1d565a6735",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{-2.50462606288666: 1, 3.50462606288666: 1}"
+      ]
+     },
+     "execution_count": 78,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "roots(z, t)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 101,
+   "id": "6927383c-e62a-436e-bbc9-79bbe7f4273f",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0.3⋅t⋅(1 - t) + 0.7\n",
+      "───────────────────\n",
+      "0.9⋅t⋅(1 - t) + 7.9\n",
+      "----------------------------------------------------------------------------------------------------\n",
+      "0.3⋅t⋅(t - 1) - 0.7\n",
+      "───────────────────\n",
+      "0.9⋅t⋅(t - 1) - 7.9\n",
+      "----------------------------------------------------------------------------------------------------\n",
+      "               2                                                          \n",
+      "          0.3⋅t                    0.3⋅t                     0.7          \n",
+      "- ────────────────────── + ────────────────────── + ──────────────────────\n",
+      "         2                        2                        2              \n",
+      "  - 0.9⋅t  + 0.9⋅t + 7.9   - 0.9⋅t  + 0.9⋅t + 7.9   - 0.9⋅t  + 0.9⋅t + 7.9\n",
+      "----------------------------------------------------------------------------------------------------\n",
+      "                   ⎛                   2                            ⎞\n",
+      "0.0886075949367088⋅⎝0.428571428571429⋅t  - 0.428571428571429⋅t - 1.0⎠\n",
+      "─────────────────────────────────────────────────────────────────────\n",
+      "                              2                                      \n",
+      "           0.113924050632911⋅t  - 0.113924050632911⋅t - 1.0          \n",
+      "----------------------------------------------------------------------------------------------------\n",
+      "t⋅(0.3 - 0.3⋅t) + 0.7\n",
+      "─────────────────────\n",
+      "t⋅(0.9 - 0.9⋅t) + 7.9\n",
+      "----------------------------------------------------------------------------------------------------\n",
+      "     2              \n",
+      "0.3⋅t  - 0.3⋅t - 0.7\n",
+      "────────────────────\n",
+      "     2              \n",
+      "0.9⋅t  - 0.9⋅t - 7.9\n",
+      "----------------------------------------------------------------------------------------------------\n",
+      "                                   0.244725738396624                \n",
+      "0.333333333333333 + ────────────────────────────────────────────────\n",
+      "                                       2                            \n",
+      "                    0.113924050632911⋅t  - 0.113924050632911⋅t - 1.0\n",
+      "----------------------------------------------------------------------------------------------------\n"
+     ]
+    }
+   ],
+   "source": [
+    "e = (0.7 + 0.3 * t * T)/z;\n",
+    "for ei in (e, simplify(e), expand(e), factor(e), collect(e, t), cancel(e), apart(e)):\n",
+    "    pprint(ei, use_unicode=True)\n",
+    "    print(100*\"-\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 102,
+   "id": "ba9fc88f-028c-476f-aa94-22c2e15d6e2c",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "sympy.core.add.Add"
+      ]
+     },
+     "execution_count": 102,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "z.func"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 103,
+   "id": "b4bc5ad7-3bb9-44d0-9b70-d27d6c9876f2",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(7.90000000000000, 0.9*t*(1 - t))"
+      ]
+     },
+     "execution_count": 103,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "z.args"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "74c9de1c-01fe-42b6-9b92-a787cfc246b2",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.9.15"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
@@ -0,0 +1,88 @@
+from pgmpy.readwrite import BIFReader
+import networkx as nx
+import pydot
+from networkx.drawing.nx_pydot import graphviz_layout
+
+def summary_dag(filename):
+    file = BIFReader(filename)
+    model = file.get_model() # Cria um objeto BayesianModel
+    nodes = model.nodes()
+    
+    asc = 0
+    des = 0
+    nodes_in = 0
+    nodes_out = 0
+
+    for x in nodes:
+        # Obtem os nós pais e filhos de cada nós
+        parents = model.get_parents(x)
+        children = model.get_children(x)
+        
+        if len(parents) != 0:
+            des += 1
+        if len(children) != 0:
+            asc += 1
+        nodes_in += len(children)
+        nodes_out += len(parents)
+    
+    return {
+        'dag_file': filename, 
+        'nodes': nodes, 
+        'count_parents': des, 
+        'count_children': asc, 
+        'count_in_edges': nodes_in, 
+        'count_out_edges': nodes_out,
+        'bnmodel': model
+        }
+
+def summary_str(d):
+    s = "DAG file:               " + d["dag_file"] + "\n"
+    s += "    Number of nodes:    " + str(len(d["nodes"])) + "\n"
+    s += "    Number of parents:  " + str(d["count_parents"]) + "\n"
+    s += "    Number of children: " + str(d["count_children"]) + "\n"
+    s += "    Average in-degree:  " + str(d["count_in_edges"]/len(d["nodes"])) + "\n"
+    s += "    Average out-degree: " + str(d["count_out_edges"]/len(d["nodes"])) + "\n"
+    s += "    Nodes:" + "\n"
+    nodes = d["nodes"]
+    for x in nodes:
+        s += "        " + str(x) + "\n"
+    return s
+
+def show_dag(model):
+    G = nx.DiGraph()
+    dot = pydot.Dot(graph_type='digraph')
+    for node in model['nodes']:
+        G.add_node(node)
+        dot.add_node(pydot.Node(node))
+    for edge in model['bnmodel'].edges:
+        G.add_edge(edge[0], edge[1])
+        dot.add_edge(pydot.Edge(edge[0], edge[1]))
+    # objeto pydot
+    
+    # defina o layout do grafo
+    pos = graphviz_layout(G, prog='dot')
+    # desenhe o grafo usando o networkx
+    nx.draw(G, pos, with_labels=True)
+    
+def save_dag(model, target_filename):
+    G = nx.DiGraph()
+    for node in model.nodes:
+        G.add_node(node)
+    for edge in model.edges:
+        G.add_edge(edge[0], edge[1])
+    # objeto pydot
+    dot = pydot.Dot(graph_type='digraph')
+    for node in model.nodes:
+        dot.add_node(pydot.Node(node))
+    for edge in model.edges:
+        dot.add_edge(pydot.Edge(edge[0], edge[1]))
+    # defina o layout do grafo
+    pos = graphviz_layout(G, prog='dot')
+    # renderize o grafo com o pydot
+    graph = pydot.graph_from_dot_data(dot.to_string())[0]
+    graph.write_png(target_filename)
+
+if __name__ == "__main__":
+    summary = summary_dag("asia2.bif")
+    message = summary_str(summary)
+    print(message)
 \ No newline at end of file
@@ -0,0 +1,38 @@
+from pgmpy.readwrite import BIFReader
+import networkx as nx
+import pydot
+from networkx.drawing.nx_pydot import graphviz_layout
+from bninput import *
+
+def get_yes_probabilities(model):
+    nodes = model.nodes()
+
+    yes_probabilities = {}
+
+    for node in nodes:
+        cpd = model.get_cpds(node)
+        state_names = cpd.state_names[node]
+        yes_index = state_names.index("yes")
+        yes_prob = cpd.values[yes_index]
+        yes_probabilities[node] = yes_prob
+
+    return yes_probabilities
+
+if __name__ == "__main__":
+    summary = summary_dag("asia2.bif")
+    model = summary["bnmodel"]
+    probabilities = get_yes_probabilities(model)
+    for node, yes_prob in probabilities.items():
+        parents = model.get_parents(node)
+        s = ""
+        if len(parents) == 0:
+            s += str(yes_prob) + " :: " + str(node)
+            print(s)
+        elif len(parents) == 1:
+            s += str(yes_prob[0]) + " :: " + str(node) + " <- " + str(parents) + "\n"
+            s += str(yes_prob[1]) + " :: " + str(node) + " <- -" + str(parents) + "\n"
+            print(s)
+        else:
+            print("Node:", node)
+            print("Probability of 'yes':", yes_prob)
+            print()
@@ -38,6 +38,7 @@
 \usepackage[nice]{nicefrac}
 \usepackage{stmaryrd}
 \usepackage{acronym}
+\usepackage{multicol}
 \usepackage{cleveref}
 %
 % Local commands
@@ -68,34 +69,54 @@
 \newcommand{\rep}[2]{\left\langle #1 \middle| #2 \right\rangle}
 \newcommand{\inconsistent}{\bot}
 \newcommand{\given}{\ensuremath{~\middle|~}}
+\newcommand{\emptyevent}{\ensuremath{\vartriangle}}
+\newcommand{\indepclass}{\ensuremath{\Diamond}}
+\newcommand{\probfact}[2]{\ensuremath{#1\!::\!#2}}
+
 \newcommand{\todo}[1]{{\color{red!50!black}(\emph{#1})}}
-\newcommand{\remark}[2]{\dashuline{#1}~{\color{green!40!black}\emph{#2}}}
+\newcommand{\remark}[2]{\uwave{#1}~{\color{green!40!black}(\emph{#2})}}
 \newcommand{\replace}[2]{\sout{#1}/{\color{green!20!black}#2}}
-
+\newcommand{\franc}[1]{{\color{orange!60!black}#1}}
 \newcommand{\bruno}{\color{red!60!blue}}
 %
 %   ACRONYMS
 %
 \acrodef{BK}[BK]{background knowledge}
 \acrodef{ASP}[ASP]{answer set program}
-\acrodef{NP}[NP]{normal (logic) program}
+\acrodef{NP}[NP]{normal program}
 \acrodef{DS}[DS]{distribution semantics}
 \acrodef{PF}[PF]{probabilistic fact}
 \acrodef{TC}[TC]{total choice}
 \acrodef{SM}[SM]{stable model}
 \acrodef{SC}[SC]{stable core}
+\acrodef{KL}[KL]{Kullback-Leibler}
 %
 %
 %
 \title{Zugzwang\\\emph{Logic and Artificial Intelligence}\\{\bruno Why this title?}}
 \author{
-    \begin{tabular}{cc}
-        Francisco Coelho & Bruno Dinis\\
-        \texttt{fc@uevora.pt} & \texttt{bruno.dinis@uevora.pt}\\
-        \begin{minipage}{0.5\textwidth}\centering
-            Universidade de Évora and NOVA\textbf{LINCS}
-        \end{minipage}
-        & Universidade de Évora
+    \begin{tabular}{ccc}
+        Francisco Coelho
+        \footnote{Universidade de Évora}
+        & Bruno Dinis
+        \footnote{Universidade de Évora}
+        & Salvador Abreu
+        \footnote{Universidade de Évora}
+        \\
+        \texttt{fc@uevora.pt}
+        & \texttt{bruno.dinis@uevora.pt}
+        & \texttt{spa@uevora.pt}
+        % \\
+        % \begin{minipage}{0.3\textwidth}\centering
+        %     Universidade de Évora and NOVA\textbf{LINCS}
+        % \end{minipage}
+        % & 
+        % \begin{minipage}{0.3\textwidth}\centering
+        %     Universidade de Évora
+        % \end{minipage}
+        % & \begin{minipage}{0.3\textwidth}\centering
+        %     Universidade de Évora and NOVA\textbf{LINCS}
+        % \end{minipage}
     \end{tabular}    
 }
@@ -116,10 +137,10 @@
 \todo{Define and/or give references to all necessary concepts used in the paper}
 \todo{state of the art; references}
-\Acf{ASP} is a logic programming paradigm based on the \ac{SM} semantics of \ac{NP} that can be implemented using the latest advances in SAT solving technology. Unlike ProLog, \ac{ASP} is a truly declarative language that supports language constructs such as disjunction in the head of a clause, choice rules, and hard and weak constraints.
+\Acf{ASP} is a logic programming paradigm based on the \ac{SM} semantics of \acp{NP} that can be implemented using the latest advances in SAT solving technology. Unlike ProLog, \ac{ASP} is a truly declarative language that supports language constructs such as disjunction in the head of a clause, choice rules, and hard and weak constraints.
 \todo{references}
-The \ac{DS} is a key approach to extend logical representations with probabilistic reasoning. \Acp{PF} are the most basic stochastic \ac{DS} primitive and they take the form of logical facts, $a$, labelled with a probability, $p$, such as $p::a$; Each \ac{PF} represents a boolean random variable that is true with probability $p$ and false with probability $\co{p} = 1 - p$. A (consistent) combination of the \acp{PF} defines a \acf{TC} $c = \set{p::a, \ldots}$ such that
+The \ac{DS} is a key approach to extend logical representations with probabilistic reasoning. \Acp{PF} are the most basic \ac{DS} stochastic primitives and take the form of logical facts, $a$, labelled with probabilities, $p$, such as $\probfact{p}{a}$; Each \ac{PF} represents a boolean random variable that is true with probability $p$ and false with probability $\co{p} = 1 - p$. A (consistent) combination of the \acp{PF} defines a \acf{TC} $c = \set{\probfact{p}{a}, \ldots}$ such that
 \begin{equation}
     \pr{C = c} = \prod_{a\in c} p \prod_{a \not\in c} \co{p}.
@@ -141,9 +162,11 @@ Our goal is to extend this probability, from \acp{TC}, to cover the \emph{specif
 %
 Our idea to extend probabilities starts with the stance that a specification describes an \emph{observable system} and that observed events must be related with the \acp{SM} of that specification. From here, probabilities must be extended from \aclp{TC} to \acp{SM} and then from \acp{SM} to any event. 
-Extending probability from \acp{TC} to \acp{SM} faces a critical problem, illustrated by the  example in \cref{sec:example.1}, concerning situations where multiple \acp{SM}, $ab$ and $ac$, result from a given \ac{TC}, $a$, but there is not enough information to assign a single probability to each \ac{SM}. We propose to address this issue by using algebraic variables to describe that lack of information and then estimate the value of those variables from empirical data.
+Extending probability from \acp{TC} to \acp{SM} faces a critical problem, illustrated by the  example in \cref{sec:example.1}, concerning situations where multiple \acp{SM}, $ab$ and $ac$, result from a single \ac{TC}, $a$, but there is not enough information (in the specification) to assign a single probability to each \ac{SM}. We propose to address this issue by using algebraic variables to describe that lack of information and then estimate the value of those variables from empirical data.
+
+In a related work, \cite{verreet2022inference}, epistemic uncertainty (or model uncertainty) is considered as a lack of knowledge about the underlying model, that may be mitigated via further observations. This seems to presuppose a Bayesian approach to imperfect knowledge in the sense that having further observations allows to improve/correct the model. Indeed, the approach in that work uses Beta distributions in order to be able to learn the full distribution. This approach seems to be specially fitted to being able to tell when some probability lies beneath some given value. \todo{Our approach seems to be similar in spirit. If so, we should mention this in the introduction.}
-In a related work, \cite{verreet2022inference}, epistemic uncertainty (or model uncertainty) is considered as a lack of knowledge about the underlying model. This lack of knowledge can be mitigated via further observations. This seems to presuppose a Bayesian approach to imperfect knowledge in the sense that having further observations allows to improve/correct the model. Indeed, the approach in the paper uses Beta distributions in order to be able to learn the full distribution. This approach seems to be specially fitted to being able to tell when some probability lies beneath some given value. \todo{Our approach seems to be similar in spirit. If so, we should mention this in the introduction.}
+\todo{cite \cite{sympy} \franc{--- why? but cite \cite{cozman2020joy} and relate with our work.}}
 \todo{Discuss the least informed strategy and the corolary that \aclp{SM} should be conditionally independent on the \acl{TC}.}
@@ -157,12 +180,12 @@ In a related work, \cite{verreet2022inference}, epistemic uncertainty (or model 
 Consider the following specification
 \begin{equation}
     \begin{aligned}
-        0.3::a&,\cr
+        \probfact{0.3}{a}&,\cr
         b \vee c& \leftarrow a.
     \end{aligned}
     \label{eq:example.1}
 \end{equation}
-This specification has three stable models, $\co{a}, ab$ and $ac$ (see Figure~\ref{F:stableexample}). While it is straightforward to set $P(\co{a})=0.7$, there is \emph{no further information} to assign values to $P(ab)$ and $P(ac)$. Assuming that the \acf{SM} are (probabilistically) independent, we can use a parameter \replace{$\lambda$}{$\theta$} such that
+This specification has three stable models, $\co{a}, ab$ and $ac$ (see \cref{fig:running.example}). While it is straightforward to set $P(\co{a})=0.7$, there is no further information to assign values to $P(ab)$ and $P(ac)$. Assuming that the \acfp{SM} are (probabilistically) independent, we can use a parameter $\theta$ such that
 $$
 \begin{aligned}
 P(ab) &= 0.3 \theta,\cr
@@ -171,15 +194,15 @@ P(ac) &amp;= 0.3 (1 - \theta).
 $$
 \end{example} 
-While uncertainty is inherent to the specification it can be mitigated with the help of a dataset: the parameter $\theta$ can be estimated from a empirical distribution \todo{or we can have a distribution of $\theta$}.
+While uncertainty is inherent to the specification it can be mitigated with the help of a dataset: the parameter $\theta$ can be estimated from a empirical distribution \todo{or we can have a distribution of $\theta$}. \todo{point to examples of this in following sections.}
 In summary, if an \ac{ASP} specification is intended to describe some observable system then:
 \begin{enumerate}
     \item Observations can be used to estimate the value of the parameters (such as $\theta$ above and others entailed from further clauses).
     \item \todo{What about the case where we already know a distribution of $\theta$?} 
-    \item With a probability set for the stable models, we want to extend it to all the events of the \replace{specification}{domain}. 
-    \item This extended probability can then be related to the \emph{empirical distribution}, using a probability divergence, such as Kullback-Leibler; and the divergence value used as a \emph{performance} measure of the specification with respect to the observations.
+    \item With a probability set for the \aclp{SM}, we want to extend it to all the events of the specification domain. 
+    \item This extended probability can then be related to the \emph{empirical distribution}, using a probability divergence, such as \ac{KL}; and the divergence value used as a \emph{performance} measure of the specification with respect to the observations.
     \item If that specification is only but one of many possible candidates then that performance measure can be used, \emph{e.g.} as fitness, by algorithms searching (optimal) specifications of a dataset of observations.
 \end{enumerate}
@@ -189,20 +212,20 @@ In summary, if an \ac{ASP} specification is intended to describe some observable
 Currently, we are addressing the problem of extending a probability function (possibly using parameters such as $\theta$), defined on the \acp{SM} of a specification, to all the events of that specification. Of course, this extension must satisfy the Kolmogorov axioms of probability so that probabilistic reasoning is consistent with the \ac{ASP} specification. 
-Conditional independence of stable worlds asserts a least informed strategy that we discussed in the introduction and make explicit here:
+The conditional independence of stable worlds asserts the least informed strategy that we discussed in the introduction and make explicit here:
 \begin{assumption}\label{assumption:smodels.independence}
-    Stable models are conditionally independent, given their total choices.
+    \Acl{SM} are conditionally independent, given their total choices.
 \end{assumption}
-The stable models $ab, ac$ from \cref{running.example} result from the clause $b \vee c \leftarrow a$ and the total choice $a$. These formulas alone impose no relation between $b$ and $c$ (given $a$), so none should be assumed. Dependence relations are further discussed in \cref{subsec:dependence}.
+The stable models $ab, ac$ from \cref{running.example} result from the clause $b \vee c \leftarrow a$ and the total choice $a$. These formulas alone imposes no relation between $b$ and $c$ (given $a$), so none should be assumed. Dependence relations are further discussed in \cref{subsec:dependence}.
 \section{Extending Probabilities}\label{sec:extending.probalilities}
 \begin{figure}[t]
     \begin{center}
         \begin{tikzpicture}
-            \node[event] (E) {$\set{}$};
+            \node[event] (E) {$\emptyevent$};
             \node[tchoice, above left = of E] (a) {$a$};
             \node[smodel, above left = of a] (ab) {$ab$};
             \node[smodel, above right = of a] (ac) {$ac$};
@@ -237,14 +260,16 @@ The stable models $ab, ac$ from \cref{running.example} result from the clause $b
             \draw[doubt] (c) to[bend right] (Ac);
         \end{tikzpicture}
     \end{center}
-    \caption{Events related to the stable models of \cref{running.example}. The circle nodes are the \acp{TC} and the shaded nodes are the \acp{SM}.}
+    \caption{Events related to the stable models of \cref{running.example}. The circle nodes are \aclp{TC} and shaded nodes are \aclp{SM}. The \emph{empty event}, with no literals, is denoted by $\emptyevent$. Notice that the event $bc$ is not related with any stable model.}
     % \caption{Extending probabilities from total choice nodes to stable models and then to general events in a \emph{node-wise} process quickly leads to coherence problems concerning probability, with no clear systematic approach --- Instead, weight extension can be based in \emph{the relation an event has with the stable models}.{\bruno Why is this comment on the caption?}}
-    \label{F:stableexample}
+    \label{fig:running.example}
 \end{figure}
 \todo{Somewhere, we need to shift the language from extending \emph{probabilities} to extending \emph{measures}}
-The diagram in \cref{F:stableexample} illustrates the problem of extending probabilities from total choice nodes to stable models and then to general events in a \emph{node-wise} process. This quickly leads to coherence problems concerning probability, with no clear systematic approach --- Instead, weight extension can be based in the relation an event has with the stable models.
+\note{$\emptyevent$ notation introduced in \cref{fig:running.example}.}
+
+The diagram in \cref{fig:running.example} illustrates the problem of extending probabilities from total choice nodes to stable models and then to general events in a \emph{node-wise} process. This quickly leads to coherence problems concerning probability, with no clear systematic approach --- Instead, weight extension can be based in the relation an event has with the stable models.
 \subsection{An Equivalence Relation}\label{subsec:equivalence.relation}
@@ -259,7 +284,7 @@ Given an ASP specification
 Our path starts with a perspective of stable models as playing a role similar to \emph{prime} factors. 
 The stable models of a specification are the irreducible events entailed from that specification and any event must be \replace{interpreted}{considered} under its relation with the stable models.
-\remark{\todo{Introduce a structure with worlds, events, and stable models}}{seems irrelevant}
+%\remark{\todo{Introduce a structure with worlds, events, and stable models}}{seems irrelevant}
 This focus on the \acp{SM} leads to the following definition:
 \begin{definition}\label{def:stable.structure}
@@ -269,7 +294,7 @@ This focus on the \acp{SM} leads to the following definition:
 \todo{expand this text to explain how the stable models form the basis of the equivalence relation}. %This \replace{stance}{} leads to definition \ref{def:rel.events}:
-\begin{definition}\label{def:rel.events}
+\begin{definition}\label{def:stable.core}
     The \emph{\ac{SC}} of the event $e\in \fml{E}$ is
     % \begin{equation}
@@ -282,12 +307,12 @@ This focus on the \acp{SM} leads to the following definition:
     %     \stablecore{e} = \uset{e} \cup \lset{e} \label{def:stable.core}
     % \end{equation}
     \begin{equation}
-        \stablecore{e} := \set{s \in \fml{S} \given e \subseteq s \vee s \subseteq e} \label{eq:stable.core}
+        \stablecore{e} := \set{s \in \fml{S} \given s \subseteq e \vee e \subseteq s} \label{eq:stable.core}
     \end{equation}
     \end{definition}
-We now define an equivalence relation, $\sim$, so that two events are related if they are either both inconsistent or both consistent with the same stable core.
+We now define an equivalence relation, $\sim$, so that two events are related if either both are inconsistent or both are consistent with the same stable core.
 \begin{definition}\label{def:equiv.rel}
 For a given specification, let $u, v \in \fml{E}$. The equivalence relation $\sim$ is defined by
@@ -296,7 +321,7 @@ For a given specification, let $u, v \in \fml{E}$. The equivalence relation $\si
     \end{equation}
 \end{definition}
-Observe that the minimality of stable models implies that, in \cref{def:stable.core}, either $e$ is a stable model or one of $e \subseteq s, s \subseteq e$ is never true. 
+Observe that the minimality of stable models implies that, in \cref{def:stable.core}, either $e$ is a stable model or one of $s \subseteq e, e \subseteq s$ is never true. 
 %
 % \begin{definition}\label{def:smodel.events}
 %     For $\set{s_1, \ldots, s_n} \subseteq \fml{S}$ define
@@ -333,17 +358,17 @@ This relation defines a partition of the events space, where each class holds a 
 %         \class{e} = \class{s_1 \cup \cdots \cup s_n}.
 %     \end{equation}
 %     We simplify the notation with $\class{s_1, \ldots, s_n} := \class{s_1 \cup \cdots \cup s_n}$. 
-%     \todo{This only works for consistent $s_1, \ldots, s_n$: $\set{\set{}} = \class{\co{a}, ab, ac} \not= \class{a\co{a}bc} = \inconsistent$.}
+%     \todo{This only works for consistent $s_1, \ldots, s_n$: $\set{\emptyevent} = \class{\co{a}, ab, ac} \not= \class{a\co{a}bc} = \inconsistent$.}
 % \end{theorem}
 % \begin{proof}
 % \todo{tbd}
 % \end{proof}
-The subsets of the stable models, together with $\inconsistent$, form a set of representatives. Consider again Example~\ref{running.example}. As previously mentioned, the stable models are $\fml{S} = \co{a}, ab, ac$ so the quotient set of this relation is $\class{\fml{E}}:$
+The subsets of the stable models, together with $\inconsistent$, form a set of representatives. Consider again Example~\ref{running.example}. As previously mentioned, the stable models are $\fml{S} = \co{a}, ab, ac$ so the quotient set of this relation is:
 \begin{equation}
-    \set{
+    \class{\fml{E}} = \set{
         \inconsistent,
-        \emptyset,
+        \indepclass,
         \class{\co{a}},
         \class{ab},
         \class{ac},
@@ -359,34 +384,73 @@ The subsets of the stable models, together with $\inconsistent$, form a set of r
 %     & \stablecore{\co{a}, ab, ac}.
 % \end{aligned}
 \end{equation}
-
-For example, 
+where $\indepclass$ denotes both the class of \emph{independent} events $e$ such that $\stablecore{e} = \emptyset$ and its core (which is the emptyset). We have:
 \begin{equation*}
-    \begin{aligned}
-        \class{\set{}} &= \class{\co{a}, ab, ac},
-            & \class{a} &= \class{ab, ac},
-            & \class{b} &= \class{ab},
-            & \class{\co{b}} &= \emptyset, 
-        \\ \class{a\co{c}} &= \emptyset, 
-            & \class{ab} &= \emptyset,
-            & \class{b\co{b}} &= \inconsistent, 
-            & \class{\co{a}b} &=\class{\co{a}}, 
-        \\ \class{\co{bc}} &=\emptyset,
-            & \class{abc} &=  \class{ab, ac},
-            & \class{a\co{b}c} &=  \class{ac},
-            & \class{\co{a}bc} &=  \class{\co{a}},
-        % & \class{\co{a}} &= \class{\co{a}},
-        %         &  \class{\set{}} &= \class{\co{a}, ac, ab}
-    \end{aligned}
+    \begin{array}{l|lr}
+        \text{\textbf{Core}} 
+        & \text{\textbf{Class}} 
+        & \text{\textbf{Size}}\\
+        \hline 
+        %
+        \inconsistent
+        & \text{inconsistent events}
+        & 37
+        \\
+        %
+        \indepclass 
+        & \co{b}, \co{c}, bc, \co{b}a, \co{b}c, \co{b}\co{c}, \co{c}a, \co{c}b, \co{b}\co{c}a 
+        & 9
+        \\
+        %
+        \co{a} 
+        & \co{a}, \co{a}b, \co{a}c, \co{a}\co{b}, \co{a}\co{c}, \co{a}bc, \co{a}b\co{c}, \co{a}\co{b}c, \co{a}\co{b}\co{c}
+        & 9
+        \\
+        %
+        ab
+        & b, ab, ab\co{c}
+        & 3
+        \\
+        %
+        ac
+        & c, ac, a\co{b}c 
+        & 3
+        \\
+        %
+        \co{a}, ab
+        & \emptyset
+        & 0
+        \\
+        %
+        \co{a}, ac
+        & \emptyset
+        & 0
+        %
+        \\
+        %
+        ab, ac
+        & a, abc
+        & 2
+        \\
+        %
+        \co{a}, ab, ac
+        & \emptyevent
+        & 1
+        \\
+        %
+        \hline
+        \Omega
+        & \text{all events}
+        & 64
+    \end{array}
 \end{equation*}
-
 \begin{itemize}
     \item Since all events within an equivalence class are in relation with a specific set of stable models, \emph{weights, including probability, should be constant within classes}:
     \[
     \forall u\in \class{e} \left(\pr{u} = \pr{e} \right).
     \]
-    \item So, instead of dealing with $64 = 2^6$ events, we need only to handle $9 = 2^3 + 1$ classes, well defined in terms of combinations of the stable models. 
+    \item So, instead of dealing with $64 = 2^6$ events, we need only to \replace{handle}{handle, at most,} $9 = 2^3 + 1$ classes, well defined in terms of combinations of the stable models. \remark{This seems fine}{but, in general, we will get \emph{much more} stable models than literals, so the number of these classes is much larger than of events!} {\bruno Nevertheless, the equivalence classes alow us to propagate probabilities from total choices to events, as explained in the next subsection.}
     % \item The extended probability \emph{events} are the \emph{classes}.
 \end{itemize}
@@ -410,10 +474,10 @@ The ``extension&#39;&#39; phase, traced by equations (\ref{eq:prob.total.choice}) and (\
     %
     \item[Stable Models.] Each total choice $c$, together with the rules and the other facts of a specification, defines a set of stable models associated with that choice, that we denote by $S_c$.
-    Given a stable model $s \in \fml{S}$, a total choice $c$, and variables/values $\theta_{s,c} \in \intcc{0, 1}$, 
+    Given a stable model $s \in \fml{S}$, a total choice $c$, and variables/values $\theta_{s,c} \in \intcc{0, 1}$,
     \begin{equation}
         \pw{s, c} := \begin{cases}
-            \pw{c}\theta_{s,c} & \text{if~} s \in S_c\cr
+            \theta_{s,c}\remark{\pw{c}}{\text{maybe not!}} & \text{if~} s \in S_c\cr
             0&\text{otherwise}
         \end{cases}
         \label{eq:weight.stablemodel}
@@ -429,10 +493,10 @@ The ``extension&#39;&#39; phase, traced by equations (\ref{eq:prob.total.choice}) and (\
         \end{equation}
         \item \textbf{Independent Class.} A world that neither contains nor is contained in a stable model describes a case that, according to the specification, should never be observed. So the respective weight is set to zero:
         \begin{equation}
-            \pw{\class{e}, c} := 0,~\text{if}~\stablecore{e} = \emptyset.
+            \pw{\indepclass, c} := 0.
             \label{eq:weight.class.independent}
         \end{equation}
-        \item \textbf{Other Classes.} The extension must be constant within a class, its value should result from the elements in the stable core, and respect the assumption \ref{assumption:smodels.independence}:
+        \item \textbf{Other Classes.} The extension must be constant within a class, its value should result from the elements in the stable core, and respect the assumption \ref{assumption:smodels.independence} (stable models independence):
         \begin{equation}
             \pw{\class{e}, c} := \prod_{k=1}^{n}\pw{s_k, c},~\text{if}~\stablecore{e} = \set{s_1, \ldots, s_n}.
             \label{eq:weight.class.other}
@@ -449,6 +513,15 @@ The ``extension&#39;&#39; phase, traced by equations (\ref{eq:prob.total.choice}) and (\
         \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}.
         \label{eq:weight.events.unconditional}
     \end{equation}
+    \remark{instead of that equation}{if we set $\pw{s,c} := \theta_{s,c}$ in equation \eqref{eq:weight.stablemodel} here we do: 
+        $$
+        \pw{e} := \sum_{c\in\fml{C}} \pw{e, c}\pw{c}.
+        $$
+    By the way, this is the \emph{marginalization + bayes theorem} in statistics:
+    $$
+    P(A) = \sum_b P(A | B=b)P(B=b) 
+    $$
+    }
 \end{description}
 % PARAMETERS FOR UNCERTAINTY
@@ -459,7 +532,7 @@ The ``extension&#39;&#39; phase, traced by equations (\ref{eq:prob.total.choice}) and (\
 \end{itemize}
-The $\theta_{s,c}$ parameters in equation \eqref{eq:weight.stablemodel} express the \emph{specification's} lack of knowledge about the weight assignment, when a single total choice entails more than one stable model. In that case, how to distribute the respective weights? Our \replace{answer}{proposal} to \replace{}{address} this problem consists in assigning an unknown weight, $\theta_{s,c}$, conditional {\bruno depending???} on the total choice, $c$, to each stable model $s$. This approach allows the expression of an unknown quantity and future estimation, given observed data.
+The $\theta_{s,c}$ parameters in equation \eqref{eq:weight.stablemodel} express the \emph{specification's} lack of knowledge about the weight assignment, when a single total choice entails more than one stable model. In that case, how to distribute the respective weights? Our proposal to address this problem consists in assigning an unknown weight, $\theta_{s,c}$, conditional on the total choice, $c$, to each stable model $s$. This approach allows the expression of an unknown quantity and future estimation, given observed data.
 % SUPERSET
 Equation \eqref{eq:weight.class.other} results from conditional independence of stable models. 
@@ -471,97 +544,281 @@ Equation \eqref{eq:weight.class.other} results from conditional independence of 
 We continue with the specification from Equation \eqref{eq:example.1}. 
-\textbf{Total Choices.} The total choices, and respective stable models, are
-\begin{center}
-    \begin{tabular}{l|r|r}
-        Total Choice ($c$) & $\pw{c}$ & Stable Models ($s$)\\
-        \hline 
-        $a$ & $0.3$ & $ab$ and $ac$.\\
-        $\co{a} = \neg a$ & $\co{0.3} = 0.7$ & $\co{a}$.
-    \end{tabular}
-\end{center}
+\begin{description}
+    %
+    \item[Total Choices.] The total choices, and respective stable models, are
+    \begin{center}
+        \begin{tabular}{llr}
+            \textbf{Total Choice} & \textbf{Stable Models} & \textbf{$\pw{\cdot}$}\\
+            \hline 
+            $a$ & $ab, ac$ & $0.3$\\
+            $\co{a} = \neg a$ & $\co{a}$ & $\co{0.3} = 0.7$
+        \end{tabular}
+    \end{center}
+    %
+    \item[Stable Models.] \todo{Enter the $\theta$ parameters.}
+    \begin{equation*}
+        \begin{array}{ll|r}
+            \text{\textbf{Stable Model}}
+            & \text{\textbf{Total Choice}} 
+            & \pw{s,c}
+            \\
+            \hline
+            \co{a}
+            & \co{a}
+            & 1.0
+            \\
+            ab
+            & a
+            & \theta
+            \\
+            ac
+            & a
+            & \co{\theta}
+        \end{array}
+    \end{equation*}
+    \item[Classes.] Following the definitions in \cref{eq:stable.core,eq:equiv.rel,eq:event.class} and in \cref{eq:weight.class.inconsistent,eq:weight.class.independent,eq:weight.class.other} we get the following quotient set, and weights:
+    \begin{equation*}
+        \begin{array}{l|rrr|r}
+            \text{\textbf{Core}} 
+            & \pw{c} 
+            & \pw{s,c} 
+            & \pw{\class{e}, c}
+            & \pw{e}
+            \\
+            \hline 
+            %
+            \inconsistent
+            &
+            & 0.0
+            & 0.0
+            & 0.0
+            \\
+            %
+            \indepclass
+            &
+            & 0.0
+            & 0.0
+            & 0.0
+            \\
+            %
+            \co{a} 
+            & 0.7 
+            & 1.0
+            & 1.0
+            & 0.7
+            \\
+            %
+            ab
+            & 0.3
+            & \theta
+            & \theta
+            & 0.3\theta
+            \\
+            %
+            ac
+            & 0.3
+            & \co{\theta}
+            & \co{\theta}
+            & 0.3\co{\theta}
+            \\
+            %
+            \co{a}, ab
+            & 0.7, 0.3
+            & 1.0, \theta
+            & 1.0, \theta
+            & 0.7 + 0.3\theta
+            \\
+            %
+            \co{a}, ac
+            & 0.7, 0.3
+            & 1.0, \co{\theta}
+            & 1.0, \co{\theta}
+            & 0.7 + 0.3\co{\theta}
+            \\
+            %
+            ab, ac
+            & 0.3, 0.3
+            & \theta, \co{\theta}
+            & \theta\co{\theta}
+            & 0.3\theta\co{\theta}
+            \\
+            %
+            \co{a}, ab, ac
+            & 0.7, 0.3, 0.3
+            & 1.0, \theta, \co{\theta}
+            & 1.0, \theta\co{\theta}
+            & 0.7 + 0.3\theta\co{\theta}
+        \end{array}
+    \end{equation*}
+    \item[Normalization.] To get a weight that sums up to one, we compute the \emph{normalization factor}
+    \begin{equation*}
+        Z := \sum_{e\in\fml{E}} \pw{e}
+        = \sum_{e\in\fml{E}} \sum_{c\in\fml{C}}\pw{e,c}\pw{c}
+        = \sum_{e\in\fml{E}} \sum_{c\in\fml{C}} \del{\prod_{s\in\stablecore{e}}\pw{s,c}}\pw{c}
+    \end{equation*}
+    to divide the weight function into a normalized weight:
+    \begin{equation*}
+        \nu\at{e} := \frac{\pw{e}}{Z}.
+    \end{equation*}
+    Since $\pw{\cdot}$ is constant on classes we have:
+    \begin{equation*}
+        \begin{array}{lr|r}
+            \text{\textbf{Core}} 
+            & \text{\textbf{Size}} 
+            & \pw{\class{\cdot}} 
+            \\
+            \hline 
+            %
+            \inconsistent
+            & 37
+            & 0.0
+            \\
+            %
+            \indepclass 
+            & 9
+            & 0.0
+            \\
+            %
+            \co{a}
+            & 9
+            & 0.7
+            \\
+            %
+            ab
+            & 3
+            & 0.3\theta
+            \\
+            %
+            ac
+            & 3
+            & 0.3\co{\theta}
+            \\
+            %
+            \co{a}, ab
+            & 0
+            & 0.3\theta + 0.7
+            \\
+            %
+            \co{a}, ac
+            & 0
+            & 0.3\co{\theta} + 0.7
+            %
+            \\
+            %
+            ab, ac
+            & 2
+            & 0.3\theta\co{\theta}
+            \\
+            %
+            \co{a}, ab, ac
+            & 1
+            & 0.3\theta\co{\theta} + 0.7
+            \\
+            %
+            \hline
+            Z
+            &
+            & 0.9\theta\co{\theta} + 7.9
+        \end{array}
+    \end{equation*}
+\end{description}
+%
+%
+%
+\subsection{A Not So Simple Example}
+
+\todo{this subsection}
+
+In this section we see how the classical example of the Burglary, Earthquake, Alarm \cite{Judea88} works in our setting. This example is a commonly used example in Bayesian networks because it illustrates reasoning under uncertainty.  The gist of example is given in \cref{Figure_Alarm}. It involves a simple network of events and conditional probabilities. 
+
+The events are: Burglary ($B$), Earthquake ($E$), Alarm ($A$), Mary calls ($M$) and John calls ($J$). The initial events $B$ and $E$ are assumed to be independent events that occur with probabilities $P(B)$ and $P(E)$, respectively. There is an alarm system that can be triggered by either of the initial events $B$ and $E$. The probability of the alarm going off is a conditional probability depending on whether $B$, or $E$ as occurred. One denotes these probabilities, as per usual,  by $P(A|B)$, and $P(A|E)$. There are two neighbours, Mary and John who have agreed to call if they hear the alarm. The probability that they do actually call is also a conditional probability denoted by $P(M|A)$ and $P(J|A)$, respectively.
+
+Considering the probabilities given in \cref{Figure_Alarm} we obtain the following specification
-\textbf{Stable Models.} The weights from equation \eqref{eq:weight.stablemodel} can be tabulated by a matrix where a column represents a total choice and a row a stable model. We get: 
-\begin{multline*}
-    \pw{s, c} = 
-    \begin{pmatrix}
-        \pw{ab, a} & \pw{ab, \co{a}} \\
-        \pw{ac, a} & \pw{ac, \co{a}} \\
-        \pw{\co{a}, a} & \pw{\co{a}, \co{a}}
-    \end{pmatrix}
-    = 
-    \begin{pmatrix}
-        \pw{a} \theta_{ab, a} & \pw{\co{a}} \theta_{ab, \co{a}} \\
-        \pw{a} \theta_{ac, a} & \pw{\co{a}} \theta_{ac, \co{a}} \\
-        \pw{a} \theta_{\co{a}, a} & \pw{\co{a}} \theta_{\co{a}, \co{a}}
-    \end{pmatrix}
-    \\
-    = 
-    \begin{pmatrix}
-        0.3 \times \theta_{ab, a} &  0.7 \times 0.0 \\
-        0.3 \times \theta_{ac, a} &  0.7 \times 0.0 \\
-        0.3 \times 0.0 & 0.7 \times 1.0
-    \end{pmatrix}
-    = 
-    \begin{pmatrix}
-        0.3\theta & 0.0 \\
-        0.3\co{\theta} &  0.0 \\
-        0.0 & 7.0
-    \end{pmatrix}
-\end{multline*} 
-where we set $\theta = \theta_{ab, a}$ to simplify the notation.
-
-\textbf{Classes.} Following the definitions in \cref{eq:stable.core,eq:equiv.rel,eq:event.class} and in \cref{eq:weight.class.inconsistent,eq:weight.class.independent,eq:weight.class.other} we get the following quotient set, and weights:
 \begin{equation*}
-    \begin{array}{ll|rr}
-        \text{\textbf{Core}} & \text{\textbf{Class}} & \text{\textbf{Parameters}} & \text{\textbf{Weights}}\\
-        \hline 
-        %
-        \inconsistent
-        & \text{inconsistent events}
-        & 0.0
-        \\
-        %
-        \emptyset 
-        & \co{b}, \co{c}, bc, \co{b}a, \co{b}c, \co{b}\co{c}, \co{c}a, \co{c}b, \co{b}\co{c}a 
-        & 0.0
-        \\
-        %
-        \co{a} 
-        & \co{a}, \co{a}b, \co{a}c, \co{a}\co{b}, \co{a}\co{c}, \co{a}bc, \co{a}b\co{c}, \co{a}\co{b}c, \co{a}\co{b}\co{c}
-        & 1.0 
-        & 0.7
-        \\
-        %
-        ab
-        & b, ab, ab\co{c}
-        & \theta
-        & 0.3
-        \\
-        %
-        ac
-        & c, ac, a\co{b}c 
-        & \co{\theta}
-        & 0.3
-        \\
-        %
-        ab, ac
-        & a, abc
-        & \co{\theta}\theta
-        & 0.3
-        \\
-        %
-        \co{a}, ab, ac
-        & \set{}
-        & \co{\theta}\theta, 1.0
-        & 0.3, 0.7
-    \end{array}
+    \begin{aligned}
+        \probfact{0.001}{B}&,\cr
+        \probfact{0.002}{E}&,\cr
+        \probfact{??}{A}&,\cr
+        ... &. \cr
+    \end{aligned}
+    \label{eq:not_so_simple_example}
 \end{equation*}
-\remark{fc}{I really don't like those squares}
-\subsection{A Not So Simple Example}
-\todo{this subsection}
+
+
+
+\begin{figure}
+\begin{center}
+\begin{tikzpicture}[node distance=2.5cm]
+
+% Nodes
+\node[smodel, circle] (A) {A};
+\node[tchoice, above right of=A] (B) {B};
+\node[tchoice, above left of=A] (E) {E};
+\node[tchoice, below left of=A] (M) {M};
+\node[tchoice, below right of=A] (J) {J};
+
+% Edges
+\draw[->] (B) to[bend left] (A) node[right,xshift=1.1cm,yshift=0.8cm] {\footnotesize{$P(B)=0,001$}} ;
+\draw[->] (E) to[bend right] (A) node[left, xshift=-1.4cm,yshift=0.8cm] {\footnotesize{$P(E)=0,002$}} ;
+\draw[->] (A) to[bend right] (M) node[left,xshift=0.2cm,yshift=0.7cm] {\footnotesize{$P(M|A)$}};
+\draw[->] (A) to[bend left] (J) node[right,xshift=-0.2cm,yshift=0.7cm] {\footnotesize{$P(J|A)$}} ;
+\end{tikzpicture}
+\end{center}
+
+\begin{multicols}{3}
+
+\footnotesize{
+ \begin{equation*}
+ \begin{split}
+ &P(M|A)\\
+      &  \begin{array}{c|cc}
+        A & T & F \\
+            \hline 
+        T & 0,9 & 0,1\\
+        F& 0,05 & 0,95    
+        \end{array}
+        \end{split}
+\end{equation*}   
+}
+
+\footnotesize{
+ \begin{equation*}
+ \begin{split}
+ &P(J|A)\\
+      &  \begin{array}{c|cc}
+        A & T & F \\
+            \hline 
+        T & 0,7 & 0,3\\
+        F& 0,01 & 0,99    
+        \end{array}
+        \end{split}
+\end{equation*}   
+}
+\footnotesize{
+ \begin{equation*}
+ \begin{split}
+ P(A|B \vee E)\\
+        \begin{array}{c|c|cc}
+         B & E& T & F \\
+            \hline 
+        T & T & 0,95 & 0,05\\
+        T & F & 0,94 & 0,06\\
+        F & T & 0,29 & 0,71\\
+        F & F & 0,001 & 0,999     
+        \end{array}
+\end{split}
+\end{equation*} 
+}
+\end{multicols}
+\caption{The Earthquake, Burglary, Alarm model}
+\label{Figure_Alarm}
+\end{figure}
+
+
 \section{Discussion}
@@ -610,7 +867,7 @@ Our basic assertion about dependence relations between atoms of the underlying s
 For example, a dependence relation between $b$ and $c$ can be expressed by $b \leftarrow c \wedge d$, where $d$ is an atomic choice that explicitly expresses the dependence between $b$ and $c$. One would get, for example, a specification such as
 $$
-0.3::a, b \vee c \leftarrow a, 0.2::d, b \leftarrow c \wedge d.
+\probfact{0.3}{a}, b \vee c \leftarrow a, \probfact{0.2}{d}, b \leftarrow c \wedge d.
 $$
 with stable models
 $
@@ -647,7 +904,7 @@ $$
                 &\not= \pr{W = b}\pr{W = c}
 \end{aligned}
 $$
-\emph{i.e.} the events $W = b$ and $W = c$ are dependent and that dependence results directly from the segment $0.2::d, b \leftarrow c \wedge d$ in the specification.
+\emph{i.e.} the events $W = b$ and $W = c$ are dependent and that dependence results directly from the segment $\probfact{0.2}{d}, b \leftarrow c \wedge d$ in the specification.
 {\bruno Why does this not contradict Assumption 1?}
+@book {Judea88,
+    AUTHOR = {Pearl, Judea},
+     TITLE = {Probabilistic reasoning in intelligent systems: networks of
+              plausible inference},
+    SERIES = {The Morgan Kaufmann Series in Representation and Reasoning},
+ PUBLISHER = {Morgan Kaufmann, San Mateo, CA},
+      YEAR = {1988},
+     PAGES = {xx+552},
+      ISBN = {0-934613-73-7},
+   MRCLASS = {68-02 (68T01 92A25)},
+  MRNUMBER = {965765},
+MRREVIEWER = {V. Yu. Trakhtman},
+}
+
+ @article{sympy,
+     title = {SymPy: symbolic computing in Python},
+     author = {Meurer, Aaron and Smith, Christopher P. and Paprocki, Mateusz and \v{C}ert\'{i}k, Ond\v{r}ej and Kirpichev, Sergey B. and Rocklin, Matthew and Kumar, AMiT and Ivanov, Sergiu and Moore, Jason K. and Singh, Sartaj and Rathnayake, Thilina and Vig, Sean and Granger, Brian E. and Muller, Richard P. and Bonazzi, Francesco and Gupta, Harsh and Vats, Shivam and Johansson, Fredrik and Pedregosa, Fabian and Curry, Matthew J. and Terrel, Andy R. and Rou\v{c}ka, \v{S}t\v{e}p\'{a}n and Saboo, Ashutosh and Fernando, Isuru and Kulal, Sumith and Cimrman, Robert and Scopatz, Anthony},
+     year = 2017,
+     month = jan,
+     keywords = {Python, Computer algebra system, Symbolics},
+     abstract = {
+                SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.
+             },
+     volume = 3,
+     pages = {e103},
+     journal = {PeerJ Computer Science},
+     issn = {2376-5992},
+     url = {https://doi.org/10.7717/peerj-cs.103},
+     doi = {10.7717/peerj-cs.103}
+    }
+
 @inproceedings{verreet2022inference,
   title={Inference and learning with model uncertainty in probabilistic logic programs},
   author={Verreet, Victor and Derkinderen, Vincent and Dos Martires, Pedro Zuidberg and De Raedt, Luc},
@@ -46,3 +77,20 @@
   year={2012},
   publisher={Morgan \& Claypool Publishers}
 }
+
+@article{10.7717/peerj-cs.103,
+     title = {SymPy: symbolic computing in Python},
+     author = {Meurer, Aaron and Smith, Christopher P. and Paprocki, Mateusz and \v{C}ert\'{i}k, Ond\v{r}ej and Kirpichev, Sergey B. and Rocklin, Matthew and Kumar, AMiT and Ivanov, Sergiu and Moore, Jason K. and Singh, Sartaj and Rathnayake, Thilina and Vig, Sean and Granger, Brian E. and Muller, Richard P. and Bonazzi, Francesco and Gupta, Harsh and Vats, Shivam and Johansson, Fredrik and Pedregosa, Fabian and Curry, Matthew J. and Terrel, Andy R. and Rou\v{c}ka, \v{S}t\v{e}p\'{a}n and Saboo, Ashutosh and Fernando, Isuru and Kulal, Sumith and Cimrman, Robert and Scopatz, Anthony},
+     year = 2017,
+     month = jan,
+     keywords = {Python, Computer algebra system, Symbolics},
+     abstract = {
+                SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.
+             },
+     volume = 3,
+     pages = {e103},
+     journal = {PeerJ Computer Science},
+     issn = {2376-5992},
+     url = {https://doi.org/10.7717/peerj-cs.103},
+     doi = {10.7717/peerj-cs.103}
+    }
	@@ -0,0 +1,88 @@		@@ -0,0 +1,88 @@
		1	+from pgmpy.readwrite import BIFReader
		2	+import networkx as nx
		3	+import pydot
		4	+from networkx.drawing.nx_pydot import graphviz_layout
		5	+
		6	+def summary_dag(filename):
		7	+ file = BIFReader(filename)
		8	+ model = file.get_model() # Cria um objeto BayesianModel
		9	+ nodes = model.nodes()
		10	+
		11	+ asc = 0
		12	+ des = 0
		13	+ nodes_in = 0
		14	+ nodes_out = 0
		15	+
		16	+ for x in nodes:
		17	+ # Obtem os nós pais e filhos de cada nós
		18	+ parents = model.get_parents(x)
		19	+ children = model.get_children(x)
		20	+
		21	+ if len(parents) != 0:
		22	+ des += 1
		23	+ if len(children) != 0:
		24	+ asc += 1
		25	+ nodes_in += len(children)
		26	+ nodes_out += len(parents)
		27	+
		28	+ return {
		29	+ 'dag_file': filename,
		30	+ 'nodes': nodes,
		31	+ 'count_parents': des,
		32	+ 'count_children': asc,
		33	+ 'count_in_edges': nodes_in,
		34	+ 'count_out_edges': nodes_out,
		35	+ 'bnmodel': model
		36	+ }
		37	+
		38	+def summary_str(d):
		39	+ s = "DAG file: " + d["dag_file"] + "\n"
		40	+ s += " Number of nodes: " + str(len(d["nodes"])) + "\n"
		41	+ s += " Number of parents: " + str(d["count_parents"]) + "\n"
		42	+ s += " Number of children: " + str(d["count_children"]) + "\n"
		43	+ s += " Average in-degree: " + str(d["count_in_edges"]/len(d["nodes"])) + "\n"
		44	+ s += " Average out-degree: " + str(d["count_out_edges"]/len(d["nodes"])) + "\n"
		45	+ s += " Nodes:" + "\n"
		46	+ nodes = d["nodes"]
		47	+ for x in nodes:
		48	+ s += " " + str(x) + "\n"
		49	+ return s
		50	+
		51	+def show_dag(model):
		52	+ G = nx.DiGraph()
		53	+ dot = pydot.Dot(graph_type='digraph')
		54	+ for node in model['nodes']:
		55	+ G.add_node(node)
		56	+ dot.add_node(pydot.Node(node))
		57	+ for edge in model['bnmodel'].edges:
		58	+ G.add_edge(edge[0], edge[1])
		59	+ dot.add_edge(pydot.Edge(edge[0], edge[1]))
		60	+ # objeto pydot
		61	+
		62	+ # defina o layout do grafo
		63	+ pos = graphviz_layout(G, prog='dot')
		64	+ # desenhe o grafo usando o networkx
		65	+ nx.draw(G, pos, with_labels=True)
		66	+
		67	+def save_dag(model, target_filename):
		68	+ G = nx.DiGraph()
		69	+ for node in model.nodes:
		70	+ G.add_node(node)
		71	+ for edge in model.edges:
		72	+ G.add_edge(edge[0], edge[1])
		73	+ # objeto pydot
		74	+ dot = pydot.Dot(graph_type='digraph')
		75	+ for node in model.nodes:
		76	+ dot.add_node(pydot.Node(node))
		77	+ for edge in model.edges:
		78	+ dot.add_edge(pydot.Edge(edge[0], edge[1]))
		79	+ # defina o layout do grafo
		80	+ pos = graphviz_layout(G, prog='dot')
		81	+ # renderize o grafo com o pydot
		82	+ graph = pydot.graph_from_dot_data(dot.to_string())[0]
		83	+ graph.write_png(target_filename)
		84	+
		85	+if __name__ == "__main__":
		86	+ summary = summary_dag("asia2.bif")
		87	+ message = summary_str(summary)
		88	+ print(message)
0	\ No newline at end of file	89	\ No newline at end of file
@@ -0,0 +1,38 @@		@@ -0,0 +1,38 @@
	1	+from pgmpy.readwrite import BIFReader
	2	+import networkx as nx
	3	+import pydot
	4	+from networkx.drawing.nx_pydot import graphviz_layout
	5	+from bninput import *
	6	+
	7	+def get_yes_probabilities(model):
	8	+ nodes = model.nodes()
	9	+
	10	+ yes_probabilities = {}
	11	+
	12	+ for node in nodes:
	13	+ cpd = model.get_cpds(node)
	14	+ state_names = cpd.state_names[node]
	15	+ yes_index = state_names.index("yes")
	16	+ yes_prob = cpd.values[yes_index]
	17	+ yes_probabilities[node] = yes_prob
	18	+
	19	+ return yes_probabilities
	20	+
	21	+if __name__ == "__main__":
	22	+ summary = summary_dag("asia2.bif")
	23	+ model = summary["bnmodel"]
	24	+ probabilities = get_yes_probabilities(model)
	25	+ for node, yes_prob in probabilities.items():
	26	+ parents = model.get_parents(node)
	27	+ s = ""
	28	+ if len(parents) == 0:
	29	+ s += str(yes_prob) + " :: " + str(node)
	30	+ print(s)
	31	+ elif len(parents) == 1:
	32	+ s += str(yes_prob[0]) + " :: " + str(node) + " <- " + str(parents) + "\n"
	33	+ s += str(yes_prob[1]) + " :: " + str(node) + " <- -" + str(parents) + "\n"
	34	+ print(s)
	35	+ else:
	36	+ print("Node:", node)
	37	+ print("Probability of 'yes':", yes_prob)
	38	+ print()
		1	+@book {Judea88,
		2	+ AUTHOR = {Pearl, Judea},
		3	+ TITLE = {Probabilistic reasoning in intelligent systems: networks of
		4	+ plausible inference},
		5	+ SERIES = {The Morgan Kaufmann Series in Representation and Reasoning},
		6	+ PUBLISHER = {Morgan Kaufmann, San Mateo, CA},
		7	+ YEAR = {1988},
		8	+ PAGES = {xx+552},
		9	+ ISBN = {0-934613-73-7},
		10	+ MRCLASS = {68-02 (68T01 92A25)},
		11	+ MRNUMBER = {965765},
		12	+MRREVIEWER = {V. Yu. Trakhtman},
		13	+}
		14	+
		15	+ @article{sympy,
		16	+ title = {SymPy: symbolic computing in Python},
		17	+ author = {Meurer, Aaron and Smith, Christopher P. and Paprocki, Mateusz and \v{C}ert\'{i}k, Ond\v{r}ej and Kirpichev, Sergey B. and Rocklin, Matthew and Kumar, AMiT and Ivanov, Sergiu and Moore, Jason K. and Singh, Sartaj and Rathnayake, Thilina and Vig, Sean and Granger, Brian E. and Muller, Richard P. and Bonazzi, Francesco and Gupta, Harsh and Vats, Shivam and Johansson, Fredrik and Pedregosa, Fabian and Curry, Matthew J. and Terrel, Andy R. and Rou\v{c}ka, \v{S}t\v{e}p\'{a}n and Saboo, Ashutosh and Fernando, Isuru and Kulal, Sumith and Cimrman, Robert and Scopatz, Anthony},
		18	+ year = 2017,
		19	+ month = jan,
		20	+ keywords = {Python, Computer algebra system, Symbolics},
		21	+ abstract = {
		22	+ SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.
		23	+ },
		24	+ volume = 3,
		25	+ pages = {e103},
		26	+ journal = {PeerJ Computer Science},
		27	+ issn = {2376-5992},
		28	+ url = {https://doi.org/10.7717/peerj-cs.103},
		29	+ doi = {10.7717/peerj-cs.103}
		30	+ }
		31	+
1	@inproceedings{verreet2022inference,	32	@inproceedings{verreet2022inference,
2	title={Inference and learning with model uncertainty in probabilistic logic programs},	33	title={Inference and learning with model uncertainty in probabilistic logic programs},
3	author={Verreet, Victor and Derkinderen, Vincent and Dos Martires, Pedro Zuidberg and De Raedt, Luc},	34	author={Verreet, Victor and Derkinderen, Vincent and Dos Martires, Pedro Zuidberg and De Raedt, Luc},
	@@ -46,3 +77,20 @@		@@ -46,3 +77,20 @@
46	year={2012},	77	year={2012},
47	publisher={Morgan \& Claypool Publishers}	78	publisher={Morgan \& Claypool Publishers}
48	}	79	}
		80	+
		81	+@article{10.7717/peerj-cs.103,
		82	+ title = {SymPy: symbolic computing in Python},
		83	+ author = {Meurer, Aaron and Smith, Christopher P. and Paprocki, Mateusz and \v{C}ert\'{i}k, Ond\v{r}ej and Kirpichev, Sergey B. and Rocklin, Matthew and Kumar, AMiT and Ivanov, Sergiu and Moore, Jason K. and Singh, Sartaj and Rathnayake, Thilina and Vig, Sean and Granger, Brian E. and Muller, Richard P. and Bonazzi, Francesco and Gupta, Harsh and Vats, Shivam and Johansson, Fredrik and Pedregosa, Fabian and Curry, Matthew J. and Terrel, Andy R. and Rou\v{c}ka, \v{S}t\v{e}p\'{a}n and Saboo, Ashutosh and Fernando, Isuru and Kulal, Sumith and Cimrman, Robert and Scopatz, Anthony},
		84	+ year = 2017,
		85	+ month = jan,
		86	+ keywords = {Python, Computer algebra system, Symbolics},
		87	+ abstract = {
		88	+ SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.
		89	+ },
		90	+ volume = 3,
		91	+ pages = {e103},
		92	+ journal = {PeerJ Computer Science},
		93	+ issn = {2376-5992},
		94	+ url = {https://doi.org/10.7717/peerj-cs.103},
		95	+ doi = {10.7717/peerj-cs.103}
		96	+ }