01-k2_algorithm.ipynb

{
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   "cell_type": "markdown",
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   "source": [
    "# **K2 algorithm** | implementation\n",
    "\n",
    "Project for *Advanced Statistics for Physics Analysis*, University of Padua, A.Y. 2021/22<br>\n",
    "**Learning the topology of a Bayesian Network from a database of cases using the K2 algorithm**\n",
    "\n",
    "**Students** | Francesco Pio Barone, Gianmarco Nagaro Quiroz, Daniele Ninni\n",
    "\n",
    "***\n",
    "1) [**Introduction**](#sec1)\n",
    "2) [**Implementation of the K2 algorithm in R**](#sec2)\n",
    "    - 2.A [function $\\texttt{k2}$](#sec2.A)\n",
    "    - 2.B [function $\\texttt{f}$](#sec2.B)\n",
    "    - 2.C [function $\\texttt{log.f}$](#sec2.C)\n",
    "    - 2.D [function $\\texttt{k2.iter}$](#sec2.D)\n",
    "3) [**Test of the implementation**](#sec3)\n",
    "    - 3.A [dummy dataset](#sec3.A)\n",
    "    - 3.B [*ASIA* dataset](#sec3.B)\n",
    "    - 3.C [*SACHS* dataset](#sec3.C)\n",
    "    - 3.D [*CHILD* dataset](#sec3.D)\n",
    "***"
   ]
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   "source": [
    "# load packages\n",
    "library(tidyverse) |> suppressPackageStartupMessages()\n",
    "library(bnlearn)   |> suppressPackageStartupMessages()\n",
    "library(Rgraphviz) |> suppressPackageStartupMessages()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec1'></a>\n",
    "## **1** - Introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bayesian networks\n",
    "\n",
    "A **Bayesian network** (also known as a **belief network**) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG).\n",
    "\n",
    "Such DAGs consist of:\n",
    "\n",
    "- $\\color{red}{\\text{nodes}}\\leftrightarrow$ variables in the Bayesian sense (observable quantities, latent variables, unknown parameters or hypotheses)\n",
    "    - unconnected nodes $\\leftrightarrow$ conditionally independent variables\n",
    "    - each node $\\leftrightarrow$ probability function (INPUT: a particular set of values for the node's parent variables, OUTPUT: probability of the variable represented by the node)\n",
    "- $\\color{blue}{\\text{edges}}\\leftrightarrow$ conditional dependencies\n",
    "\n",
    "Given a database of records, it is interesting to construct a Bayesian network which can provide insights into the probabilistic dependencies existing among the variables in the database itself.\n",
    "Such network can be further used to predict the future behaviour of the system thus modeled.\n",
    "\n",
    "### Topology learning\n",
    "\n",
    "In the simplest case, a Bayesian network has a small number of nodes and therefore can be specified manually by an expert.\n",
    "Even in the case of a large number of nodes, an expert could be able to assess the network structure, if his knowledge of the system is sufficient.\n",
    "In general, it is crucial to learn the network topology following a data-driven approach.\n",
    "\n",
    "Automatically learning the graph structure of a Bayesian network is a challenge pursued within machine learning.\n",
    "For instance, $\\color{green}{\\text{score-based}}$ algorithms use optimization-based search according to:\n",
    "\n",
    "- a scoring function\n",
    "- a search strategy\n",
    "\n",
    "Such score-based algorithms focus on the DAG as a whole and approach the topology learning task as an optimization problem:\n",
    "\n",
    "1) each candidate DAG is assigned a score which represents a statistical measurement of how well the DAG itself mirrors the dependence structure of the data\n",
    "2) the output DAG is the one corresponding to the highest score\n",
    "\n",
    "### K2 algorithm\n",
    "\n",
    "The K2 algorithm is a score-based algorithm introduced by G. F. Cooper and E. Herskovits in 1992.\n",
    "It allows to learn the topology of a Bayesian network, i.e. to find the most probable belief network structure, given a database of cases.\n",
    "\n",
    "The algorithm holds on the following assumptions:\n",
    "\n",
    "- there is an ordering on the domain variables\n",
    "- a priori, all structures are considered equally likely\n",
    "\n",
    "The algorithm is based on the use of a greedy-search method. In particular, for each node:\n",
    "\n",
    "1) it begins by making the assumption that the current node has no parents\n",
    "2) it adds incrementally that parent whose addition most increases the probability of the resulting structure\n",
    "3) when the addition of no single parent can increase the probability, it stops adding parents to the current node and moves on to the next node"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec2'></a>\n",
    "## **2** - Implementation of the K2 algorithm in R"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**INPUT**:\n",
    "\n",
    "- a set of $n$ nodes\n",
    "- an ordering on the nodes\n",
    "- an upper bound $u$ on the number of parents a node may have\n",
    "- a database $D$ containing $m$ cases\n",
    "\n",
    "**OUTPUT**:\n",
    "\n",
    "- for each node, a printout of the parents of the node\n",
    "\n",
    "**PSEUDOCODE**:\n",
    "\n",
    "```\n",
    "for i := 1 to n do\n",
    "\n",
    "    π_i     := ∅\n",
    "    P_old   := f(i, π_i)\n",
    "    proceed := true\n",
    "\n",
    "    while proceed and |π_i| < u do\n",
    "\n",
    "        z     := node in Pred(x_i) - π_i that maximizes f(i, π_i ∪ {z})\n",
    "        P_new := f(i, π_i ∪ {z})\n",
    "\n",
    "        if P_new > P_old then\n",
    "\n",
    "            P_old := P_new\n",
    "            π_i   := π_i ∪ {z}\n",
    "\n",
    "        else proceed := false\n",
    "\n",
    "    end {while}\n",
    "\n",
    "    write('Node:', x_i, 'Parents of this node:', π_i)\n",
    "    \n",
    "end {for}\n",
    "```"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec2.A'></a>\n",
    "### **2.A** function $\\texttt{k2}$"
   ]
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    "k2 <- function(dataset, parents.nmax, f, ...) {\n",
    "\n",
    "        nodes   <- names(dataset)           # node names\n",
    "        net.dag <- empty.graph(nodes=nodes) # network DAG (Directed Acyclic Graph)\n",
    "\n",
    "        for (i in 2:length(nodes)) {\n",
    "\n",
    "            node           <- nodes[i]                  # current node\n",
    "            previous.nodes <- nodes[1:(i-1)]            # nodes that precede the current node\n",
    "            parents        <- c()                       # parents of the current node\n",
    "            P_old          <- f(node, parents, dataset) # old probability\n",
    "            proceed        <- T\n",
    "            \n",
    "            while (proceed & (length(parents) < parents.nmax)) {\n",
    "\n",
    "                candidates <- setdiff(previous.nodes, parents) # candidate parents of the current node \n",
    "                P_new      <- P_old                            # new probability\n",
    "\n",
    "                for (candidate in candidates) {\n",
    "\n",
    "                    candidate.score <- f(node, c(parents, candidate), dataset) # candidate parent score\n",
    "\n",
    "                    if (candidate.score > P_new) {\n",
    "\n",
    "                        candidates.best <- candidate       # best candidate parent\n",
    "                        P_new           <- candidate.score\n",
    "\n",
    "                    }\n",
    "\n",
    "                }\n",
    "\n",
    "                if (P_new > P_old) {\n",
    "\n",
    "                    P_old   <- P_new\n",
    "                    parents <- c(parents, candidates.best)\n",
    "                    net.dag <- set.arc(net.dag, from=candidates.best, to=node)\n",
    "\n",
    "                } else {\n",
    "\n",
    "                    proceed <- F\n",
    "\n",
    "                }\n",
    "\n",
    "            }\n",
    "\n",
    "        }\n",
    "\n",
    "        net.score <- score(net.dag, dataset |> mutate_all(as_factor) |> as.data.frame(), ...) # network score\n",
    "\n",
    "        return(list('dag'=net.dag, 'score'=net.score))\n",
    "\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec2.B'></a>\n",
    "### **2.B** function $\\texttt{f}$"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The scoring function $f(i, \\pi_i)$ is the probability of the database $D$ given that the set of parents of the variable $x_i$ is represented by the list of variables $\\pi_i$.\n",
    "\n",
    "$$\n",
    "f(i, \\pi_i) = \\prod_{j=1}^{q_i} \\frac{(r_i - 1)! \\prod_{k=1}^{r_i} \\alpha_{ijk}!}{(N_{ij} + r_i - 1)!} = \\prod_{j=1}^{q_i} f_j\n",
    "$$\n",
    "\n",
    "where:\n",
    "\n",
    "- $V_i =$ list of unique instantiations of $x_i$ relative to $D$\n",
    "- $r_i = |V_i|$\n",
    "- $\\phi_i =$ list of unique instantiations of $\\pi_i$ relative to $D$\n",
    "- $q_i = |\\phi_i|$\n",
    "- $\\alpha_{ijk} =$ # cases in $D$ in which $x_i$ is instantiated as the $k$-th element in $V_i$ and $\\pi_i$ is instantiated as the $j$-th element in $\\phi_i$\n",
    "- $N_{ij} = \\sum_{k=1}^{r_i} \\alpha_{ijk} =$ # cases in $D$ in which $\\pi_i$ is instantiated as the $j$-th element in $\\phi_i$"
   ]
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    "f <- function(node, parents, dataset) {\n",
    "\n",
    "    node.nunique <- dataset[node] |>\n",
    "                        unique() |>\n",
    "                        nrow()\n",
    "\n",
    "    contingency_table <- dataset[c(node, parents)] |>\n",
    "                            table() |>\n",
    "                            as.data.frame() |>\n",
    "                            as_tibble() |>\n",
    "                            filter(Freq > 0)\n",
    "\n",
    "    numerator <- contingency_table |>\n",
    "                    mutate(Freq = factorial(Freq)) |>\n",
    "                    group_by_at(parents) |>\n",
    "                    summarise(Freq = prod(Freq), .groups='drop_last') |>\n",
    "                    mutate(Freq = (Freq * factorial(node.nunique - 1))) |>\n",
    "                    ungroup() |>\n",
    "                    select(Freq) |>\n",
    "                    deframe()\n",
    "\n",
    "    denominator <- contingency_table |>\n",
    "                    group_by_at(parents) |>\n",
    "                    summarise(Freq = sum(Freq), .groups='drop_last') |>\n",
    "                    mutate(Freq = factorial(Freq + node.nunique - 1)) |>\n",
    "                    ungroup() |>\n",
    "                    select(Freq) |>\n",
    "                    deframe()\n",
    "\n",
    "    dataset.prob <- prod(numerator / denominator)\n",
    "\n",
    "    return(dataset.prob)\n",
    "\n",
    "}"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec2.C'></a>\n",
    "### **2.C** function $\\texttt{log.f}$"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is possible to improve the run-time speed of the K2 algorithm by replacing $f$ with $\\log(f)$.\n",
    "In fact, the logarithmic version of the scoring function requires only addition and subtraction, rather than multiplication and division.\n",
    "\n",
    "Note that:\n",
    "\n",
    "$$\n",
    "\\log(n!) = \\log \\left( \\prod_{x=1}^n x \\right) = \\sum_{x=1}^n \\log(x)\n",
    "$$\n",
    "\n",
    "Therefore:\n",
    "\n",
    "$$\n",
    "\\log(f) = \\log \\left( \\prod_{j=1}^{q_i} f_j \\right) = \\sum_{j=1}^{q_i} \\log(f_j)\n",
    "$$\n",
    "\n",
    "where:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\log(f_j) &= \\log \\left[ \\frac{(r_i - 1)! \\prod_{k=1}^{r_i} \\alpha_{ijk}!}{(N_{ij} + r_i - 1)!} \\right] = \\newline\n",
    "&= \\log \\left[ \\left( r_i - 1 \\right)! \\right] + \\log \\left[ \\prod_{k=1}^{r_i} \\alpha_{ijk}! \\right] - \\log \\left[ \\left( N_{ij} + r_i - 1 \\right)! \\right] = \\newline\n",
    "&= \\color{red}{\\sum_{x=1}^{r_i - 1} \\log(x)} + \\color{green}{\\sum_{k=1}^{r_i} \\sum_{y=1}^{\\alpha_{ijk}} \\log(y)} - \\color{blue}{\\sum_{z=1}^{N_{ij} + r_i - 1} \\log(z)} = \\newline\n",
    "&= \\color{red}{A} + \\color{green}{B} - \\color{blue}{C}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "The use of $\\log(f)$ as scoring function is particularly effective in the case of very large datasets, since it allows to avoid the computation of huge factorials."
   ]
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   "source": [
    "log.f <- function(node, parents, dataset) {\n",
    "\n",
    "    node.nunique <- dataset[node] |>\n",
    "                        unique() |>\n",
    "                        nrow()\n",
    "\n",
    "    contingency_table <- dataset[c(node, parents)] |>\n",
    "                            table() |>\n",
    "                            as.data.frame() |>\n",
    "                            as_tibble() |>\n",
    "                            filter(Freq > 0)\n",
    "\n",
    "    A <- 1:(node.nunique - 1) |>\n",
    "            log() |>\n",
    "            sum()\n",
    "\n",
    "    B <- contingency_table |>\n",
    "            rowwise() |>\n",
    "            mutate(Freq = ((1:Freq) |> log() |> sum())) |>\n",
    "            ungroup() |>\n",
    "            group_by_at(parents) |>\n",
    "            summarise(Freq = sum(Freq), .groups='drop_last') |>\n",
    "            ungroup() |>\n",
    "            select(Freq) |>\n",
    "            deframe()\n",
    "\n",
    "    C <- contingency_table |>\n",
    "            group_by_at(parents) |>\n",
    "            summarise(Freq = sum(Freq), .groups='drop_last') |>\n",
    "            ungroup() |>\n",
    "            rowwise() |>\n",
    "            mutate(Freq = ((1:(Freq + node.nunique - 1)) |> log() |> sum())) |>\n",
    "            ungroup() |>\n",
    "            select(Freq) |>\n",
    "            deframe()\n",
    "\n",
    "    dataset.log.prob <- sum(A + B - C)\n",
    "    \n",
    "    return(dataset.log.prob)\n",
    "\n",
    "}"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec2.D'></a>\n",
    "### **2.D** function $\\texttt{k2.iter}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The K2 algorithm requires the knowledge of the true node ordering.\n",
    "However, there are several cases in which such information is not known a priori.\n",
    "\n",
    "In these cases, a possible solution is to use the K2 algorithm iteratively, following a *brute-force* approach.\n",
    "In other words, we could perform an $\\color{red}{\\text{exhaustive}}$ search, i.e. we could test all the possible node orderings and then keep the one that leads to the highest score.\n",
    "\n",
    "The main disadvantage of exhaustive search is that it increases the computation time by a factor $n!$, where $n$ is the number of nodes.\n",
    "Therefore, in order to deal with a large number of nodes, a different approach is needed.\n",
    "\n",
    "To mitigate this problem, we propose a $\\color{green}{\\text{randomized}}$ version of the K2 algorithm.\n",
    "In particular:\n",
    "\n",
    "1) instead of testing all the possible node orderings, the number of iterations to perform `n.iter` is fixed a priori\n",
    "2) for each iteration, a random node ordering is sampled and then the corresponding optimal DAG is found\n",
    "3) among all the `n.iter` DAGs found, the one that achieves the highest score is returned as output"
   ]
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   "execution_count": 5,
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   "source": [
    "k2.iter <- function(dataset, parents.nmax, f, k2, n.iter=1, seed=1, ...) {\n",
    "\n",
    "    tic <- Sys.time()\n",
    "    set.seed(seed)\n",
    "\n",
    "    n.iter     <- max(n.iter, 1)\n",
    "    best.score <- -Inf\n",
    "\n",
    "    for (i in 1:n.iter) {\n",
    "\n",
    "        cat('Running iteration #', i, '...')\n",
    "        flush.console()\n",
    "\n",
    "        nodes.order <- if (i == 1) names(dataset) else sample(names(dataset))\n",
    "        result      <- k2(dataset[nodes.order], parents.nmax, f, ...)\n",
    "\n",
    "        if (result$score > best.score) {\n",
    "\n",
    "            best.dag   <- result$dag\n",
    "            best.score <- result$score\n",
    "            best.order <- nodes.order\n",
    "\n",
    "        }\n",
    "\n",
    "        cat(' DONE\\n')\n",
    "        flush.console()\n",
    "\n",
    "    }\n",
    "\n",
    "    toc <- Sys.time()\n",
    "    cat('\\nTotal execution time:', difftime(toc, tic, units='secs'), 's')\n",
    "\n",
    "    return(list('dag'=best.dag, 'score'=best.score, 'order'=best.order))\n",
    "\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec3'></a>\n",
    "## **3** - Test of the implementation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec3.A'></a>\n",
    "### **3.A** dummy dataset [[2]](#ref2)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 6,
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    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A tibble: 10 × 3</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>x1</th><th scope=col>x2</th><th scope=col>x3</th></tr>\n",
       "\t<tr><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td>1</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>1</td><td>1</td><td>1</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>1</td></tr>\n",
       "\t<tr><td>1</td><td>1</td><td>1</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>1</td><td>1</td></tr>\n",
       "\t<tr><td>1</td><td>1</td><td>1</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>1</td><td>1</td><td>1</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A tibble: 10 × 3\n",
       "\\begin{tabular}{lll}\n",
       " x1 & x2 & x3\\\\\n",
       " <dbl> & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\t 1 & 0 & 0\\\\\n",
       "\t 1 & 1 & 1\\\\\n",
       "\t 0 & 0 & 1\\\\\n",
       "\t 1 & 1 & 1\\\\\n",
       "\t 0 & 0 & 0\\\\\n",
       "\t 0 & 1 & 1\\\\\n",
       "\t 1 & 1 & 1\\\\\n",
       "\t 0 & 0 & 0\\\\\n",
       "\t 1 & 1 & 1\\\\\n",
       "\t 0 & 0 & 0\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A tibble: 10 × 3\n",
       "\n",
       "| x1 &lt;dbl&gt; | x2 &lt;dbl&gt; | x3 &lt;dbl&gt; |\n",
       "|---|---|---|\n",
       "| 1 | 0 | 0 |\n",
       "| 1 | 1 | 1 |\n",
       "| 0 | 0 | 1 |\n",
       "| 1 | 1 | 1 |\n",
       "| 0 | 0 | 0 |\n",
       "| 0 | 1 | 1 |\n",
       "| 1 | 1 | 1 |\n",
       "| 0 | 0 | 0 |\n",
       "| 1 | 1 | 1 |\n",
       "| 0 | 0 | 0 |\n",
       "\n"
      ],
      "text/plain": [
       "   x1 x2 x3\n",
       "1  1  0  0 \n",
       "2  1  1  1 \n",
       "3  0  0  1 \n",
       "4  1  1  1 \n",
       "5  0  0  0 \n",
       "6  0  1  1 \n",
       "7  1  1  1 \n",
       "8  0  0  0 \n",
       "9  1  1  1 \n",
       "10 0  0  0 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dummy <- tibble(\n",
    "            x1 = c(1, 1, 0, 1, 0, 0, 1, 0, 1, 0),\n",
    "            x2 = c(0, 1, 0, 1, 0, 1, 1, 0, 1, 0),\n",
    "            x3 = c(0, 1, 1, 1, 0, 1, 1, 0, 1, 0))\n",
    "\n",
    "dummy"
   ]
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  {
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   "execution_count": 7,
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running iteration # 1 ... DONE\n",
      "\n",
      "Total execution time: 0.2824841 s"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "  Random/Generated Bayesian network\n",
       "\n",
       "  model:\n",
       "   [x1][x2|x1][x3|x2] \n",
       "  nodes:                                 3 \n",
       "  arcs:                                  2 \n",
       "    undirected arcs:                     0 \n",
       "    directed arcs:                       2 \n",
       "  average markov blanket size:           1.33 \n",
       "  average neighbourhood size:            1.33 \n",
       "  average branching factor:              0.67 \n",
       "\n",
       "  generation algorithm:                  Empty \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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173+dsr1Cxj9FX2k5/fjfTnD5vvkH2x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\nRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dummy_net <- k2.iter(dataset=dummy, parents.nmax=2, f=f, k2=k2, n.iter=1, seed=1, type='bic')\n",
    "\n",
    "dummy_net$dag\n",
    "graphviz.plot(dummy_net$dag)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running iteration # 1 ... DONE\n",
      "\n",
      "Total execution time: 0.2401311 s"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "  Random/Generated Bayesian network\n",
       "\n",
       "  model:\n",
       "   [x1][x2|x1][x3|x2] \n",
       "  nodes:                                 3 \n",
       "  arcs:                                  2 \n",
       "    undirected arcs:                     0 \n",
       "    directed arcs:                       2 \n",
       "  average markov blanket size:           1.33 \n",
       "  average neighbourhood size:            1.33 \n",
       "  average branching factor:              0.67 \n",
       "\n",
       "  generation algorithm:                  Empty \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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173+dsr1Cxj9FX2k5/fjfTnD5vvkH2x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\nRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "dummy_net <- k2.iter(dataset=dummy, parents.nmax=2, f=log.f, k2=k2, n.iter=1, seed=1, type='bic')\n",
    "\n",
    "dummy_net$dag\n",
    "graphviz.plot(dummy_net$dag)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec3.B'></a>\n",
    "### **3.B** *ASIA* dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[1mRows: \u001b[22m\u001b[34m10000\u001b[39m \u001b[1mColumns: \u001b[22m\u001b[34m8\u001b[39m\n",
      "\u001b[36m--\u001b[39m \u001b[1mColumn specification\u001b[22m \u001b[36m------------------------------------------------------------------------------------------------\u001b[39m\n",
      "\u001b[1mDelimiter:\u001b[22m \",\"\n",
      "\u001b[31mchr\u001b[39m (8): asia, tub, smoke, lung, bronc, either, xray, dysp\n",
      "\n",
      "\u001b[36mi\u001b[39m Use `spec()` to retrieve the full column specification for this data.\n",
      "\u001b[36mi\u001b[39m Specify the column types or set `show_col_types = FALSE` to quiet this message.\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A tibble: 6 × 8</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>asia</th><th scope=col>tub</th><th scope=col>smoke</th><th scope=col>lung</th><th scope=col>bronc</th><th scope=col>either</th><th scope=col>xray</th><th scope=col>dysp</th></tr>\n",
       "\t<tr><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td>no</td><td>no</td><td>no </td><td>no</td><td>no </td><td>no</td><td>no</td><td>yes</td></tr>\n",
       "\t<tr><td>no</td><td>no</td><td>yes</td><td>no</td><td>no </td><td>no</td><td>no</td><td>no </td></tr>\n",
       "\t<tr><td>no</td><td>no</td><td>no </td><td>no</td><td>no </td><td>no</td><td>no</td><td>no </td></tr>\n",
       "\t<tr><td>no</td><td>no</td><td>yes</td><td>no</td><td>no </td><td>no</td><td>no</td><td>no </td></tr>\n",
       "\t<tr><td>no</td><td>no</td><td>yes</td><td>no</td><td>yes</td><td>no</td><td>no</td><td>yes</td></tr>\n",
       "\t<tr><td>no</td><td>no</td><td>yes</td><td>no</td><td>no </td><td>no</td><td>no</td><td>no </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A tibble: 6 × 8\n",
       "\\begin{tabular}{llllllll}\n",
       " asia & tub & smoke & lung & bronc & either & xray & dysp\\\\\n",
       " <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr>\\\\\n",
       "\\hline\n",
       "\t no & no & no  & no & no  & no & no & yes\\\\\n",
       "\t no & no & yes & no & no  & no & no & no \\\\\n",
       "\t no & no & no  & no & no  & no & no & no \\\\\n",
       "\t no & no & yes & no & no  & no & no & no \\\\\n",
       "\t no & no & yes & no & yes & no & no & yes\\\\\n",
       "\t no & no & yes & no & no  & no & no & no \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A tibble: 6 × 8\n",
       "\n",
       "| asia &lt;chr&gt; | tub &lt;chr&gt; | smoke &lt;chr&gt; | lung &lt;chr&gt; | bronc &lt;chr&gt; | either &lt;chr&gt; | xray &lt;chr&gt; | dysp &lt;chr&gt; |\n",
       "|---|---|---|---|---|---|---|---|\n",
       "| no | no | no  | no | no  | no | no | yes |\n",
       "| no | no | yes | no | no  | no | no | no  |\n",
       "| no | no | no  | no | no  | no | no | no  |\n",
       "| no | no | yes | no | no  | no | no | no  |\n",
       "| no | no | yes | no | yes | no | no | yes |\n",
       "| no | no | yes | no | no  | no | no | no  |\n",
       "\n"
      ],
      "text/plain": [
       "  asia tub smoke lung bronc either xray dysp\n",
       "1 no   no  no    no   no    no     no   yes \n",
       "2 no   no  yes   no   no    no     no   no  \n",
       "3 no   no  no    no   no    no     no   no  \n",
       "4 no   no  yes   no   no    no     no   no  \n",
       "5 no   no  yes   no   yes   no     no   yes \n",
       "6 no   no  yes   no   no    no     no   no  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "asia <- read_csv('https://www.ccd.pitt.edu/wiki/images/ASIA10k.csv')\n",
    "\n",
    "head(asia)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running iteration # 1 ... DONE\n",
      "Running iteration # 2 ... DONE\n",
      "Running iteration # 3 ... DONE\n",
      "Running iteration # 4 ... DONE\n",
      "Running iteration # 5 ... DONE\n",
      "\n",
      "Total execution time: 11.18496 s\n",
      "\n",
      "Best node ordering found: either tub asia bronc dysp lung smoke xray"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "  Random/Generated Bayesian network\n",
       "\n",
       "  model:\n",
       "   [either][tub|either][xray|either][asia|tub][lung|either:tub][bronc|asia]\n",
       "   [dysp|either:bronc][smoke|either:bronc]\n",
       "  nodes:                                 8 \n",
       "  arcs:                                  10 \n",
       "    undirected arcs:                     0 \n",
       "    directed arcs:                       10 \n",
       "  average markov blanket size:           2.75 \n",
       "  average neighbourhood size:            2.50 \n",
       "  average branching factor:              1.25 \n",
       "\n",
       "  generation algorithm:                  Empty \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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VKY3CBSdt78SZAihckNImUHkSyCSPl5EShFiBQmN4iUH0QyCCIV4MmgJBlSmNwg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\no+2/RgeOZgAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "asia_net <- k2.iter(dataset=head(asia, 150), parents.nmax=2, f=f, k2=k2, n.iter=5, seed=1, type='bde', iss=10)\n",
    "\n",
    "cat('\\n\\nBest node ordering found:', asia_net$order)\n",
    "asia_net$dag\n",
    "graphviz.plot(asia_net$dag)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running iteration # 1 ... DONE\n",
      "Running iteration # 2 ... DONE\n",
      "Running iteration # 3 ... DONE\n",
      "Running iteration # 4 ... DONE\n",
      "Running iteration # 5 ... DONE\n",
      "\n",
      "Total execution time: 11.2983 s\n",
      "\n",
      "Best node ordering found: asia tub smoke lung bronc either xray dysp"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "  Random/Generated Bayesian network\n",
       "\n",
       "  model:\n",
       "   [asia][tub][smoke][lung|smoke][bronc|smoke][either|tub:lung][xray|either]\n",
       "   [dysp|bronc:either]\n",
       "  nodes:                                 8 \n",
       "  arcs:                                  7 \n",
       "    undirected arcs:                     0 \n",
       "    directed arcs:                       7 \n",
       "  average markov blanket size:           2.25 \n",
       "  average neighbourhood size:            1.75 \n",
       "  average branching factor:              0.88 \n",
       "\n",
       "  generation algorithm:                  Empty \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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IA0SN/9efR3+dK2SNyRLp/+ukvfYdt9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\nAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "asia_net <- k2.iter(dataset=asia, parents.nmax=2, f=log.f, k2=k2, n.iter=5, seed=1, type='bde', iss=10)\n",
    "\n",
    "cat('\\n\\nBest node ordering found:', asia_net$order)\n",
    "asia_net$dag\n",
    "graphviz.plot(asia_net$dag)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec3.C'></a>\n",
    "### **3.C** *SACHS* dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[1mRows: \u001b[22m\u001b[34m10000\u001b[39m \u001b[1mColumns: \u001b[22m\u001b[34m11\u001b[39m\n",
      "\u001b[36m--\u001b[39m \u001b[1mColumn specification\u001b[22m \u001b[36m------------------------------------------------------------------------------------------------\u001b[39m\n",
      "\u001b[1mDelimiter:\u001b[22m \",\"\n",
      "\u001b[32mdbl\u001b[39m (11): Akt, Erk, Jnk, Mek, P38, PIP2, PIP3, PKA, PKC, Plcg, Raf\n",
      "\n",
      "\u001b[36mi\u001b[39m Use `spec()` to retrieve the full column specification for this data.\n",
      "\u001b[36mi\u001b[39m Specify the column types or set `show_col_types = FALSE` to quiet this message.\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A tibble: 6 × 11</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>PKC</th><th scope=col>Plcg</th><th scope=col>PKA</th><th scope=col>PIP3</th><th scope=col>Raf</th><th scope=col>Jnk</th><th scope=col>P38</th><th scope=col>PIP2</th><th scope=col>Mek</th><th scope=col>Erk</th><th scope=col>Akt</th></tr>\n",
       "\t<tr><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td>3</td><td>1</td><td>2</td><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr>\n",
       "\t<tr><td>2</td><td>1</td><td>2</td><td>2</td><td>1</td><td>1</td><td>2</td><td>1</td><td>1</td><td>2</td><td>1</td></tr>\n",
       "\t<tr><td>1</td><td>3</td><td>1</td><td>2</td><td>3</td><td>3</td><td>3</td><td>2</td><td>1</td><td>1</td><td>1</td></tr>\n",
       "\t<tr><td>1</td><td>1</td><td>3</td><td>3</td><td>2</td><td>1</td><td>1</td><td>1</td><td>1</td><td>2</td><td>1</td></tr>\n",
       "\t<tr><td>2</td><td>2</td><td>2</td><td>3</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>2</td><td>1</td></tr>\n",
       "\t<tr><td>2</td><td>1</td><td>2</td><td>1</td><td>2</td><td>2</td><td>1</td><td>1</td><td>2</td><td>2</td><td>1</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A tibble: 6 × 11\n",
       "\\begin{tabular}{lllllllllll}\n",
       " PKC & Plcg & PKA & PIP3 & Raf & Jnk & P38 & PIP2 & Mek & Erk & Akt\\\\\n",
       " <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl>\\\\\n",
       "\\hline\n",
       "\t 3 & 1 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\\\\n",
       "\t 2 & 1 & 2 & 2 & 1 & 1 & 2 & 1 & 1 & 2 & 1\\\\\n",
       "\t 1 & 3 & 1 & 2 & 3 & 3 & 3 & 2 & 1 & 1 & 1\\\\\n",
       "\t 1 & 1 & 3 & 3 & 2 & 1 & 1 & 1 & 1 & 2 & 1\\\\\n",
       "\t 2 & 2 & 2 & 3 & 1 & 1 & 1 & 1 & 1 & 2 & 1\\\\\n",
       "\t 2 & 1 & 2 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A tibble: 6 × 11\n",
       "\n",
       "| PKC &lt;dbl&gt; | Plcg &lt;dbl&gt; | PKA &lt;dbl&gt; | PIP3 &lt;dbl&gt; | Raf &lt;dbl&gt; | Jnk &lt;dbl&gt; | P38 &lt;dbl&gt; | PIP2 &lt;dbl&gt; | Mek &lt;dbl&gt; | Erk &lt;dbl&gt; | Akt &lt;dbl&gt; |\n",
       "|---|---|---|---|---|---|---|---|---|---|---|\n",
       "| 3 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |\n",
       "| 2 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 |\n",
       "| 1 | 3 | 1 | 2 | 3 | 3 | 3 | 2 | 1 | 1 | 1 |\n",
       "| 1 | 1 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 2 | 1 |\n",
       "| 2 | 2 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 2 | 1 |\n",
       "| 2 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 |\n",
       "\n"
      ],
      "text/plain": [
       "  PKC Plcg PKA PIP3 Raf Jnk P38 PIP2 Mek Erk Akt\n",
       "1 3   1    2   2    1   1   1   1    1   1   1  \n",
       "2 2   1    2   2    1   1   2   1    1   2   1  \n",
       "3 1   3    1   2    3   3   3   2    1   1   1  \n",
       "4 1   1    3   3    2   1   1   1    1   2   1  \n",
       "5 2   2    2   3    1   1   1   1    1   2   1  \n",
       "6 2   1    2   1    2   2   1   1    2   2   1  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sachs <- read_csv('https://www.ccd.pitt.edu/wiki/images/SACHS10k.csv')\n",
    "sachs <- sachs |> select(PKC, Plcg, PKA, PIP3, Raf, Jnk, P38, PIP2, Mek, Erk, Akt)\n",
    "\n",
    "head(sachs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running iteration # 1 ... DONE\n",
      "Running iteration # 2 ... DONE\n",
      "Running iteration # 3 ... DONE\n",
      "Running iteration # 4 ... DONE\n",
      "Running iteration # 5 ... DONE\n",
      "\n",
      "Total execution time: 26.78662 s\n",
      "\n",
      "Best node ordering found: PKC Plcg PKA PIP3 Raf Jnk P38 PIP2 Mek Erk Akt"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "  Random/Generated Bayesian network\n",
       "\n",
       "  model:\n",
       "   [PKC][Plcg][PKA|PKC][PIP3|Plcg][PIP2|Plcg][Raf|PKC:PKA][Jnk|PKC:PKA][P38|PKA]\n",
       "   [Mek|PKA:Raf][Erk|PKA:Mek][Akt|PKA:Erk]\n",
       "  nodes:                                 11 \n",
       "  arcs:                                  14 \n",
       "    undirected arcs:                     0 \n",
       "    directed arcs:                       14 \n",
       "  average markov blanket size:           2.55 \n",
       "  average neighbourhood size:            2.55 \n",
       "  average branching factor:              1.27 \n",
       "\n",
       "  generation algorithm:                  Empty \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Rx0m78Sl7YgFfkO3lP0sB2jkSxX12Gk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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "sachs_net <- k2.iter(dataset=head(sachs, 150), parents.nmax=3, f=f, k2=k2, n.iter=5, seed=1, type='bde', iss=10)\n",
    "\n",
    "cat('\\n\\nBest node ordering found:', sachs_net$order)\n",
    "sachs_net$dag\n",
    "graphviz.plot(sachs_net$dag)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running iteration # 1 ... DONE\n",
      "Running iteration # 2 ... DONE\n",
      "Running iteration # 3 ... DONE\n",
      "Running iteration # 4 ... DONE\n",
      "Running iteration # 5 ... DONE\n",
      "\n",
      "Total execution time: 49.16842 s\n",
      "\n",
      "Best node ordering found: PKC Plcg PKA PIP3 Raf Jnk P38 PIP2 Mek Erk Akt"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "  Random/Generated Bayesian network\n",
       "\n",
       "  model:\n",
       "   [PKC][Plcg][PKA|PKC][PIP3|Plcg][Raf|PKC:PKA][Jnk|PKC:PKA][P38|PKC:PKA]\n",
       "   [PIP2|Plcg:PIP3][Mek|PKC:PKA:Raf][Erk|PKA:Mek][Akt|PKA:Erk]\n",
       "  nodes:                                 11 \n",
       "  arcs:                                  17 \n",
       "    undirected arcs:                     0 \n",
       "    directed arcs:                       17 \n",
       "  average markov blanket size:           3.09 \n",
       "  average neighbourhood size:            3.09 \n",
       "  average branching factor:              1.55 \n",
       "\n",
       "  generation algorithm:                  Empty \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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Zg8gkhuRcyru38MOg7XgTfP7x5q2vYc3/PpWmUR6T4obRHp+/pZqX/jtt/xGIfq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\nBEAkAAiASAAQAJEAIAAiAUDAf6Xw0Rjec69nAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "sachs_net <- k2.iter(dataset=sachs, parents.nmax=3, f=log.f, k2=k2, n.iter=5, seed=1, type='bde', iss=10)\n",
    "\n",
    "cat('\\n\\nBest node ordering found:', sachs_net$order)\n",
    "sachs_net$dag\n",
    "graphviz.plot(sachs_net$dag)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec3.D'></a>\n",
    "### **3.D** *CHILD* dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[1mRows: \u001b[22m\u001b[34m10000\u001b[39m \u001b[1mColumns: \u001b[22m\u001b[34m20\u001b[39m\n",
      "\u001b[36m--\u001b[39m \u001b[1mColumn specification\u001b[22m \u001b[36m------------------------------------------------------------------------------------------------\u001b[39m\n",
      "\u001b[1mDelimiter:\u001b[22m \",\"\n",
      "\u001b[31mchr\u001b[39m (20): BirthAsphyxia, HypDistrib, HypoxiaInO2, CO2, ChestXray, Grunting, ...\n",
      "\n",
      "\u001b[36mi\u001b[39m Use `spec()` to retrieve the full column specification for this data.\n",
      "\u001b[36mi\u001b[39m Specify the column types or set `show_col_types = FALSE` to quiet this message.\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table class=\"dataframe\">\n",
       "<caption>A tibble: 6 × 20</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>BirthAsphyxia</th><th scope=col>Disease</th><th scope=col>Sick</th><th scope=col>DuctFlow</th><th scope=col>CardiacMixing</th><th scope=col>LungParench</th><th scope=col>LungFlow</th><th scope=col>LVH</th><th scope=col>Age</th><th scope=col>Grunting</th><th scope=col>HypDistrib</th><th scope=col>HypoxiaInO2</th><th scope=col>CO2</th><th scope=col>ChestXray</th><th scope=col>LVHreport</th><th scope=col>GruntingReport</th><th scope=col>LowerBodyO2</th><th scope=col>RUQO2</th><th scope=col>CO2Report</th><th scope=col>XrayReport</th></tr>\n",
       "\t<tr><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td>no</td><td>PAIVS </td><td>yes</td><td>Lt_to_Rt</td><td>Complete</td><td><span style=white-space:pre-wrap>Normal  </span></td><td><span style=white-space:pre-wrap>Low   </span></td><td>no </td><td>0-3_days </td><td>no </td><td>Equal</td><td>Moderate</td><td>Normal</td><td>Oligaemic</td><td>no </td><td>no </td><td>5-12</td><td>5-12</td><td>&gt;=7.5</td><td>Grd_Glass </td></tr>\n",
       "\t<tr><td>no</td><td><span style=white-space:pre-wrap>TGA   </span></td><td>yes</td><td><span style=white-space:pre-wrap>None    </span></td><td>Transp. </td><td>Abnormal</td><td><span style=white-space:pre-wrap>Low   </span></td><td>no </td><td>0-3_days </td><td>yes</td><td>Equal</td><td><span style=white-space:pre-wrap>Severe  </span></td><td>Normal</td><td>Asy/Patch</td><td>no </td><td>no </td><td>12+ </td><td><span style=white-space:pre-wrap>&lt;5  </span></td><td>&lt;7.5 </td><td>Asy/Patchy</td></tr>\n",
       "\t<tr><td>no</td><td><span style=white-space:pre-wrap>TGA   </span></td><td>yes</td><td><span style=white-space:pre-wrap>None    </span></td><td>Transp. </td><td><span style=white-space:pre-wrap>Normal  </span></td><td><span style=white-space:pre-wrap>High  </span></td><td>yes</td><td>0-3_days </td><td>no </td><td>Equal</td><td><span style=white-space:pre-wrap>Severe  </span></td><td>Normal</td><td><span style=white-space:pre-wrap>Normal   </span></td><td>yes</td><td>no </td><td>12+ </td><td>5-12</td><td>&lt;7.5 </td><td><span style=white-space:pre-wrap>Normal    </span></td></tr>\n",
       "\t<tr><td>no</td><td><span style=white-space:pre-wrap>Lung  </span></td><td>yes</td><td><span style=white-space:pre-wrap>None    </span></td><td><span style=white-space:pre-wrap>Mild    </span></td><td>Abnormal</td><td>Normal</td><td>no </td><td>0-3_days </td><td>yes</td><td>Equal</td><td>Moderate</td><td>Normal</td><td>Asy/Patch</td><td>no </td><td>yes</td><td>5-12</td><td>5-12</td><td>&gt;=7.5</td><td>Asy/Patchy</td></tr>\n",
       "\t<tr><td>no</td><td>Fallot</td><td>no </td><td>Lt_to_Rt</td><td>Complete</td><td><span style=white-space:pre-wrap>Normal  </span></td><td><span style=white-space:pre-wrap>Low   </span></td><td>no </td><td>4-10_days</td><td>no </td><td>Equal</td><td>Moderate</td><td>Normal</td><td><span style=white-space:pre-wrap>Normal   </span></td><td>no </td><td>no </td><td>5-12</td><td>5-12</td><td>&lt;7.5 </td><td><span style=white-space:pre-wrap>Normal    </span></td></tr>\n",
       "\t<tr><td>no</td><td>Fallot</td><td>no </td><td>Lt_to_Rt</td><td><span style=white-space:pre-wrap>Mild    </span></td><td><span style=white-space:pre-wrap>Normal  </span></td><td><span style=white-space:pre-wrap>Low   </span></td><td>no </td><td>4-10_days</td><td>no </td><td>Equal</td><td>Moderate</td><td>Normal</td><td>Oligaemic</td><td>no </td><td>yes</td><td>5-12</td><td><span style=white-space:pre-wrap>&lt;5  </span></td><td>&lt;7.5 </td><td>Oligaemic </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A tibble: 6 × 20\n",
       "\\begin{tabular}{llllllllllllllllllll}\n",
       " BirthAsphyxia & Disease & Sick & DuctFlow & CardiacMixing & LungParench & LungFlow & LVH & Age & Grunting & HypDistrib & HypoxiaInO2 & CO2 & ChestXray & LVHreport & GruntingReport & LowerBodyO2 & RUQO2 & CO2Report & XrayReport\\\\\n",
       " <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr> & <chr>\\\\\n",
       "\\hline\n",
       "\t no & PAIVS  & yes & Lt\\_to\\_Rt & Complete & Normal   & Low    & no  & 0-3\\_days  & no  & Equal & Moderate & Normal & Oligaemic & no  & no  & 5-12 & 5-12 & >=7.5 & Grd\\_Glass \\\\\n",
       "\t no & TGA    & yes & None     & Transp.  & Abnormal & Low    & no  & 0-3\\_days  & yes & Equal & Severe   & Normal & Asy/Patch & no  & no  & 12+  & <5   & <7.5  & Asy/Patchy\\\\\n",
       "\t no & TGA    & yes & None     & Transp.  & Normal   & High   & yes & 0-3\\_days  & no  & Equal & Severe   & Normal & Normal    & yes & no  & 12+  & 5-12 & <7.5  & Normal    \\\\\n",
       "\t no & Lung   & yes & None     & Mild     & Abnormal & Normal & no  & 0-3\\_days  & yes & Equal & Moderate & Normal & Asy/Patch & no  & yes & 5-12 & 5-12 & >=7.5 & Asy/Patchy\\\\\n",
       "\t no & Fallot & no  & Lt\\_to\\_Rt & Complete & Normal   & Low    & no  & 4-10\\_days & no  & Equal & Moderate & Normal & Normal    & no  & no  & 5-12 & 5-12 & <7.5  & Normal    \\\\\n",
       "\t no & Fallot & no  & Lt\\_to\\_Rt & Mild     & Normal   & Low    & no  & 4-10\\_days & no  & Equal & Moderate & Normal & Oligaemic & no  & yes & 5-12 & <5   & <7.5  & Oligaemic \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A tibble: 6 × 20\n",
       "\n",
       "| BirthAsphyxia &lt;chr&gt; | Disease &lt;chr&gt; | Sick &lt;chr&gt; | DuctFlow &lt;chr&gt; | CardiacMixing &lt;chr&gt; | LungParench &lt;chr&gt; | LungFlow &lt;chr&gt; | LVH &lt;chr&gt; | Age &lt;chr&gt; | Grunting &lt;chr&gt; | HypDistrib &lt;chr&gt; | HypoxiaInO2 &lt;chr&gt; | CO2 &lt;chr&gt; | ChestXray &lt;chr&gt; | LVHreport &lt;chr&gt; | GruntingReport &lt;chr&gt; | LowerBodyO2 &lt;chr&gt; | RUQO2 &lt;chr&gt; | CO2Report &lt;chr&gt; | XrayReport &lt;chr&gt; |\n",
       "|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n",
       "| no | PAIVS  | yes | Lt_to_Rt | Complete | Normal   | Low    | no  | 0-3_days  | no  | Equal | Moderate | Normal | Oligaemic | no  | no  | 5-12 | 5-12 | &gt;=7.5 | Grd_Glass  |\n",
       "| no | TGA    | yes | None     | Transp.  | Abnormal | Low    | no  | 0-3_days  | yes | Equal | Severe   | Normal | Asy/Patch | no  | no  | 12+  | &lt;5   | &lt;7.5  | Asy/Patchy |\n",
       "| no | TGA    | yes | None     | Transp.  | Normal   | High   | yes | 0-3_days  | no  | Equal | Severe   | Normal | Normal    | yes | no  | 12+  | 5-12 | &lt;7.5  | Normal     |\n",
       "| no | Lung   | yes | None     | Mild     | Abnormal | Normal | no  | 0-3_days  | yes | Equal | Moderate | Normal | Asy/Patch | no  | yes | 5-12 | 5-12 | &gt;=7.5 | Asy/Patchy |\n",
       "| no | Fallot | no  | Lt_to_Rt | Complete | Normal   | Low    | no  | 4-10_days | no  | Equal | Moderate | Normal | Normal    | no  | no  | 5-12 | 5-12 | &lt;7.5  | Normal     |\n",
       "| no | Fallot | no  | Lt_to_Rt | Mild     | Normal   | Low    | no  | 4-10_days | no  | Equal | Moderate | Normal | Oligaemic | no  | yes | 5-12 | &lt;5   | &lt;7.5  | Oligaemic  |\n",
       "\n"
      ],
      "text/plain": [
       "  BirthAsphyxia Disease Sick DuctFlow CardiacMixing LungParench LungFlow LVH\n",
       "1 no            PAIVS   yes  Lt_to_Rt Complete      Normal      Low      no \n",
       "2 no            TGA     yes  None     Transp.       Abnormal    Low      no \n",
       "3 no            TGA     yes  None     Transp.       Normal      High     yes\n",
       "4 no            Lung    yes  None     Mild          Abnormal    Normal   no \n",
       "5 no            Fallot  no   Lt_to_Rt Complete      Normal      Low      no \n",
       "6 no            Fallot  no   Lt_to_Rt Mild          Normal      Low      no \n",
       "  Age       Grunting HypDistrib HypoxiaInO2 CO2    ChestXray LVHreport\n",
       "1 0-3_days  no       Equal      Moderate    Normal Oligaemic no       \n",
       "2 0-3_days  yes      Equal      Severe      Normal Asy/Patch no       \n",
       "3 0-3_days  no       Equal      Severe      Normal Normal    yes      \n",
       "4 0-3_days  yes      Equal      Moderate    Normal Asy/Patch no       \n",
       "5 4-10_days no       Equal      Moderate    Normal Normal    no       \n",
       "6 4-10_days no       Equal      Moderate    Normal Oligaemic no       \n",
       "  GruntingReport LowerBodyO2 RUQO2 CO2Report XrayReport\n",
       "1 no             5-12        5-12  >=7.5     Grd_Glass \n",
       "2 no             12+         <5    <7.5      Asy/Patchy\n",
       "3 no             12+         5-12  <7.5      Normal    \n",
       "4 yes            5-12        5-12  >=7.5     Asy/Patchy\n",
       "5 no             5-12        5-12  <7.5      Normal    \n",
       "6 yes            5-12        <5    <7.5      Oligaemic "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "child <- read_csv('https://www.ccd.pitt.edu/wiki/images/CHILD10k.csv')\n",
    "child <- child |> select(BirthAsphyxia, Disease, Sick, DuctFlow, CardiacMixing, LungParench, LungFlow, LVH, Age, Grunting, HypDistrib, HypoxiaInO2, CO2, ChestXray, LVHreport, GruntingReport, LowerBodyO2, RUQO2, CO2Report, XrayReport)\n",
    "\n",
    "head(child)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running iteration # 1 ... DONE\n",
      "Running iteration # 2 ... DONE\n",
      "Running iteration # 3 ... DONE\n",
      "Running iteration # 4 ... DONE\n",
      "Running iteration # 5 ... DONE\n",
      "\n",
      "Total execution time: 70.8791 s\n",
      "\n",
      "Best node ordering found: BirthAsphyxia Disease Sick DuctFlow CardiacMixing LungParench LungFlow LVH Age Grunting HypDistrib HypoxiaInO2 CO2 ChestXray LVHreport GruntingReport LowerBodyO2 RUQO2 CO2Report XrayReport"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "  Random/Generated Bayesian network\n",
       "\n",
       "  model:\n",
       "   [BirthAsphyxia][Disease][Sick|Disease][DuctFlow|Disease]\n",
       "   [CardiacMixing|Disease][LungParench|Disease][LungFlow|Disease][LVH|Disease]\n",
       "   [Age|Disease][Grunting|Sick:LungParench][HypDistrib|DuctFlow]\n",
       "   [HypoxiaInO2|CardiacMixing][CO2|LungParench][ChestXray|LungParench:LungFlow]\n",
       "   [LVHreport|LVH][GruntingReport|Grunting][LowerBodyO2|HypoxiaInO2]\n",
       "   [RUQO2|HypoxiaInO2][CO2Report|CO2][XrayReport|ChestXray]\n",
       "  nodes:                                 20 \n",
       "  arcs:                                  20 \n",
       "    undirected arcs:                     0 \n",
       "    directed arcs:                       20 \n",
       "  average markov blanket size:           2.20 \n",
       "  average neighbourhood size:            2.00 \n",
       "  average branching factor:              1.00 \n",
       "\n",
       "  generation algorithm:                  Empty \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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VU4XcOpgoRcYVOtdMLpGvYTtoZB8hRO13CqIGHX11AtnXC6hv2EzfldC3p11fXS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\nSIQAYJAIAcAgEQKAQSIEAINECID/AKk9mbgrbNg2AAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "child_net <- k2.iter(dataset=head(child, 150), parents.nmax=2, f=f, k2=k2, n.iter=5, seed=1, type='bde', iss=10)\n",
    "\n",
    "cat('\\n\\nBest node ordering found:', child_net$order)\n",
    "child_net$dag\n",
    "graphviz.plot(child_net$dag)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "vscode": {
     "languageId": "r"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running iteration # 1 ... DONE\n",
      "Running iteration # 2 ... DONE\n",
      "Running iteration # 3 ... DONE\n",
      "Running iteration # 4 ... DONE\n",
      "Running iteration # 5 ... DONE\n",
      "\n",
      "Total execution time: 78.97155 s\n",
      "\n",
      "Best node ordering found: BirthAsphyxia Disease Sick DuctFlow CardiacMixing LungParench LungFlow LVH Age Grunting HypDistrib HypoxiaInO2 CO2 ChestXray LVHreport GruntingReport LowerBodyO2 RUQO2 CO2Report XrayReport"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\n",
       "  Random/Generated Bayesian network\n",
       "\n",
       "  model:\n",
       "   [BirthAsphyxia][Disease|BirthAsphyxia][Sick|Disease][DuctFlow|Disease]\n",
       "   [CardiacMixing|Disease][LungParench|Disease][LungFlow|Disease][LVH|Disease]\n",
       "   [Age|Disease:Sick][Grunting|Sick:LungParench]\n",
       "   [HypDistrib|DuctFlow:CardiacMixing][HypoxiaInO2|CardiacMixing:LungParench]\n",
       "   [CO2|LungParench][ChestXray|LungParench:LungFlow][LVHreport|LVH]\n",
       "   [GruntingReport|Grunting][LowerBodyO2|HypDistrib:HypoxiaInO2]\n",
       "   [RUQO2|HypoxiaInO2][CO2Report|CO2][XrayReport|ChestXray]\n",
       "  nodes:                                 20 \n",
       "  arcs:                                  25 \n",
       "    undirected arcs:                     0 \n",
       "    directed arcs:                       25 \n",
       "  average markov blanket size:           3.00 \n",
       "  average neighbourhood size:            2.50 \n",
       "  average branching factor:              1.25 \n",
       "\n",
       "  generation algorithm:                  Empty \n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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tE/42d9v2dmVfSxhLOX7f\n9bWsDV/avj1sl0/Z3n04zrUZaa+2i/lPcZfDmqChkXYr+0KmJfL13ddGqu1vh/BiksOom5V9\nLWEsWyP91uudkZYbvsv7eEe3kT7rKXEb/yyyprHeXS38ZKTdQzuEkZZTRU1/O4S/Rvp8dX79\nbFT2tYSx7M9IdUbar2IjI21vfTPSXqrnjHQQPBhJHPn67pORNv+997dPuBzUikTZ1xLGslvP\nm27eP7TbblfKfbtFRjr8d2WkzdF081+18LOR1pX0K/fOry2Rb+7e93xX1Ht/O4R3Ripb+TZl\nZ0sYyi+Lsj3f/5r4+/H1YsNn98mI1UYqy2Xf5eayplN+928vNqzuK9sN6oSrjXTpbWHku7t3\n1wN/j6mq+isX3hyatv1sV/a2hJE0ZtFw8O87frfl0RD5xUjflbSu762FX65/GxRct4+C8LXZ\nOoUH4SOLhjQuFtLbqtQRbtu388QhxqJgneX8HuXy0V+/8BgmT8PUSD1x0xWMEx7D5Gm4HW+6\ngmmkIaQbb7qCaaQhpBtvuoJppCGkG2+6gmmkMdjkUREVJZyuYJzwEObOw/F40xVMIw0h3XjT\nFUwjDSHdeNMVTCONwSIR2DtMUK/Iuy4YJzyCqRNxPd50BdNIQ0g33nQF00hj0M+kMiJKOF3B\nOOEBzJyJ8/GmK5hGGoN2KtXxUMLpCsYJ2zNxKu7Hm65gGmkMurk0REMJpysYJ2yOp1xUk2mK\nhRJOVzBO2Jppkwkx3nQF00hj0MumMRJKOF3BOGFjZs0myHjTFUwjjUErneY4KOF0BeOEbXGW\njlI+gigo4XQF44RN8ZaPSkKiGCjhdAXjhC1xl5BCRsIIKOF0BeOEDfGXUXdK4v1RwukKxgnb\n4TGlvpzku6OE0xWME7bDYUp9SXUVhBJOVzBO2AqXSXVk1VkPSjhdwThhI3xmBXu6Ee55TtiC\nccI2OE0LdgEs2JW3wAXjhE3wmpfkCaXOc1CUcLqCccIWuE2suWl6PUYJpysYJ6yP49TaklMt\nBCWcrmCcsDauk6s/BGkfq1DC6QrGCSvjO7vK9ln0GCWcrmCcsCre8/t8Xj7F+/X+cMLpCsYJ\nK+I/w8/n6pPMX+4ILpyuYJywFiGS/I9Sdi09/HNC4XQF44Q1CJPoP8pCFuF0BeOE+wiWLiE+\noZEIUYBGIkQBGokQBWgkQhSgkQhRgEYiRAEaiRAFaCRCFKCRCFGARiJEARqJEAVoJEIUoJEI\nUYBGIkQBGokQBWgkQhSgkQhRgEYiRAEaiRAF/gdftpk0ZOLJyQAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "child_net <- k2.iter(dataset=child, parents.nmax=2, f=log.f, k2=k2, n.iter=5, seed=1, type='bde', iss=10)\n",
    "\n",
    "cat('\\n\\nBest node ordering found:', child_net$order)\n",
    "child_net$dag\n",
    "graphviz.plot(child_net$dag)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## **References**\n",
    "\n",
    "<a id='ref1'></a>\n",
    "**[1]** M. Scutari and J. B. Denis, *Bayesian Networks*, CRC Press, 2022, Taylor and Francis Group\n",
    "\n",
    "<a id='ref2'></a>\n",
    "**[2]** G. F. Cooper and E. Herskovits, *A Bayesian Method for the Induction of Probabilistic Networks from Data*, Machine Learning 9, (1992) 309\n",
    "\n",
    "<a id='ref3'></a>\n",
    "**[3]** C. Ruiz, *Illustration of the K2 Algorithm for learning Bayes Net Structures*, http://web.cs.wpi.edu/~cs539/s11/Projects/k2_algorithm.pdf\n",
    "\n",
    "<a id='ref4'></a>\n",
    "**[4]** A. Franzin et al., *$\\texttt{bnstruct}$: an R package for Bayesian Network structure learning in the presence of missing data*, Bioinformatics 33(8) (2017) 1250\n",
    "\n",
    "<a id='ref5'></a>\n",
    "**[5]** F. Sambo and A. Franzin, *$\\texttt{bnstruct}$: an R package for Bayesian Network Structure Learning with missing data*, December 12, 2016\n",
    "\n",
    "<a id='ref6'></a>\n",
    "**[6]** Wikipedia, *Bayesian network*, https://en.wikipedia.org/wiki/Bayesian_network"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "\n",
    "Code by *Daniele Ninni*, last review: 17 July 2022, at 09:15 <br>**Workgroup**: Barone Francesco, Nagaro Gianmarco, Ninni Daniele.\n",
    "\n",
    "University of Padua, **Advanced Statistics for Physics Analysis**."
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
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   "codemirror_mode": "r",
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   "name": "R",
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 },
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