Adds Bayes Model

Author: samagra14

notebooks/ProbabilisticReasoning.ipynb

@@ -268,8 +268,95 @@
     "\n",
     "Then we repeat the process, reducing each conjunctive probability to a conditional probability\n",
     "and a smaller conjunction. We end up with one big product:\n",
+    "$$P (x_1 , . . . , x_n ) = P (x_n | x_{n−1} , . . . , x_1 )P (x_{n−1} | x_{n−2} , . . . , x_1 ) · · · P (x_2 | x_1 )P (x_1 )$$\n",
+    "$$ = \\prod_{i = 1}^{n}P(x_i|x_{i-1},...,x_1)$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "This identity is called the chain rule. It holds for any set of random variables. We can conclude that the specification of the joint distribution is equivalent to the\n",
+    "general assertion that, for every variable $X_i$ in the network\n",
+    "\n",
+    "$$\\textbf{P}(X_i | X_{i−1} , . . . , X_1 ) = \\textbf{P}(X_i | Parents(X_i ))$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "provided that $Parents(X_i) \\subseteq {X_{i−1} , . . . , X_1 }$. This last condition is satisfied by numbering\n",
+    "the nodes in a way that is consistent with the partial order implicit in the graph structure."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Since we can represent the full joint distribution of a particular system using Bayesian Networks, therefore we can consider the BayesianNetworks as Probability Models and can use them for inference procedures in the same way as we used the `FullJointProbabilityModel`. This model can be created using the `FiniteBayesModel` class from the repository. this class asks for a Bayesian Network and an Inference Procedure. If no inference procedure is mentioned, the `EnumerationAsk` algorithm is used as the default inference procedure. From now on, I will be using the `BayesNetExampleFactory` to creste the ToothAcheCavityCatch example. Let's have a look at how we can manipulate the `BayesNetModel` to perform inference using uncertain knowledge."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "The random variables in the model = [Cavity, Toothache, Catch]\n",
+      "The prior probability of having a toothache = 0.2\n",
+      "The prior probability of having a cavity = 0.2\n",
+      "The probability of having a cavity and toothache simultaneously is = 0.12\n",
+      "The probability of having a toothache given that the person has a cavity(causal direction) is = 0.6\n",
+      "The probability of having a cavity given that the person is experiencing toothache(diagnostic direction) is = 0.6\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "null"
+      ]
+     },
+     "execution_count": 18,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "package aima.notebooks.probabilisticreasoning;\n",
+    "\n",
+    "import aima.core.probability.example.*;\n",
+    "import aima.core.probability.bayes.*;\n",
+    "import aima.core.probability.bayes.impl.*;\n",
+    "import aima.core.probability.bayes.impl.*;\n",
+    "import aima.core.probability.bayes.model.*;\n",
+    "import aima.core.probability.proposition.*;\n",
+    "\n",
+    "// Load the network from the network factory.\n",
+    "BayesianNetwork cavityNet = BayesNetExampleFactory.constructToothacheCavityCatchNetwork();\n",
+    "// Construct the BayesModel from the BayesNet\n",
+    "// We have not passed any inference procedure. Hence, the default inference procedure will be used.\n",
+    "FiniteBayesModel model = new FiniteBayesModel(cavityNet);\n",
+    "// Now we are ready to answer all sorts of questions.\n",
     "\n",
-    "$$P (x 1 , . . . , x n ) = P (x n | x n−1 , . . . , x 1 )P (x n−1 | x n−2 , . . . , x 1 ) · · · P (x 2 | x_1 )P (x_1 )$$"
+    "// Let's define a few assignments\n",
+    "AssignmentProposition toothache = new AssignmentProposition(ExampleRV.TOOTHACHE_RV,true);\n",
+    "AssignmentProposition cavity = new AssignmentProposition(ExampleRV.CAVITY_RV,true);\n",
+    "// Now let's have a look at what we can do with the model.\n",
+    "// To print the random variables in the model\n",
+    "System.out.println(\"The random variables in the model = \" + model.getRepresentation());\n",
+    "// We can calculate the prior probabilities of a variety of combinations of random variables\n",
+    "System.out.println(\"The prior probability of having a toothache = \"+ model.prior(toothache));\n",
+    "System.out.println(\"The prior probability of having a cavity = \"+ model.prior(cavity));\n",
+    "System.out.println(\"The probability of having a cavity and toothache simultaneously is = \"+ model.prior(toothache, cavity));\n",
+    "// We can also calculate a variety of posterior probabilities from the model as follows\n",
+    "System.out.println(\"The probability of having a toothache given that the person has a cavity(causal direction) is = \"+ \n",
+    "                  model.posterior(toothache,cavity));\n",
+    "System.out.println(\"The probability of having a cavity given that the person is experiencing toothache(diagnostic direction) is = \"\n",
+    "                  + model.posterior(cavity,toothache));"
    ]
   },
   {