Updated nyga distribution theory.

examples/nyga_distribution.ipynb

@@ -88,6 +88,18 @@
    ],
    "id": "122bdb92c9030631"
   },
+  {
+   "metadata": {},
+   "cell_type": "markdown",
+   "source": [
+    "In case of a mixture of two uniform distributions, the likelihood is given by\n",
+    "\n",
+    "$$\n",
+    "p(x_i) = w_{left} \\cdot U_{left} + w_{right} \\cdot U_{right}\n",
+    "$$\n"
+   ],
+   "id": "d9b477da487ef67d"
+  },
   {
    "metadata": {},
    "cell_type": "markdown",
@@ -150,8 +162,13 @@
     "Plugging it in we get the likelihood for the split\n",
     "\n",
     "\\begin{align*}\n",
-    "log \\left( L(\\mathcal{D}| d_1, d_{split}, d_N) \\right) &= \\sum_{i=1}^k log(w_i) + log(p(x_i)) =  \\\\\n",
-    "&= \\sum_{i=1}^k log(w_i) + log(p_{left}(x) + p_{right}(x)) \\\\ &= \\sum_{i=1}^k log(w_i) + \\underbrace{log(p_{left}(x))}_{\\text{Constant for every sample in the left split}}  + \\sum_{i=k+1}^N log(w_i) + \\underbrace{log(p_{right}(x))}_{\\text{Constant for every sample in the right split}}\n",
+    "log \\left( L(\\mathcal{D}| d_1, d_{split}, d_N) \\right) &= \\sum_{i=1}^N log(w_i) + log(p(x_i))\\\\\n",
+    "&=  \\sum_{i=1}^{left} log(w_i) + log(p(x_i)) + \\sum_{i=right}^{N} log(w_i) + log(p(x_i)) \\\\\n",
+    "&=  \\sum_{i=1}^{left} log(w_i) + log(w_{left} \\cdot p_{left}(x_i)) + \\sum_{i=right}^{N} log(w_i) + log(w_{right} \\cdot p_{right}(x_i)) \\\\\n",
+    "&=  \\underbrace{\\sum_{i=1}^{left} log(w_i) + log(w_{left}) + log(U_{left})}_{\\text{Left Hand Side (LHS)}} + \\underbrace{\\sum_{i=right}^{N} log(w_i) + log(w_{right}) + log(U_{right})}_{\\text{Right Hand Side (RHS)}} \\\\\n",
+    "LHS &=  L \\cdot ( log(w_{left}) + log(U_{left})) + \\sum_{i=1}^{left} log(w_i)\\\\\n",
+    "&= L \\cdot \\left( log \\left( \\sum_{i=1}^{left} w_i \\right) - log \\left( \\sum_{i=1}^{N} w_i \\right) - log(D_{split} - D_{left}) \\right) + \\sum_{i=1}^{left} log(w_i)\\\\\n",
+    "RHS &= (N - R) \\cdot \\left( log \\left( \\sum_{i=right}^{N} w_i \\right) - log \\left( \\sum_{i=1}^{N} w_i \\right) - log(D_{right} - D_{split}) \\right) + \\sum_{i=right}^{N} log(w_i)\\\\\n",
     "\\end{align*}\n"
    ],
    "id": "97c049e0191e5577"