update ipynb

seminars/day4/AdaGram/AdaGram seminar.ipynb

@@ -267,7 +267,19 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "**Answer**: $\\mathbb{E} K = $"
+    "**Solution**: \n",
+    "\n",
+    "We can express the number of tables $K$ as the number of times a new table has been created:\n",
+    "$$\n",
+    "    K = \\sum_{i=1}^n \\mathbb{1}[ \\text{table created at } i].\n",
+    "$$\n",
+    "Then recall that expectation of an indicator is the probability of it to be true, hence\n",
+    "$$\n",
+    "    \\mathbb{E} K = \\sum_{i=1}^n \\frac{\\alpha}{i - 1 + \\alpha}.\n",
+    "$$\n",
+    "This formula can be recognized as harmonic series, see https://en.wikipedia.org/wiki/Harmonic_number. \n",
+    "We can use this fact and after some calculations we get\n",
+    "$\\mathbb{E} K = \\alpha \\log(n) + \\text{const}$ when $n \\rightarrow \\infty$"
    ]
   },
   {
@@ -298,7 +310,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "**Answer**: $q^*(\\beta_k) = $"
+    "**Solution**: The full solution can be found in the paper http://cs.columbia.edu/~blei/papers/BleiJordan2004.pdf, equation 18. In the paper's notation $\\beta_k$ is $V_k$."
    ]
   },
   {