added 3.10.7

BDA 3.10.4.ipynb

@@ -31,14 +31,10 @@
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": null,
+   "cell_type": "markdown",
    "metadata": {},
-   "outputs": [],
    "source": [
-    "#(a).  Since p0 and p1 are binomially distributed and independent,\n",
-    "# the noninformative prior is the beta(1,1) distribution and the posterior \n",
-    "# distributions are p0~beta(636,40) and p1~(659,23)."
+    "# Here we use the beta(1,1) uniform prior"
    ]
   },
   {
@@ -81,7 +77,6 @@
     "ax[2].hist(odds_ratio,bins=50)\n",
     "o=ax[2].set_title('Odds Ratios')\n",
     "plt.show()\n",
-    "\n",
     "print('odds summary:',np.mean(odds_ratio),np.var(odds_ratio),np.percentile(odds_ratio,[.025,25,50,75,97.5]))\n",
     "\n"
    ]
@@ -90,7 +85,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Here we use the beta(0,0) improper prior; the effect is pretty minimal"
+    "# Here we use the beta(0,0) improper prior; the difference from the previous case is small"
    ]
   },
   {

BDA 3.10.7.ipynb

@@ -0,0 +1,71 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Problem 3.10.7\n",
+    "\n",
+    "Poisson and binomial distributions: a student sits on a street corner for an hour and records the number of bicycles $ b $ and the number of other vehicles $v$ that go by. Two models are considered: \n",
+    "\n",
+    "* The outcomes $b$ and $v$ have independent Poisson distributions, with unknown means $\\theta_b$ and $\\theta_v$ . \n",
+    "\n",
+    "* The outcome $b$ has a binomial distribution, with unknown probability $p$ and sample size $b + v$. \n",
+    "\n",
+    "Show that the two models have the same likelihood if we define $p = \\theta_b/( \\theta_b +\\theta_v)$.\n",
+    "\n",
+    "Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; Rubin, Donald B.. Bayesian Data Analysis, Third Edition (Chapman & Hall/CRC Texts in Statistical Science) (Page 81). CRC Press. Kindle Edition. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "This problem has no computational element, it's a fact about poisson distributions.  In the first case we have\n",
+    "$$\n",
+    "P(b=b_0)=\\frac{\\theta_b^{b_0}e^{-\\theta_b)}}{b_0!}\n",
+    "$$\n",
+    "and\n",
+    "$$\n",
+    "P(v=v_0)=\\frac{\\theta_v^{v_0}e^{-\\theta_v)}}{v_0!}\n",
+    "$$\n",
+    "It's also a fact that the sum of two poisson variables with rates $\\theta_v$ and $\\theta_b$ is poisson with \n",
+    "rate $\\theta_v+\\theta_b$.  \n",
+    "\n",
+    "A direct calculation gives \n",
+    "$$\n",
+    "P(b=b_0,v=v_0|b_0+v_0=N)=\\binom{N}{b_0}\\frac{\\theta_b^{b_0}\\theta_v^{N-b_0}}{(\\theta_b+\\theta_v)^{N}}\n",
+    "$$\n",
+    "which is what we're supposed to show."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "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": "3.6.4"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}