From 1715a6d08308697b3c286fc2368abd85d934f85d Mon Sep 17 00:00:00 2001
From: Jeremy Teitelbaum <jeremy.teitelbaum@uconn.edu>
Date: Fri, 27 Apr 2018 12:01:51 -0400
Subject: [PATCH] updated useful formulae

---
 Useful Formulae.ipynb | 89 +++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 86 insertions(+), 3 deletions(-)

diff --git a/Useful Formulae.ipynb b/Useful Formulae.ipynb
index fd0c133..3fc65f4 100644
--- a/Useful Formulae.ipynb	
+++ b/Useful Formulae.ipynb	
@@ -21,20 +21,103 @@
     "and the posterior mean is\n",
     "$$\n",
     "\\mu_1=\\frac{\\frac{\\mu_0}{\\tau_0^2}+\\frac{n\\overline{y}}{\\sigma^2}}{\\frac{1}{\\tau_{1}^2}}\n",
-    "$$\n"
+    "$$\n",
+    "\n",
+    "The posterior sampling distribution $\\theta$ is\n",
+    "$$\n",
+    "p( z |y)=\\int_{\\theta} p(z|\\theta) d\\theta\n",
+    "$$\n",
+    "is a normal distribution with mean equal to the posterior mean $\\mu_1$ and variance equal to $\\sigma^2+\\tau_1^2$\n",
+    "where $\\tau_1$ is the posterior variance.\n",
+    "\n",
+    "See Pages 39-42 of BDA (Section 2.5) for more information."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 24,
    "metadata": {},
    "outputs": [],
    "source": [
+    "from scipy.stats import norm\n",
+    "import numpy as np\n",
     "def posterior(prior_mean,prior_variance,sample_mean,pop_variance,n):\n",
     "    post_var=1/((1/prior_variance) + n/pop_variance)\n",
     "    post_mean=(prior_mean/prior_variance+sample_mean*n/pop_variance)/(1/post_var)\n",
-    "    return post_mean, post_var"
+    "    return post_mean, post_var\n",
+    "\n",
+    "def post_sample(y,prior_mean,prior_variance,sample_mean,pop_variance,n):\n",
+    "    post_mean,post_var=posterior(prior_mean,prior_variance,sample_mean,pop_variance,n)\n",
+    "    return norm.pdf(y,post_mean,np.sqrt(pop_variance+post_var))\n",
+    "    "
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0.7403867575800461"
+      ]
+     },
+     "execution_count": 26,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "post_sample(-.25,1,.25,-.25,1,10)+post_sample(-.25,-1,.25,-.25,1,10)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0.05095226579074726"
+      ]
+     },
+     "execution_count": 29,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    ".1*post_sample(-.25,-1,.25,-.25,1,10)/(post_sample(-.25,1,.25,-.25,1,10)+post_sample(-.25,-1,.25,-.25,1,10))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0.4414296078832747"
+      ]
+     },
+     "execution_count": 30,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    ".9*post_sample(-.25,1,.25,-.25,1,10)/(post_sample(-.25,1,.25,-.25,1,10)+post_sample(-.25,-1,.25,-.25,1,10))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {