Added initial text on Jeffreys priors

Author: zaxtax

Chapter6_Priorities/Priors.ipynb

@@ -1,6 +1,6 @@
 {
  "metadata": {
-  "name": "Priors"
+  "name": ""
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -1393,7 +1393,17 @@
       "\n",
       "## Jefferys Priors\n",
       "\n",
-      "TODO."
+      "Earlier, we talked about objective priors rarely are *objective*. Partly what we mean by this is that we want a prior that doesn't bias our posterior estimates. The flat prior seems like a reasonable choice as it assigns equal probability to all values. \n",
+      "\n",
+      "But the flat prior is not transformation invariant. What does this mean? Suppose we have a random variable $ \\bf X $ from Bernoulli($\\theta$). We define the prior on  $p(\\theta) = 1$. \n",
+      "\n",
+      "PUT PLOT OF THETA HERE\n",
+      "\n",
+      "Now, let's transform $\\theta$ with the function $\\psi = log \\frac{\\theta}{1-\\theta}$. This is just a function to stretch $\\theta$ across the real line. Now how likely are different values of $\\psi$ under our transformation.\n",
+      "\n",
+      "PUT PLOT OF PSI HERE\n",
+      "\n",
+      "Oh no! Our function is no longer flat. It turns out flat priors do carry information in them after all. The point of Jeffreys Priors is to create priors that don't accidentally become informative when you transform the variables you placed them originally on."
      ]
     },
     {