Update horseshoe notebook

docs/notebooks/horseshoe_prior.ipynb

@@ -29,7 +29,29 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We are simulating 100 observations with the 50 explanatory variables, also simulated. "
+    "Here is what we did:\n",
+    "\n",
+    "* We defined an intercept.\n",
+    "* We defined a vector of 50 betas, 5 of which were drawn from a normal(5,1) distribution, and then assigned a random sign.\n",
+    "* We created the design matrix with normal(0,1) entries and set $\\sigma$ to 1.\n",
+    "* We calculated the deterministic means $\\mu$ using the intercept and the design matrix multiplied by the betas.\n",
+    "* We simulated 100 response variables (observations) from a normal distribution with mean $\\mu$ and standard deviation $\\sigma$.\n",
+    "\n",
+    "Next, we proceeded with the Bayesian estimation of the model. We proposed the horseshoe prior, for which the following parameters were calculated:\n",
+    "\n",
+    "$$\\mu_i = \\alpha + \\beta_1 x_{1i} + \\beta_2 x_{2i} + ... + \\beta_p x_{pi}$$\n",
+    "\n",
+    "$$y_i \\sim N(\\mu_i, \\sigma^2)$$\n",
+    "\n",
+    "$$\\alpha \\sim N(0,1)$$\n",
+    "\n",
+    "$$\\beta_j \\sim N(0,\\lambda_j^2 \\tau^2)$$\n",
+    "\n",
+    "$$\\lambda_j \\sim C^+(0,1)$$\n",
+    "\n",
+    "$$\\tau \\sim T^+(df=3)$$\n",
+    "\n",
+    "$$\\sigma^2 \\sim N^+(0,1)$$"
    ]
   },
   {
@@ -62,7 +84,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Then we create the dataframe and the term name for the set of variables, to define the formula. "
+    "Here we create the dataframe and the term name for the set of variables, to define the formula. "
    ]
   },
   {