small updates to notebook

Author: bwengals

docs/source/notebooks/GP-Latent.ipynb

@@ -19,7 +19,7 @@
    "source": [
     "## `.prior`\n",
     "\n",
-    "Concretely, what this means is that with some data set of finite size, the `prior` method places a multivariate normal prior distribution on the vector of function values, $\\mathbf{f}$,\n",
+    "With some data set of finite size, the `prior` method places a multivariate normal prior distribution on the vector of function values, $\\mathbf{f}$,\n",
     "\n",
     "$$\n",
     "\\mathbf{f} \\sim \\text{MvNormal}(\\mathbf{m}_{x},\\, \\mathbf{K}_{xx}) \\,,\n",
@@ -42,10 +42,10 @@
     "    gp = pm.gp.Latent(cov_func=cov_func)\n",
     "    \n",
     "    # Place a GP prior over the function f.\n",
-    "    f = gp.prior(\"f\", n_points=10, X=X)\n",
+    "    f = gp.prior(\"f\", X=X)\n",
     "```\n",
     "\n",
-    "By default, PyMC3 reparameterizes the prior on `f` by rotating it with the Cholesky factor.  This helps to reduce covariances in the posterior of the transformed GP, `v`.  The reparameterized model is,\n",
+    "By default, PyMC3 reparameterizes the prior on `f` under the hood by rotating it with the Cholesky factor of its covariance matrix.  This helps to reduce covariances in the posterior of the transformed random variable, `v`.  The reparameterized model is,\n",
     "\n",
     "$$\n",
     "\\begin{aligned}\n",
@@ -64,29 +64,23 @@
    "source": [
     "## `.conditional`\n",
     "\n",
-    "The conditional method implements the \"predictive\" distribution for function values that were not part of the original data set.  This distribution is,\n",
+    "The conditional method implements the predictive distribution for function values that were not part of the original data set.  This distribution is,\n",
     "\n",
     "$$\n",
     "\\mathbf{f}_* \\mid \\mathbf{f} \\sim \\text{MvNormal} \\left(\n",
     "  \\mathbf{m}_* + \\mathbf{K}_{*x}\\mathbf{K}_{xx}^{-1} \\mathbf{f} ,\\,\n",
     "  \\mathbf{K}_{**} - \\mathbf{K}_{*x}\\mathbf{K}_{xx}^{-1}\\mathbf{K}_{x*} \\right)\n",
     "$$\n",
     "\n",
-    "Using the same `gp` object we defined above, this is specified as,\n",
+    "Using the same `gp` object we defined above, we can construct a random variable with this\n",
+    "distribution by,\n",
     "\n",
     "```python\n",
     "# vector of new X points we want to predict the function at\n",
     "X_star = np.linspace(0, 2, 100)[:, None]\n",
     "\n",
     "with latent_gp_model:\n",
     "    f_star = gp.conditional(\"f_star\", n_points=100, X_star)\n",
-    "```\n",
-    "\n",
-    "If `gp` is part of a sum of GP objects, it can be conditioned on different components of that sum using the optional keyword argument `given`,\n",
-    "\n",
-    "```python\n",
-    "    f_star_diff = gp.conditional(\"f_star_diff\", n_points=100, X_star, \n",
-    "                                  gp=a_different_gp)\n",
     "```"
    ]
   },