Update docs to make Predictive behavior more clear (#1850)

* Update docs to make Predictive behavior more clear

* Reorder imports to hopefully fix CI issue

* Another attempt to revert autoformatting applied by vscode

Author: tomwallis

notebooks/source/bayesian_hierarchical_linear_regression.ipynb

@@ -427,6 +427,13 @@
     "samples_predictive = predictive(random.PRNGKey(0), patient_code, Weeks, None)"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Note that for [`Predictive`](http://num.pyro.ai/en/latest/utilities.html#numpyro.infer.util.Predictive) to work as expected, the response variable of the model (in this case, `FVC_obs`) must be set to `None`."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},

notebooks/source/bayesian_regression.ipynb

@@ -1507,6 +1507,14 @@
     "ax.set(xlabel=\"Marriage rate\", ylabel=\"Divorce rate\", title=\"Predictions with 90% CI\");"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Note that for `Predictive` to work as expected, the response variable of the model (in this case, `divorce`) must be set to `None`.\n",
+    "In the code above this is done implicitly by not passing a value for `divorce` to the model in the call to `prior_predictive`, which due to the model definition, sets `divorce=None`."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {

numpyro/infer/util.py

@@ -857,6 +857,9 @@ class Predictive(object):
         The interface for the `Predictive` class is experimental, and
         might change in the future.
 
+    Note that for the predictive distribution to be returned as intended, observed
+    variables in the model (constraining the likelihood term) must be set to `None` (see Example).
+
     :param model: Python callable containing Pyro primitives.
     :param dict posterior_samples: dictionary of samples from the posterior.
     :param callable guide: optional guide to get posterior samples of sites not present
@@ -908,6 +911,10 @@ def model(X, y=None):
         predictive = Predictive(model, num_samples=1000)
         y_pred = predictive(rng_key, X)["obs"]
 
+    Note how above, no value for `y` is passed to `predictive`, resulting in `y`
+    being set to `None`. Setting the observed variable(s) to `None` when using
+    `Predictive` is required for the method to function as expected.
+
     If you also have posterior samples, you can sample from the posterior predictive::
 
         predictive = Predictive(model, posterior_samples=posterior_samples)