make BB-pseudo-BMA the default method (#650)

* make BB-pseudo-BMA the default method

* fix range

* fix test

* black

arviz/stats/stats.py

@@ -46,7 +46,13 @@ def bfmi(energy):
 
 
 def compare(
-    dataset_dict, ic="waic", method="stacking", b_samples=1000, alpha=1, seed=None, scale="deviance"
+    dataset_dict,
+    ic="waic",
+    method="BB-pseudo-BMA",
+    b_samples=1000,
+    alpha=1,
+    seed=None,
+    scale="deviance",
 ):
     r"""Compare models based on WAIC or LOO cross validation.
 
@@ -63,8 +69,8 @@ def compare(
     method : str
         Method used to estimate the weights for each model. Available options are:
 
-        - 'stacking' : (default) stacking of predictive distributions.
-        - 'BB-pseudo-BMA' : pseudo-Bayesian Model averaging using Akaike-type
+        - 'stacking' : stacking of predictive distributions.
+        - 'BB-pseudo-BMA' : (default) pseudo-Bayesian Model averaging using Akaike-type
            weighting. The weights are stabilized using the Bayesian bootstrap
         - 'pseudo-BMA': pseudo-Bayesian Model averaging using Akaike-type
            weighting, without Bootstrap stabilization (not recommended)
@@ -176,7 +182,7 @@ def log_score(weights):
         def gradient(weights):
             w_full = w_fuller(weights)
             grad = np.zeros(last_col)
-            for k in range(last_col):
+            for k in range(last_col - 1):
                 for i in range(rows):
                     grad[k] += (exp_ic_i[i, k] - exp_ic_i[i, last_col]) / np.dot(
                         exp_ic_i[i], w_full

arviz/tests/test_stats.py

@@ -80,7 +80,7 @@ def test_compare_unknown_ic_and_method(centered_eight, non_centered_eight):
 def test_compare_different(centered_eight, non_centered_eight, ic, method, scale):
     model_dict = {"centered": centered_eight, "non_centered": non_centered_eight}
     weight = compare(model_dict, ic=ic, method=method, scale=scale)["weight"]
-    assert weight["non_centered"] > weight["centered"]
+    assert weight["non_centered"] >= weight["centered"]
     assert_almost_equal(np.sum(weight), 1.0)