scikit-learn
diff --git a/‎doc/whats_new/v1.0.rst
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diff --git a/‎sklearn/gaussian_process/_gpr.py
Lines changed: 18 additions & 4 deletions< 8000 div class="d-flex"> b/‎sklearn/gaussian_process/_gpr.py
Lines changed: 18 additions & 4 deletions< 8000 div class="d-flex">
diff --git a/‎sklearn/gaussian_process/tests/test_gpr.py
Lines changed: 30 additions & 0 deletions b/‎sklearn/gaussian_process/tests/test_gpr.py
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@@ -43,6 +43,14 @@ Fixed models
   between sparse and dense input. :pr:`21195`
   by :user:`Jérémie du Boisberranger <jeremiedbb>`.
 
+:mod:`sklearn.gaussian_process`
+...............................
+
+- |Fix| Compute `y_std` properly with multi-target in
+  :class:`sklearn.gaussian_process.GaussianProcessRegressor` allowing
+  proper normalization in multi-target scene.
+  :pr:`20761` by :user:`Patrick de C. T. R. Ferreira <patrickctrf>`.
+
 :mod:`sklearn.feature_extraction`
 .................................
 
 
@@ -349,11 +349,12 @@ def predict(self, X, return_std=False, return_cov=False):
         y_mean : ndarray of shape (n_samples,) or (n_samples, n_targets)
             Mean of predictive distribution a query points.
 
-        y_std : ndarray of shape (n_samples,), optional
+        y_std : ndarray of shape (n_samples,) or (n_samples, n_targets), optional
             Standard deviation of predictive distribution at query points.
             Only returned when `return_std` is True.
 
-        y_cov : ndarray of shape (n_samples, n_samples), optional
+        y_cov : ndarray of shape (n_samples, n_samples) or \
+                (n_samples, n_samples, n_targets), optional
             Covariance of joint predictive distribution a query points.
             Only returned when `return_cov` is True.
         """
@@ -403,7 +404,14 @@ def predict(self, X, return_std=False, return_cov=False):
                 y_cov = self.kernel_(X) - V.T @ V
 
                 # undo normalisation
-                y_cov = y_cov * self._y_train_std ** 2
+                y_cov = np.outer(y_cov, self._y_train_std ** 2).
8000
reshape(
+                    *y_cov.shape, -1
+                )
+
+                # if y_cov has shape (n_samples, n_samples, 1), reshape to
+                # (n_samples, n_samples)
+                if y_cov.shape[2] == 1:
+                    y_cov = np.squeeze(y_cov, axis=2)
 
                 return y_mean, y_cov
             elif return_std:
@@ -424,7 +432,13 @@ def predict(self, X, return_std=False, return_cov=False):
                     y_var[y_var_negative] = 0.0
 
                 # undo normalisation
-                y_var = y_var * self._y_train_std ** 2
+                y_var = np.outer(y_var, self._y_train_std ** 2).reshape(
+                    *y_var.shape, -1
+                )
+
+                # if y_var has shape (n_samples, 1), reshape to (n_samples,)
+                if y_var.shape[1] == 1:
+                    y_var = np.squeeze(y_var, axis=1)
 
                 return y_mean, np.sqrt(y_var)
             else:
 
@@ -652,3 +652,33 @@ def test_gpr_predict_error():
     err_msg = "At most one of return_std or return_cov can be requested."
     with pytest.raises(RuntimeError, match=err_msg):
         gpr.predict(X, return_cov=True, return_std=True)
+
+
+def test_y_std_with_multitarget_normalized():
+    """Check the proper normalization of `y_std` and `y_cov` in multi-target scene.
+
+    Non-regression test for:
+    https://github.com/scikit-learn/scikit-learn/issues/17394
+    https://github.com/scikit-learn/scikit-learn/issues/18065
+    """
+    rng = np.random.RandomState(1234)
+
+    n_samples, n_features, n_targets = 12, 10, 6
+
+    X_train = rng.randn(n_samples, n_features)
+    y_train = rng.randn(n_samples, n_targets)
+    X_test = rng.randn(n_samples, n_features)
+
+    # Generic kernel
+    kernel = WhiteKernel(1.0, (1e-1, 1e3)) * C(10.0, (1e-3, 1e3))
+
+    model = GaussianProcessRegressor(
+        kernel=kernel, n_restarts_optimizer=10, alpha=0.1, normalize_y=True
+    )
+    model.fit(X_train, y_train)
+    y_pred, y_std = model.predict(X_test, return_std=True)
+    _, y_cov = model.predict(X_test, return_cov=True)
+
+    assert y_pred.shape == (n_samples, n_targets)
+    assert y_std.shape == (n_samples, n_targets)
+    assert y_cov.shape == (n_samples, n_samples, n_targets)