diff --git a/integration_tests/conftest.py b/integration_tests/conftest.py
index 506e7a7d..11c939aa 100644
--- a/integration_tests/conftest.py
+++ b/integration_tests/conftest.py
@@ -201,35 +201,34 @@ def _calculate_bayes_factor(
 
     @cache
     def _fit_beta_distribution_to_ui(self, lb: float, ub: float) -> Tuple[float, float]:
-        assert lb > 0 or ub < 1
+        assert lb > 0 and ub < 1
         # Inspired by https://stats.stackexchange.com/a/112671/
         def objective(x):
             # np.exp ensures they are always positive
             a, b = np.exp(x)
             dist = scipy.stats.beta(a=a, b=b)
 
-            squared_error_lower = (
-                self._quantile_squared_error(dist, lb, 0.025) if lb > 0 else 0
-            )
-            squared_error_upper = (
-                self._quantile_squared_error(dist, ub, 0.975) if ub < 1 else 0
-            )
+            squared_error_lower = self._quantile_squared_error(dist, lb, 0.025)
+            squared_error_upper = self._quantile_squared_error(dist, ub, 0.975)
 
             return squared_error_lower + squared_error_upper
 
         # It is quite important to start with a reasonable guess.
         ui_midpoint = (lb + ub) / 2
         # TODO: Is this reasonable!?
-        first_guess_concentration = 10
-        optimization_result = scipy.optimize.minimize(
-            objective,
-            x0=[
-                np.log(ui_midpoint * first_guess_concentration),
-                np.log((1 - ui_midpoint) * first_guess_concentration),
-            ],
-        )
-        # Sometimes it warns that it may not have found a good solution,
-        # but the solution is very accurate.
+        for first_guess_concentration in [1_000, 100, 10]:
+            optimization_result = scipy.optimize.minimize(
+                objective,
+                x0=[
+                    np.log(ui_midpoint * first_guess_concentration),
+                    np.log((1 - ui_midpoint) * first_guess_concentration),
+                ],
+            )
+            # Sometimes it warns that it may not have found a good solution,
+            # but the solution is very accurate.
+            if optimization_result.success or optimization_result.fun < 1e-05:
+                break
+
         assert optimization_result.success or optimization_result.fun < 1e-05
 
         result = np.exp(optimization_result.x)
diff --git a/integration_tests/test_domestic_migration.py b/integration_tests/test_domestic_migration.py
index 157ecf8d..50494437 100644
--- a/integration_tests/test_domestic_migration.py
+++ b/integration_tests/test_domestic_migration.py
@@ -114,7 +114,7 @@ def test_individuals_move_into_new_households(
             "Domestic migration joining recently created households",
             # This case -- people who joined a just-created household -- should be exceedingly rare.
             joining_recently_created_households,
-            target_value_lb=0,
+            target_value_lb=(1 / 1_000_000_000),
             target_value_ub=0.001,
             name_addl=f"Time step {time_step}",
         )
@@ -160,7 +160,7 @@ def test_individuals_move_into_new_households(
         "Domestic migration joining recently created households",
         # This case -- people who joined a just-created household -- should be exceedingly rare.
         all_time_joining_recently_created_households,
-        target_value_lb=0,
+        target_value_lb=(1 / 1_000_000_000),
         target_value_ub=0.001,
         name_addl="All time steps",
     )