diff --git a/package/samplers/value_at_risk/_gp/acqf.py b/package/samplers/value_at_risk/_gp/acqf.py
index e52a6632..40ff03d0 100644
--- a/package/samplers/value_at_risk/_gp/acqf.py
+++ b/package/samplers/value_at_risk/_gp/acqf.py
@@ -11,10 +11,11 @@
 if TYPE_CHECKING:
     from typing import Protocol
 
-    from optuna._gp.gp import GPRegressor
     from optuna._gp.search_space import SearchSpace
     import torch
 
+    from .gp import GPRegressor
+
     class SobolGenerator(Protocol):
         def __call__(self, dim: int, n_samples: int, seed: int | None) -> torch.Tensor:
             raise NotImplementedError
diff --git a/package/samplers/value_at_risk/requirements.txt b/package/samplers/value_at_risk/requirements.txt
index e3f6b6a4..fa41cc4b 100644
--- a/package/samplers/value_at_risk/requirements.txt
+++ b/package/samplers/value_at_risk/requirements.txt
@@ -1,2 +1,3 @@
 scipy
 torch
+typing-extensions
diff --git a/package/samplers/value_at_risk/sampler.py b/package/samplers/value_at_risk/sampler.py
index fd54324e..75d13867 100644
--- a/package/samplers/value_at_risk/sampler.py
+++ b/package/samplers/value_at_risk/sampler.py
@@ -1,7 +1,9 @@
 from __future__ import annotations
 
 from typing import Any
+from typing import cast
 from typing import TYPE_CHECKING
+from typing import TypedDict
 
 import numpy as np
 import optuna
@@ -17,6 +19,7 @@
 from optuna.study import StudyDirection
 from optuna.trial import FrozenTrial
 from optuna.trial import TrialState
+from typing_extensions import NotRequired
 
 
 if TYPE_CHECKING:
@@ -39,6 +42,11 @@
 EPS = 1e-10
 
 
+class _NoiseKWArgs(TypedDict):
+    uniform_input_noise_rads: NotRequired[torch.Tensor]
+    normal_input_noise_stdevs: NotRequired[torch.Tensor]
+
+
 def _standardize_values(values: np.ndarray) -> tuple[np.ndarray, np.ndarray, np.ndarray]:
     clipped_values = gp.warn_and_convert_inf(values)
     means = np.mean(clipped_values, axis=0)
@@ -123,7 +131,10 @@ def __init__(
         self._independent_sampler = independent_sampler or optuna.samplers.RandomSampler(seed=seed)
         self._intersection_search_space = optuna.search_space.IntersectionSearchSpace()
         self._n_startup_trials = n_startup_trials
-        self._log_prior: Callable[[gp.GPRegressor], torch.Tensor] = prior.default_log_prior
+        # We assume gp.GPRegressor is compatible with optuna._gp.gp.GPRegressor
+        self._log_prior: Callable[[gp.GPRegressor], torch.Tensor] = cast(
+            Callable[[gp.GPRegressor], torch.Tensor], prior.default_log_prior
+        )
         self._minimum_noise: float = prior.DEFAULT_MINIMUM_NOISE_VAR
         # We cache the kernel parameters for initial values of fitting the next time.
         # TODO(nabenabe): Make the cache lists system_attrs to make GPSampler stateless.
@@ -180,7 +191,8 @@ def _optimize_acqf(
         # Particularly, we may remove this function in future refactoring.
         assert best_params is None or len(best_params.shape) == 2
         normalized_params, _acqf_val = optim_mixed.optimize_acqf_mixed(
-            acqf,
+            # We assume acqf_module.BaseAcquisitionFunc is compatible with optuna._gp.acqf.BaseAcquisitionFunc
+            cast(optuna._gp.acqf.BaseAcquisitionFunc, acqf),
             warmstart_normalized_params_array=best_params,
             n_preliminary_samples=self._n_preliminary_samples,
             n_local_search=self._n_local_search,
@@ -267,7 +279,7 @@ def _get_scaled_input_noise_params(
                 scaled_input_noise_params[i] = input_noise_param / (dist.high - dist.low)
             return scaled_input_noise_params
 
-        noise_kwargs: dict[str, torch.Tensor] = {}
+        noise_kwargs: _NoiseKWArgs = {}
         if self._uniform_input_noise_rads is not None:
             scaled_input_noise_params = _get_scaled_input_noise_params(
                 self._uniform_input_noise_rads, "uniform_input_noise_rads"
diff --git a/package/samplers/value_at_risk/tests/test_samplers.py b/package/samplers/value_at_risk/tests/test_samplers.py
index 727467c5..4a9e46a6 100644
--- a/package/samplers/value_at_risk/tests/test_samplers.py
+++ b/package/samplers/value_at_risk/tests/test_samplers.py
@@ -280,8 +280,8 @@ def sample() -> float:
 @parametrize_noise_type
 def test_sample_relative_numerical(
     relative_sampler_class: Callable[[], BaseSampler],
-    x_distribution: BaseDistribution,
-    y_distribution: BaseDistribution,
+    x_distribution: FloatDistribution | IntDistribution,
+    y_distribution: FloatDistribution | IntDistribution,
     noise_type: str,
 ) -> None:
     can_x_be_noisy = x_distribution.step is None and not x_distribution.log