diff --git a/.gitignore b/.gitignore
index 2b1aa21f..f5571df5 100644
--- a/.gitignore
+++ b/.gitignore
@@ -9,7 +9,6 @@ tf_pwa/version.py
 *.npy
 *.json
 *.log
-*.yml
 *.png
 *.pdf
 *.pptx
diff --git a/examples/ex2_particle_amplitude.py b/examples/ex2_particle_amplitude.py
index eb2b3444..3061508d 100644
--- a/examples/ex2_particle_amplitude.py
+++ b/examples/ex2_particle_amplitude.py
@@ -15,8 +15,11 @@
     "R1": {"J": 0, "P": 1, "mass": 1.0, "width": 0.07},
     "R2": {"J": 1, "P": -1, "mass": 1.225, "width": 0.08},
 }
-
-a, b, c, d = [get_particle(i, J=0, P=-1) for i in "ABCD"]
+m_A, m_B, m_C, m_D = 1.8, 0.18, 0.18, 0.18
+a, b, c, d = [
+    get_particle(i, J=0, P=-1, mass=m)
+    for i, m in zip("ABCD", [m_A, m_B, m_C, m_D])
+]
 r1, r2, r3 = [get_particle(i, **resonances[i]) for i in resonances.keys()]
 
 
diff --git a/examples/ex_params_trans.py b/examples/ex_params_trans.py
index 8fe0ba7f..c83fca46 100644
--- a/examples/ex_params_trans.py
+++ b/examples/ex_params_trans.py
@@ -78,23 +78,52 @@
 print(a2_x.numpy(), pt.get_error(a2_x).numpy())
 
 # %%
-# We can also calculate some more complex examples, such as the ratio in mass range (0.75, 0.85) over full phace space.
+# Uncertainties of fit fractions
+# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+# We can also calculate some more complex examples, such as the fit fractions of all in `C+D`.
 # Even further, we can get the error of error in the meaning of error propagation.
 
-from tf_pwa.data import data_mask
-
-m_R2 = phsp.get_mass("(B, D)")
-cut_cond = (m_R2 < 0.85) & (m_R2 > 0.75)
-
 amp = config.get_amplitude()
 
 with config.params_trans() as pt1:
     with config.params_trans() as pt:
         int_mc = tf.reduce_sum(amp(phsp))
-        cut_phsp = data_mask(phsp, cut_cond)
-        cut_int_mc = tf.reduce_sum(amp(cut_phsp))
-        ratio = cut_int_mc / int_mc
+        with amp.temp_used_res(["R1_a", "R1_b"]):
+            part_int_mc = tf.reduce_sum(amp(phsp))
+        ratio = part_int_mc / int_mc
     error = pt.get_error(ratio)
 
 print(ratio.numpy(), "+/-", error.numpy())
 print(error.numpy(), "+/-", pt1.get_error(error).numpy())
+
+# %%
+# For large data size it would be some problem named OOM (out of memory).
+# TFPWA provide `vm.batch_sum_var` to do sum of large samples
+
+int_mc_v = config.vm.batch_sum_var(amp, phsp, batch=5000)
+
+with amp.temp_used_res(["R1_a", "R1_b"]):
+    part_int_mc_v = config.vm.batch_sum_var(amp, phsp, batch=5000)
+
+# %%
+# It will store the pre-calculated gradients as
+
+print(int_mc_v.grad, part_int_mc_v.grad)
+
+# %%
+# Then, we can use it as a function to do error propagation:
+
+with config.params_trans() as pt:
+    ratio = part_int_mc_v() / int_mc_v()
+error = pt.get_error(ratio)
+
+print(ratio.numpy(), "+/-", error.numpy())
+
+# %%
+# Besides the error propagation, there would be some additional uncertainties.
+# For example, the uncertainty from the integration sample size is often defined as the sum of square as
+
+with amp.temp_used_res(["R1_a", "R1_b"]):
+    int_square = tf.reduce_sum((amp(phsp) / int_mc) ** 2)
+
+print(ratio.numpy(), "+/-", error.numpy(), "+/-", tf.sqrt(int_square).numpy())
diff --git a/fit.py b/fit.py
index 45bc90b3..6c119572 100644
--- a/fit.py
+++ b/fit.py
@@ -55,7 +55,7 @@ def print_fit_result_roofit(config, fit_result):
     value = fit_result.params
     params_name = config.vm.trainable_vars
     n_par = len(params_name)
-    name_size = max(len(i) for i in params_name)
+    name_size = max([5] + [len(i) for i in params_name])
     # fcn = config.get_fcn()
     # _, grad = fcn.nll_grad(fit_result.params)
     # edm = np.dot(np.dot(config.inv_he, grad), grad)
diff --git a/tf_pwa/amp/core.py b/tf_pwa/amp/core.py
index ad303c63..109c727d 100644
--- a/tf_pwa/amp/core.py
+++ b/tf_pwa/amp/core.py
@@ -259,6 +259,11 @@ def add_var(self, names, is_complex=False, shape=(), **kwargs):
         """
         if not hasattr(self, "_variables_map"):
             self._variables_map = {}
+        if True:
+            default_config = getattr(self, "default_params", {}).get(names, {})
+            if isinstance(default_config, (float, int)):
+                default_config = {"value": default_config}
+            kwargs.update(default_config)
         name = self.get_variable_name(names)
         var = Variable(name, shape, is_complex, **kwargs)
         self._variables_map[names] = var
diff --git a/tf_pwa/applications.py b/tf_pwa/applications.py
index b9ebc871..86a27b90 100644
--- a/tf_pwa/applications.py
+++ b/tf_pwa/applications.py
@@ -48,17 +48,24 @@ def fit_fractions(
     :param mcdata: MCdata array.
     :param inv_he: The inverse of Hessian matrix. If it's not given, the errors will not be calculated.
     :return frac: Dictionary of fit fractions for each resonance.
-    :return err_frac: Dictionary of their errors. If ``inv_he`` is ``None``, it will be a dictionary of ``None``.
+    :return err_frac: Dictionary of their errors. If ``inv_he`` is ``None``, it will be a dictionary of ``{}``.
     """
     if params is None:
         params = {}
     err_frac = {}
     if method == "old":
-        with amp.temp_params(params):
-            frac, grad = cal_fitfractions(amp, mcdata, res=res, batch=batch)
         if inv_he is not None:
+            with amp.temp_params(params):
+                frac, grad = cal_fitfractions(
+                    amp, mcdata, res=res, batch=batch
+                )
             for i in frac:
                 err_frac[i] = np.sqrt(np.dot(np.dot(inv_he, grad[i]), grad[i]))
+        else:
+            with amp.temp_params(params):
+                frac = cal_fitfractions_no_grad(
+                    amp, mcdata, res=res, batch=batch
+                )
         return frac, err_frac
     else:
         ret = FitFractions(amp, res)
@@ -289,6 +296,8 @@ def cal_hesse_error(
     :return hesse_error: List of errors.
     :return inv_he: The inverse Hessian matrix.
     """
+    if len(fcn.vm.trainable_vars) == 0:
+        return {}, None
     t = time.time()
     nll, g, h = fcn.nll_grad_hessian(
         params
diff --git a/tf_pwa/config_loader/config_loader.py b/tf_pwa/config_loader/config_loader.py
index 1c14f4f2..020c38bf 100644
--- a/tf_pwa/config_loader/config_loader.py
+++ b/tf_pwa/config_loader/config_loader.py
@@ -262,8 +262,6 @@ def add_constraints(self, amp):
             constrains = {}
         self.add_decay_constraints(amp, constrains.get("decay", {}))
         self.add_particle_constraints(amp, constrains.get("particle", {}))
-        self.add_fix_var_constraints(amp, constrains.get("fix_var", {}))
-        self.add_free_var_constraints(amp, constrains.get("free_var", []))
         self.add_var_range_constraints(amp, constrains.get("var_range", {}))
         self.add_var_equal_constraints(amp, constrains.get("var_equal", []))
         self.add_pre_trans_constraints(amp, constrains.get("pre_trans", None))
@@ -273,6 +271,8 @@ def add_constraints(self, amp):
         self.add_gauss_constr_constraints(
             amp, constrains.get("gauss_constr", {})
         )
+        self.add_fix_var_constraints(amp, constrains.get("fix_var", {}))
+        self.add_free_var_constraints(amp, constrains.get("free_var", []))
         for k, v in self.extra_constrains.items():
             v(amp, constrains.get(k, {}))
 
@@ -832,8 +832,8 @@ def get_params_error(
             params = {}
         if correct_params is None:
             correct_params = []
-            if method is None:
-                method = "correct"
+        if len(correct_params) > 0 and method is None:
+            method = "correct"
         if hasattr(params, "params"):
             params = getattr(params, "params")
         if not using_cached:
@@ -1016,10 +1016,10 @@ def set_params(self, params, neglect_params=None):
             neglect_params = self._neglect_when_set_params
         if len(neglect_params) != 0:
             for v in params:
-                if v in self._neglect_when_set_params:
+                if v in neglect_params:
                     warnings.warn(
                         "Neglect {} when setting params.".format(
-                            neglect_params
+                            [i for i in params if i in neglect_params]
                         )
                     )
                     del ret[v]
diff --git a/tf_pwa/data_trans/helicity_angle.py b/tf_pwa/data_trans/helicity_angle.py
index 2d67ec90..219532fc 100644
--- a/tf_pwa/data_trans/helicity_angle.py
+++ b/tf_pwa/data_trans/helicity_angle.py
@@ -3,6 +3,7 @@
 
 from tf_pwa.amp import get_relative_p
 from tf_pwa.angle import LorentzVector as lv
+from tf_pwa.data import data_index
 
 
 class HelicityAngle1:
@@ -73,7 +74,9 @@ def get_all_mass(self, replace_mass):
         for i in self.decay_chain:
             for j in [i.core] + list(i.outs):
                 if j not in ms:
-                    if str(j) in replace_mass:
+                    if j in replace_mass:
+                        ms[j] = replace_mass[j]
+                    elif str(j) in replace_mass:
                         ms[j] = tf.convert_to_tensor(
                             replace_mass[str(j)], tf.float64
                         )
@@ -90,7 +93,9 @@ def generate_p_mass(self, name, m, random=False):
         for i in self.decay_chain:
             data[i] = {}
             data[i]["|p|"] = get_relative_p(
-                ms[i.core], ms[i.outs[0]], ms[i.outs[1]]
+                data_index(ms, i.core),
+                data_index(ms, i.outs[0]),
+                data_index(ms, i.outs[1]),
             )
             if random:
                 costheta = np.random.random(m.shape) * 2 - 1
@@ -108,10 +113,13 @@ def generate_p_mass(self, name, m, random=False):
     def build_data(self, ms, costheta, phi):
         """generate monmentum with M_name = m"""
         data = {}
+        ms = self.get_all_mass(ms)
         for j, i in enumerate(self.decay_chain):
             data[i] = {}
             data[i]["|p|"] = get_relative_p(
-                ms[i.core], ms[i.outs[0]], ms[i.outs[1]]
+                data_index(ms, i.core),
+                data_index(ms, i.outs[0]),
+                data_index(ms, i.outs[1]),
             )
             costheta_i = costheta[j]
             phi_i = phi[j]
diff --git a/tf_pwa/fit.py b/tf_pwa/fit.py
index 6fc0734a..9e6496a6 100644
--- a/tf_pwa/fit.py
+++ b/tf_pwa/fit.py
@@ -278,7 +278,12 @@ def callback(x):
         f_g = Cached_FG(f_g, grad_scale=grad_scale)
         # print(f_g)
         x0 = np.array(fcn.vm.get_all_val(True))
-        # print(x0, fcn.vm.get_all_dic())
+        if len(x0) == 0:
+            min_nll, _ = f_g(x0)
+            params = fcn.get_params()
+            return FitResult(
+                params, fcn, min_nll, ndf=0, success=True, hess_inv=None
+            )
         # s = minimize(f_g, x0, method='trust-constr', jac=True, hess=BFGS(), options={'gtol': 1e-4, 'disp': True})
         if method == "test":
             try:
diff --git a/tf_pwa/fitfractions.py b/tf_pwa/fitfractions.py
index 993aa470..cd472f5c 100644
--- a/tf_pwa/fitfractions.py
+++ b/tf_pwa/fitfractions.py
@@ -6,20 +6,32 @@
 from tf_pwa.data import LazyCall, data_split
 
 
-def eval_integral(
+def _eval_integral(
     f, data, var, weight=None, args=(), no_grad=False, kwargs=None
 ):
     kwargs = {} if kwargs is None else kwargs
     weight = 1.0 if weight is None else weight
     if no_grad:
-        ret = tf.reduce_sum(f(data, *args, **kwargs) * weight)
+        fx = f(data, *args, **kwargs) * weight
+        ret = tf.reduce_sum(fx)
         ret_grad = np.zeros((len(var),))
     else:
         with tf.GradientTape() as tape:
-            ret = tf.reduce_sum(f(data, *args, **kwargs) * weight)
+            fx = f(data, *args, **kwargs) * weight
+            ret = tf.reduce_sum(fx)
         ret_grad = tape.gradient(ret, var, unconnected_gradients="zero")
-        ret_grad = np.stack([i.numpy() for i in ret_grad])
-    return ret.numpy(), ret_grad
+        if len(ret_grad) == 0:
+            ret_grad = np.array([])
+        else:
+            ret_grad = np.stack([i.numpy() for i in ret_grad])
+    int_square = tf.reduce_sum(fx**2)
+    return ret.numpy(), ret_grad, int_square.numpy()
+
+
+def force_list(x):
+    if isinstance(x, (list, tuple)):
+        return x
+    return [x]
 
 
 class FitFractions:
@@ -30,6 +42,7 @@ def __init__(self, amp, res):
         self.res = res
         self.cached_int = {}
         self.cached_grad = {}
+        self.cached_square = {}
         self.cached_int_total = 0.0
         self.cached_grad_total = np.zeros((self.n_var,))
         self.error_matrix = np.diag(np.zeros((self.n_var,)))
@@ -46,6 +59,7 @@ def init_res_table(self):
                 else:
                     name = (str(self.res[i]), str(self.res[j]))
                 self.cached_int[name] = 0.0
+                self.cached_square[name] = 0.0
                 self.cached_grad[name] = np.zeros_like((self.n_var,))
 
     def integral(self, mcdata, *args, batch=None, no_grad=False, **kwargs):
@@ -62,7 +76,7 @@ def append_int(self, mcdata, *args, weight=None, no_grad=False, **kwargs):
             mcdata = mcdata.eval()
         if weight is None:
             weight = mcdata.get("weight", 1.0)
-        int_mc, g_int_mc = eval_integral(
+        int_mc, g_int_mc, _ = _eval_integral(
             self.amp,
             mcdata,
             var=self.var,
@@ -74,15 +88,16 @@ def append_int(self, mcdata, *args, weight=None, no_grad=False, **kwargs):
         self.cached_grad_total += g_int_mc
         cahced_res = self.amp.used_res
         amp_tmp = self.amp
+        fl = force_list
         for i in range(len(self.res)):
             for j in range(i, -1, -1):
                 if i == j:
                     name = str(self.res[i])
-                    amp_tmp.set_used_res([self.res[i]])
+                    amp_tmp.set_used_res(fl(self.res[i]))
                 else:
                     name = (str(self.res[i]), str(self.res[j]))
-                    amp_tmp.set_used_res([self.res[i], self.res[j]])
-                int_tmp, g_int_tmp = eval_integral(
+                    amp_tmp.set_used_res(fl(self.res[i]) + fl(self.res[j]))
+                int_tmp, g_int_tmp, int_square = _eval_integral(
                     amp_tmp,
                     mcdata,
                     var=self.var,
@@ -92,6 +107,9 @@ def append_int(self, mcdata, *args, weight=None, no_grad=False, **kwargs):
                 )
                 self.cached_int[name] = self.cached_int[name] + int_tmp
                 self.cached_grad[name] = self.cached_grad[name] + g_int_tmp
+                self.cached_square[name] = (
+                    self.cached_square[name] + int_square
+                )
 
         self.amp.set_used_res(cahced_res)
 
@@ -130,10 +148,9 @@ def get_frac_grad(self, sum_diag=True):
             g_fit_frac["sum_diag"] = sum(
                 [g_fit_frac[str(i)] for i in self.res]
             )
-        print(fit_frac)
         return fit_frac, g_fit_frac
 
-    def get_frac(self, error_matrix=None, sum_diag=True):
+    def get_frac(self, error_matrix=None, sum_diag=True, add_int_error=False):
         if error_matrix is None:
             error_matrix = self.error_matrix
         fit_frac, g_fit_frac = self.get_frac_grad(sum_diag=sum_diag)
@@ -143,6 +160,9 @@ def get_frac(self, error_matrix=None, sum_diag=True):
         for k, v in g_fit_frac.items():
             e = np.sqrt(np.dot(np.dot(error_matrix, v), v))
             fit_frac_err[k] = e
+            if add_int_error and k in self.cached_square:
+                scale = 1 / self.cached_int_total**2
+                fit_frac_err[k] = np.sqrt(e**2 + self.cached_square[k] * scale)
         return fit_frac, fit_frac_err
 
     def __iter__(self):
diff --git a/tf_pwa/tests/config_no_params.yml b/tf_pwa/tests/config_no_params.yml
new file mode 100644
index 00000000..fc04a00f
--- /dev/null
+++ b/tf_pwa/tests/config_no_params.yml
@@ -0,0 +1,49 @@
+data:
+  dat_order: [B, C, D]
+  extra_var:
+    c:
+      default: 1
+  data: [["toy_data/data.dat"]]
+  bg: [["toy_data/bg.dat"]]
+  phsp: [["toy_data/PHSP.dat"]]
+  phsp_c: ["toy_data/phsp_eff_value.dat"]
+  bg_weight: [0.1]
+  random_z: False
+  r_boost: False
+  cached_amp: True
+  use_mix_likelihood: True
+
+decay:
+  A:
+    - [R_BC, D, ls_list: [[0, 1], [2, 1]]]
+    - [R_BD, C, l_list: [0, 2]]
+    - [R_CD, B]
+  R_BC: [B, C]
+  R_BD: [B, D]
+  R_CD: [C, D]
+
+particle:
+  $top:
+    A: { J: 1, P: -1, spins: [-1, 1] }
+  $finals:
+    B: { J: 1, P: -1 }
+    C: { J: 1, P: -1 }
+    D: { J: 0, P: -1 }
+  R_BC: [] # { J: 1, Par: 1, m0: 4.16, g0: 0.1, model: BWR2 }
+  R_BD: { J: 1, Par: 1, m0: 2.43, g0: 0.3, model: BW }
+  R_CD: [] # { J: 1, Par: 1, m0: 2.42, g0: 0.03 }
+
+constrains:
+  fix_var:
+    A->R_BD.C_g_ls_1r: 0.0
+    A->R_BD.C_g_ls_1i: 0.0
+    A->R_BD.C_g_ls_2r: 0.0
+    A->R_BD.C_g_ls_2i: 0.0
+    R_BD->B.D_g_ls_1r: 0.0
+    R_BD->B.D_g_ls_1i: 0.0
+
+plot:
+  mass:
+    R_BC: { display: "$M_{BC}$" }
+    R_BD: { display: "$M_{BD}$" }
+    R_CD: { display: "$M_{CD}$" }
diff --git a/tf_pwa/tests/test_full.py b/tf_pwa/tests/test_full.py
index e64f49f0..35f8ec64 100644
--- a/tf_pwa/tests/test_full.py
+++ b/tf_pwa/tests/test_full.py
@@ -144,6 +144,14 @@ def test_save_model(toy_config):
     config.save_tensorflow_model("toy_data/model")
 
 
+def test_fit_no_params(gen_toy):
+    config = ConfigLoader(f"{this_dir}/config_no_params.yml")
+    config.fit()
+    config.get_params_error(method="default")
+    config.cal_fitfractions(method="old")
+    config.cal_fitfractions(method="new")
+
+
 def test_cfit(gen_toy):
     config = ConfigLoader(f"{this_dir}/config_cfit.yml")
     config.set_params(f"{this_dir}/gen_params.json")
@@ -333,6 +341,9 @@ def pw_f(x, **kwargs):
     fit_result.save_as("toy_data/final_params.json")
     fit_frac, frac_err = toy_config.cal_fitfractions()
     fit_frac, frac_err = toy_config.cal_fitfractions(method="new")
+    fit_frac_obj = toy_config.cal_fitfractions(
+        method="new", res=["R_BC", ["R_BD", "R_CD"]]
+    )
     save_frac_csv("toy_data/fit_frac.csv", fit_frac)
 
     with toy_config.params_trans() as pt: