diff --git a/.github/workflows/main.yml b/.github/workflows/main.yml
index c4fda92..c39c51b 100644
--- a/.github/workflows/main.yml
+++ b/.github/workflows/main.yml
@@ -226,7 +226,7 @@ jobs:
       - name: nominal fit
         run: >- 
           rabbit_fit.py $RABBIT_OUTDIR/test_tensor.hdf5 -o $RABBIT_OUTDIR/ --postfix composite
-          -t 0 --unblind '.*' --doImpacts --globalImpacts
+          -t -1 0 --unblind '.*' --doImpacts --globalImpacts
           --saveHists --saveHistsPerProcess --computeHistErrors --computeHistErrorsPerProcess
           --computeHistCov --computeHistImpacts --computeVariations
           --compositeMapping -m Project ch1 a -m Project ch1 b 
@@ -234,12 +234,12 @@ jobs:
       - name: nominal fit blinded
         run: >- 
           rabbit_fit.py $RABBIT_OUTDIR/test_tensor.hdf5 -o $RABBIT_OUTDIR/
-          -t 0 --postfix blinded --doImpacts --globalImpacts
+          -t 0 --postfix blinded --setConstraintMinimum bkg_2_norm 0.5 --doImpacts --globalImpacts
 
       - name: sparse tensor fit 
         run: >- 
           rabbit_fit.py $RABBIT_OUTDIR/test_tensor_sparse.hdf5 -o $RABBIT_OUTDIR/ --postfix sparse
-          -t -0 --noBinByBinStat --doImpacts --globalImpacts --computeVariations
+          -t 0 --noBinByBinStat --doImpacts --globalImpacts --computeVariations
           --saveHists --saveHistsPerProcess --computeHistErrors --computeHistErrorsPerProcess
           -m Project ch1 a -m Project ch1 b
 
diff --git a/rabbit/fitter.py b/rabbit/fitter.py
index faba62a..22a5e11 100644
--- a/rabbit/fitter.py
+++ b/rabbit/fitter.py
@@ -141,12 +141,27 @@ def __init__(
 
         self.parms = np.concatenate([self.poi_model.pois, self.indata.systs])
 
+        # tf tensor containing default constraint minima
+        theta0default = np.zeros(self.indata.nsyst)
+        for parm, val in options.setConstraintMinimum:
+            idx = np.where(self.indata.systs.astype(str) == parm)[0]
+            if len(idx) != 1:
+                raise RuntimeError(
+                    f"Expect to find exactly one match for {parm} to set constraint minimum, but found {len(idx)}"
+                )
+            theta0default[idx[0]] = val
+
+        self.theta0default = tf.convert_to_tensor(
+            theta0default, dtype=self.indata.dtype
+        )
+
         # tf variable containing all fit parameters
-        thetadefault = tf.zeros([self.indata.nsyst], dtype=self.indata.dtype)
         if self.poi_model.npoi > 0:
-            xdefault = tf.concat([self.poi_model.xpoidefault, thetadefault], axis=0)
+            xdefault = tf.concat(
+                [self.poi_model.xpoidefault, self.theta0default], axis=0
+            )
         else:
-            xdefault = thetadefault
+            xdefault = self.theta0default
 
         self.x = tf.Variable(xdefault, trainable=True, name="x")
 
@@ -186,7 +201,7 @@ def __init__(
 
         # constraint minima for nuisance parameters
         self.theta0 = tf.Variable(
-            tf.zeros([self.indata.nsyst], dtype=self.indata.dtype),
+            self.theta0default,
             trainable=False,
             name="theta0",
         )
@@ -475,7 +490,7 @@ def set_beta0(self, values):
         self.logbeta0.assign(tf.math.log(beta0safe))
 
     def theta0defaultassign(self):
-        self.theta0.assign(tf.zeros([self.indata.nsyst], dtype=self.theta0.dtype))
+        self.theta0.assign(self.theta0default)
 
     def xdefaultassign(self):
         if self.poi_model.npoi == 0:
diff --git a/rabbit/parsing.py b/rabbit/parsing.py
index 24bf82c..c2dca87 100644
--- a/rabbit/parsing.py
+++ b/rabbit/parsing.py
@@ -93,6 +93,13 @@ def common_parser():
         E.g. use '--unblind ^signal$' to unblind a parameter named signal or '--unblind' to unblind all.
         """,
     )
+    parser.add_argument(
+        "--setConstraintMinimum",
+        default=[],
+        nargs=2,
+        action="append",
+        help="Set the constraint minima of specified parameter to specified value",
+    )
     parser.add_argument(
         "--freezeParameters",
         type=str,