diff --git a/aemcmc/basic.py b/aemcmc/basic.py
index d69fbf8..2a568a5 100644
--- a/aemcmc/basic.py
+++ b/aemcmc/basic.py
@@ -127,6 +127,6 @@ def construct_sampler(
         if rv not in obs_rvs_to_values
     }
 
-    return Sampler(sampling_steps, updates, parameters), {
+    return Sampler(sampling_steps, posterior_updates, parameters), {
         new_to_old_rvs[rv]: init_var for rv, init_var in rvs_to_init_vals.items()
     }
diff --git a/aemcmc/rewriting.py b/aemcmc/rewriting.py
index 9039440..81befe5 100644
--- a/aemcmc/rewriting.py
+++ b/aemcmc/rewriting.py
@@ -5,7 +5,7 @@
 from aeppl.rewriting import MeasurableConversionTracker
 from aesara.compile.builders import OpFromGraph
 from aesara.compile.mode import optdb
-from aesara.graph.basic import Apply, Variable, clone_replace, io_toposort
+from aesara.graph.basic import Apply, Constant, Variable, clone_replace, io_toposort
 from aesara.graph.features import AlreadyThere, Feature
 from aesara.graph.fg import FunctionGraph
 from aesara.graph.op import Op
@@ -186,10 +186,28 @@ def __init__(self, inputs, outputs, *args, **kwargs):
         # self.destroy_map = self.elemwise_op.destroy_map
         self.ufunc = None
         self.nfunc = None
-        OpFromGraph.__init__(self, inputs, outputs, *args, **kwargs)
+
+        used_inputs = [inp for inp in inputs if not isinstance(inp, Constant)]
+
+        OpFromGraph.__init__(self, used_inputs, outputs, *args, **kwargs)
 
     def make_node(self, *inputs):
-        node = super().make_node(*inputs)
+        # Remove constants
+        used_inputs = [inp for inp in inputs if not isinstance(inp, Constant)]
+
+        # TODO: We could make sure that the new constant inputs correspond to
+        # the originals...
+
+        # The user interface doesn't expect the shared variable inputs of the
+        # inner-graph, but, since `Op.make_node` does (and `Op.__call__`
+        # dispatches to `Op.make_node`), we need to compensate here
+        num_expected_inps = len(self.inner_inputs) - len(self.shared_inputs)
+
+        if len(used_inputs) == num_expected_inps:
+            used_inputs = used_inputs + self.shared_inputs
+
+        node = super().make_node(*used_inputs)
+
         # Remove shared variable inputs.  We aren't going to compute anything
         # with this `Op`, so they're not needed
         real_inputs = node.inputs[: len(node.inputs) - len(self.shared_inputs)]
@@ -346,8 +364,6 @@ def local_elemwise_dimshuffle_subsume(fgraph, node):
 
     new_out = new_op(*new_inputs)
 
-    assert len(new_out.owner.inputs) == len(node.inputs)
-
     return new_out.owner.outputs
 
 
diff --git a/tests/test_rewriting.py b/tests/test_rewriting.py
index ef07ba4..f616224 100644
--- a/tests/test_rewriting.py
+++ b/tests/test_rewriting.py
@@ -6,7 +6,11 @@
 from etuples import etuple, etuplize
 from unification import unify
 
-from aemcmc.rewriting import SubsumingElemwise, local_elemwise_dimshuffle_subsume
+from aemcmc.rewriting import (
+    SubsumingElemwise,
+    construct_ir_fgraph,
+    local_elemwise_dimshuffle_subsume,
+)
 
 
 def test_SubsumingElemwise_basics():
@@ -107,3 +111,23 @@ def test_local_elemwise_dimshuffle_subsume_transpose():
     # The input corresponding to `b`/`b_ds` should be equivalent to `b.T`
     assert isinstance(res.owner.inputs[1].owner.op, DimShuffle)
     assert equal_computations([b.T], [res.owner.inputs[1]])
+
+
+def test_SubsumingElemwise_constant_inputs():
+    """Make sure constant inputs are handled correctly by `SubsumingElemwise`."""
+
+    srng = at.random.RandomStream(0)
+
+    s = at.lscalar("s")
+    # The `1` is the constant input to a `true_div` `Elemwise` that should be
+    # "subsumed"
+    mu = 1 / srng.exponential(1, size=s, name="Z")
+    Y = srng.normal(mu, name="Y")
+    y = Y.clone()
+    y.name = "y"
+
+    res, *_ = construct_ir_fgraph({Y: y})
+
+    normal_node = res.outputs[1].owner
+    subelem_node = normal_node.inputs[3].owner
+    assert isinstance(subelem_node.op, SubsumingElemwise)