diff --git a/pymc/logprob/scan.py b/pymc/logprob/scan.py
index d9eb85a838..6eed6a1b35 100644
--- a/pymc/logprob/scan.py
+++ b/pymc/logprob/scan.py
@@ -50,13 +50,20 @@
 from pytensor.scan.rewriting import scan_eqopt1, scan_eqopt2
 from pytensor.scan.utils import ScanArgs
 from pytensor.tensor.random.type import RandomType
+from pytensor.tensor.rewriting.shape import ShapeFeature
 from pytensor.tensor.subtensor import Subtensor, indices_from_subtensor
 from pytensor.tensor.var import TensorVariable
 from pytensor.updates import OrderedUpdates
 
-from pymc.logprob.abstract import MeasurableVariable, _get_measurable_outputs, _logprob
+from pymc.logprob.abstract import (
+    MeasurableVariable,
+    _get_measurable_outputs,
+    _logprob,
+    get_measurable_outputs,
+)
 from pymc.logprob.joint_logprob import factorized_joint_logprob
 from pymc.logprob.rewriting import (
+    PreserveRVMappings,
     inc_subtensor_ops,
     logprob_rewrites_db,
     measurable_ir_rewrites_db,
@@ -66,6 +73,9 @@
 class MeasurableScan(Scan):
     """A placeholder used to specify a log-likelihood for a scan sub-graph."""
 
+    def __str__(self):
+        return f"Measurable({super().__str__()})"
+
 
 MeasurableVariable.register(MeasurableScan)
 
@@ -359,6 +369,12 @@ def find_measurable_scans(fgraph, node):
     )
     for n in local_fgraph_topo:
         if isinstance(n.op, MeasurableVariable):
+            measurable_outputs = get_measurable_outputs(n.op, n)
+            # This variable's source of measure is used by another inner node,
+            # So we don't need it to be an output!
+            if not measurable_outputs:
+                continue
+
             non_output_node_clients = [
                 c for c in clients[n] if c not in curr_scanargs.inner_outputs
             ]
@@ -494,6 +510,10 @@ def add_opts_to_inner_graphs(fgraph, node):
         clone=True,
         copy_inputs=False,
         copy_orphans=False,
+        features=[
+            ShapeFeature(),
+            PreserveRVMappings({}),
+        ],
     )
 
     logprob_rewrites_db.query(RewriteDatabaseQuery(include=["basic"])).rewrite(inner_fgraph)
diff --git a/tests/logprob/test_scan.py b/tests/logprob/test_scan.py
index 0b4577b1a9..551bb51471 100644
--- a/tests/logprob/test_scan.py
+++ b/tests/logprob/test_scan.py
@@ -42,9 +42,10 @@
 from pytensor import Mode
 from pytensor.raise_op import assert_op
 from pytensor.scan.utils import ScanArgs
+from scipy import stats
 
 from pymc.logprob.abstract import logprob
-from pymc.logprob.joint_logprob import factorized_joint_logprob
+from pymc.logprob.joint_logprob import factorized_joint_logprob, logp
 from pymc.logprob.scan import (
     construct_scan,
     convert_outer_out_to_in,
@@ -458,3 +459,22 @@ def test_mode_is_kept(remove_asserts):
     else:
         with pytest.raises(AssertionError):
             x_logp(x=x_test_val)
+
+
+def test_scan_non_pure_rv_output():
+    grw, _ = pytensor.scan(
+        fn=lambda xtm1: at.random.normal() + xtm1,
+        outputs_info=[at.zeros(())],
+        n_steps=10,
+        name="grw",
+    )
+
+    grw_vv = grw.clone()
+    grw_logp = logp(grw, grw_vv)
+    assert_no_rvs(grw_logp)
+
+    grw_vv_test = np.arange(10) + 1
+    np.testing.assert_array_almost_equal(
+        grw_logp.eval({grw_vv: grw_vv_test}),
+        stats.norm.logpdf(np.ones(10)),
+    )