diff --git a/blackjax/util.py b/blackjax/util.py
index d761189cf..b5bd1a89e 100644
--- a/blackjax/util.py
+++ b/blackjax/util.py
@@ -202,9 +202,12 @@ def one_step(state, xs):
 
     if progress_bar:
         one_step = progress_bar_scan(num_steps)(one_step)
+        xs = jnp.arange(num_steps), keys
+        final_state, history = lax.scan(one_step, (initial_state, -1), xs)
+    else:
+        xs = jnp.arange(num_steps), keys
+        final_state, history = lax.scan(one_step, initial_state, xs)
 
-    xs = jnp.arange(num_steps), keys
-    final_state, history = lax.scan(one_step, initial_state, xs)
     return final_state, history
 
 
@@ -212,6 +215,7 @@ def store_only_expectation_values(
     sampling_algorithm,
     state_transform=lambda x: x,
     incremental_value_transform=lambda x: x,
+    burn_in=0,
 ):
     """Takes a sampling algorithm and constructs from it a new sampling algorithm object. The new sampling algorithm has the same
      kernel but only stores the streaming expectation values of some observables, not the full states; to save memory.
@@ -262,8 +266,14 @@ def update_fn(rng_key, state_and_incremental_val):
             rng_key, state
         )  # update the state with the sampling algorithm
         averaging_state = incremental_value_update(
-            state_transform(state), averaging_state
-        )  # update the expectation value with the running average
+            state_transform(state),
+            averaging_state,
+            weight=(
+                averaging_state[0] >= burn_in
+            ),  # If we want to eliminate some number of steps as a burn-in
+            zero_prevention=1e-10 * (burn_in > 0),
+        )
+        # update the expectation value with the running average
         return (state, averaging_state), info
 
     def transform(state_and_incremental_val, info):