diff --git a/coix/api.py b/coix/api.py
index b1018c3..861b9a4 100644
--- a/coix/api.py
+++ b/coix/api.py
@@ -124,9 +124,6 @@ def propose(p, q, *, loss_fn=None, detach=False, chain=False):
   additional batch dimensions to the whole program by using `vmap`, e.g.
   `vmap(propose(p, q))`.
 
-  Note: We assume superfluous variables, which appear in `q` but not in `p`,
-  implicitly follow Delta distribution in `p`.
-
   Args:
     p: a target program
     q: a proposal program
@@ -179,10 +176,10 @@ def wrapped(*args, **kwargs):
         for name, lp in p_log_probs.items()
         if util.is_observed_site(p_trace[name]) or (name in q_trace)
     )
-    # Note: We include superfluous variables, whose `name in p_trace`.
     q_log_weight = sum(
         lp.reshape(lp.shape[:batch_ndims] + (-1,)).sum(-1)
-        for lp in q_log_probs.values()
+        for name, lp in q_log_probs.items()
+        if util.is_observed_site(q_trace[name]) or (name in p_trace)
     )
     incremental_log_weight = p_log_weight - q_log_weight
     log_weight = in_log_weight + incremental_log_weight
diff --git a/coix/loss.py b/coix/loss.py
index bc7fda4..6d4dc46 100644
--- a/coix/loss.py
+++ b/coix/loss.py
@@ -115,29 +115,17 @@ def elbo_loss(q_trace, p_trace, incoming_log_weight, incremental_log_weight):
   return loss
 
 
-def _proposal_and_target_sites(q_trace, p_trace):
-  """Gets current proposal sites and current target sites."""
-  proposal_sites = []
-  target_sites = []
-  for name in p_trace:
-    if not name.endswith("_PREV_"):
-      target_sites.append(name)
-      if name in q_trace:
-        while name + "_PREV_" in q_trace:
-          name += "_PREV_"
-        proposal_sites.append(name)
-  if not any(name.endswith("_PREV_") for name in proposal_sites):
-    proposal_sites = []
-  return proposal_sites, target_sites
-
-
 def fkl_loss(q_trace, p_trace, incoming_log_weight, incremental_log_weight):
-  """Forward KL objective."""
+  """Forward KL objective. Here we do not optimize p."""
   batch_ndims = incoming_log_weight.ndim
   q_log_probs = {
       name: util.get_site_log_prob(site) for name, site in q_trace.items()
   }
-  proposal_sites, _ = _proposal_and_target_sites(q_trace, p_trace)
+  proposal_sites = [
+      name
+      for name, site in q_trace.items()
+      if util.is_observed_site(site) or name.endswith("_PREV_")
+  ]
 
   proposal_lp = sum(
       lp.reshape(lp.shape[:batch_ndims] + (-1,)).sum(-1)
@@ -180,7 +168,11 @@ def rkl_loss(q_trace, p_trace, incoming_log_weight, incremental_log_weight):
   q_log_probs = {
       name: util.get_site_log_prob(site) for name, site in q_trace.items()
   }
-  proposal_sites, target_sites = _proposal_and_target_sites(q_trace, p_trace)
+  proposal_sites = [
+      name
+      for name, site in q_trace.items()
+      if util.is_observed_site(site) or name.endswith("_PREV_")
+  ]
 
   proposal_lp = sum(
       lp.reshape(lp.shape[:batch_ndims] + (-1,)).sum(-1)
@@ -190,7 +182,7 @@ def rkl_loss(q_trace, p_trace, incoming_log_weight, incremental_log_weight):
   target_lp = sum(
       lp.reshape(lp.shape[:batch_ndims] + (-1,)).sum(-1)
       for name, lp in p_log_probs.items()
-      if name in target_sites
+      if not name.endswith("_PREV_")
   )
 
   w1 = jax.lax.stop_gradient(jax.nn.softmax(incoming_log_weight, axis=0))
@@ -213,7 +205,11 @@ def rws_loss(q_trace, p_trace, incoming_log_weight, incremental_log_weight):
   q_log_probs = {
       name: util.get_site_log_prob(site) for name, site in q_trace.items()
   }
-  proposal_sites, target_sites = _proposal_and_target_sites(q_trace, p_trace)
+  proposal_sites = [
+      name
+      for name, site in q_trace.items()
+      if util.is_observed_site(site) or name.endswith("_PREV_")
+  ]
 
   proposal_lp = sum(
       lp.reshape(lp.shape[:batch_ndims] + (-1,)).sum(-1)
@@ -228,7 +224,7 @@ def rws_loss(q_trace, p_trace, incoming_log_weight, incremental_log_weight):
   target_lp = sum(
       lp.reshape(lp.shape[:batch_ndims] + (-1,)).sum(-1)
       for name, lp in p_log_probs.items()
-      if name in target_sites
+      if not name.endswith("_PREV_")
   )
 
   surrogate_loss = (target_lp - proposal_lp) + forward_lp
diff --git a/examples/anneal.py b/examples/anneal.py
index 91e27a4..a0f761d 100644
--- a/examples/anneal.py
+++ b/examples/anneal.py
@@ -121,6 +121,8 @@ def __call__(self, x):
 def anneal_target(network, k=0):
   x = numpyro.sample("x", dist.Normal(0, 5).expand([2]).mask(False).to_event())
   anneal_density = network.anneal_density(x, index=k)
+  # We make "anneal_density" a latent site so that it does not contribute
+  # to the likelihood weighting of the first proposal.
   numpyro.sample("anneal_density", dist.Unit(anneal_density))
   return ({"x": x},)