diff --git a/autofit/messages/interface.py b/autofit/messages/interface.py
index 3a0410624..24ae2ea08 100644
--- a/autofit/messages/interface.py
+++ b/autofit/messages/interface.py
@@ -95,7 +95,19 @@ def log_partition(self, xp=np) -> np.ndarray:
     @classmethod
     def natural_logpdf(cls, eta, t, log_base, log_partition, xp=np):
         eta_t = xp.multiply(eta, t).sum(0)
-        return xp.nan_to_num(log_base + eta_t - log_partition, nan=-xp.inf)
+        # Only NaN is replaced. ``nan=-inf`` maps an out-of-support NaN (e.g.
+        # ``log`` of a negative value under a transformed message) to zero
+        # density, which is intended -- but ``nan_to_num``'s DEFAULT ``neginf``
+        # replaces a genuine ``-inf`` with ``-1.8e308``, turning a zero-density
+        # point into a finite one. That is the opposite of the intent, and it
+        # matters: ``isfinite`` is what ``optax.apply_if_finite`` and
+        # ``non_linear.clipper`` branch on to detect a lane leaving the support.
+        return xp.nan_to_num(
+            log_base + eta_t - log_partition,
+            nan=-xp.inf,
+            neginf=-xp.inf,
+            posinf=xp.inf,
+        )
 
     def numerical_logpdf_gradient(
         self, x: np.ndarray, eps: float = 1e-6
diff --git a/test_autofit/messages/test_natural_logpdf_neginf.py b/test_autofit/messages/test_natural_logpdf_neginf.py
new file mode 100644
index 000000000..01435a3f9
--- /dev/null
+++ b/test_autofit/messages/test_natural_logpdf_neginf.py
@@ -0,0 +1,77 @@
+"""
+``AbstractMessage.natural_logpdf`` must not clamp a genuine ``-inf``.
+
+The reduction maps NaN to ``-inf`` on purpose -- an out-of-support NaN (``log`` of a
+negative value under a transformed message) is zero density. But ``nan_to_num``'s
+DEFAULT ``neginf`` replaces a real ``-inf`` with ``-1.7976931348623157e+308``, so the
+call did the opposite of its intent for exactly the inputs that were already correct.
+
+``-1.8e308`` is finite, and ``isfinite`` is what ``optax.apply_if_finite`` and
+``autofit.non_linear.clipper`` branch on to decide a lane has left the prior support.
+"""
+
+import sys
+
+import numpy as np
+import pytest
+
+import autofit as af
+
+
+@pytest.fixture(name="log_gaussian")
+def make_log_gaussian():
+    return af.LogGaussianPrior(mean=0.4, sigma=1.3)
+
+
+def test__genuine_neginf_survives_the_reduction(log_gaussian):
+    """
+    ``log(0) = -inf`` reaches the reduction as ``-inf``; it must come back as ``-inf``,
+    not as negative float max.
+    """
+    value = log_gaussian.message.logpdf(np.array(0.0))
+
+    assert value == -np.inf
+    assert not np.isfinite(value)
+    assert value != -sys.float_info.max
+
+
+def test__nan_still_maps_to_neginf(log_gaussian):
+    """
+    The other half of the same call, which was always correct: ``log(-1) = NaN``, and an
+    out-of-support NaN is zero density. Pinned so a fix to one half cannot break the other.
+    """
+    value = log_gaussian.message.logpdf(np.array(-1.0))
+
+    assert value == -np.inf
+    assert not np.isnan(value)
+
+
+@pytest.mark.parametrize("value", [1e-9, 0.001, 0.1, 0.5, 1.0, 2.0, 10.0, 1000.0])
+def test__in_support_values_are_unchanged(log_gaussian, value):
+    """
+    The fix must touch only the non-finite branch. Every in-support point is finite before
+    and after, and the density is the one the prior reports.
+    """
+    assert np.isfinite(log_gaussian.message.logpdf(np.array(value)))
+
+
+@pytest.mark.parametrize(
+    "prior, outside",
+    [
+        (af.UniformPrior(lower_limit=0.0, upper_limit=2.0), [-1.0, 3.0]),
+        (af.LogUniformPrior(lower_limit=0.01, upper_limit=100.0), [-1.0, 0.0, 1000.0]),
+        (af.LogGaussianPrior(mean=0.4, sigma=1.3), [-1.0, 0.0]),
+    ],
+    ids=["Uniform", "LogUniform", "LogGaussian"],
+)
+def test__no_prior_family_reports_a_finite_density_off_its_support(prior, outside):
+    """
+    The general form. A finite log density outside the support is a licence for any
+    ``isfinite`` consumer to treat an impossible point as merely a bad one.
+
+    ``TruncatedGaussianPrior`` is deliberately absent: its message returns finite values
+    outside its limits for an unrelated reason (``TruncatedNormalMessage`` does not gate on
+    them), which this fix does not address and must not be read as covering.
+    """
+    for value in outside:
+        assert prior.message.logpdf(np.array(value)) == -np.inf