diff --git a/src/nessai/config.py b/src/nessai/config.py
index 6999facb..607590b4 100644
--- a/src/nessai/config.py
+++ b/src/nessai/config.py
@@ -144,6 +144,12 @@ class PlottingConfig(_BaseConfig):
     Based on the default DPI in matplotlib of 100, so this will give a maximum
     size of 5000 pixels.
     """
+    clip_min: float = -1e10
+    """Minimum value to clip data to for plotting.
+
+    This is used to avoid issues with plots where the dynamic range is too
+    large for matplotlib to handle.
+    """
 
 
 @dataclass
diff --git a/src/nessai/plot.py b/src/nessai/plot.py
index ff51b874..e1c69023 100644
--- a/src/nessai/plot.py
+++ b/src/nessai/plot.py
@@ -68,6 +68,36 @@ def wrapper(*args, **kwargs):
         return decorator
 
 
+def sanitise_array(
+    a: np.ndarray,
+    /,
+    a_min: Optional[float] = None,
+    a_max: Optional[float] = None,
+):
+    """Sanitise an array for plotting.
+
+    If :code:`x_min` is not specified, it is set to the value in
+    :code:`nessai.config.plotting.clip_min`.
+
+    Parameters
+    ----------
+    x : array_like
+        Array to sanitise.
+    x_min : float, optional
+        Minimum value to clip the data to.
+    xmax : float, optional
+        Maximum value to clip the data to.
+
+    Returns
+    -------
+    np.ndarray
+        Sanitised array.
+    """
+    if a_min is None:
+        a_min = config.plotting.clip_min
+    return np.clip(a, a_min, a_max)
+
+
 @nessai_style()
 def plot_live_points(
     live_points, filename=None, bounds=None, c=None, **kwargs
diff --git a/src/nessai/samplers/nestedsampler.py b/src/nessai/samplers/nestedsampler.py
index 9b662359..0ea0545d 100644
--- a/src/nessai/samplers/nestedsampler.py
+++ b/src/nessai/samplers/nestedsampler.py
@@ -19,7 +19,7 @@
 from .. import config
 from ..evidence import _NSIntegralState
 from ..livepoint import empty_structured_array
-from ..plot import nessai_style, plot_indices, plot_trace
+from ..plot import nessai_style, plot_indices, plot_trace, sanitise_array
 from ..proposal.utils import (
     check_proposal_kwargs,
     get_flow_proposal_class,
@@ -1004,11 +1004,39 @@ def plot_state(self, filename=None):
         for a in ax:
             a.axvline(self.iteration, c="#ff9900", ls="-.")
 
-        ax[0].plot(it, self.history["min_log_likelihood"], label="Min log L")
-        ax[0].plot(it, self.history["max_log_likelihood"], label="Max log L")
+        ax[0].plot(
+            it,
+            sanitise_array(self.history["min_log_likelihood"]),
+            label="Min log L",
+        )
+        ax[0].plot(
+            it,
+            sanitise_array(self.history["max_log_likelihood"]),
+            label="Max log L",
+        )
         ax[0].set_ylabel(r"$\log L$")
         ax[0].legend(frameon=False)
 
+        ax_logl_diff = plt.twinx(ax[0])
+        logl_diff = sanitise_array(
+            np.array(self.history["max_log_likelihood"])
+            - np.array(self.history["min_log_likelihood"]),
+        )
+        ax_logl_diff.plot(
+            it,
+            logl_diff,
+            c="C2",
+            ls=config.plotting.line_styles[2],
+            label=r"$\Delta \log L$ ",
+        )
+        ax_logl_diff.set_yscale("log")
+        ax_logl_diff.set_ylabel(r"$\Delta \log L$")
+        handles, labels = ax[0].get_legend_handles_labels()
+        handles_diff, labels_diff = ax_logl_diff.get_legend_handles_labels()
+        ax[0].legend(
+            handles + handles_diff, labels + labels_diff, frameon=False
+        )
+
         logX_its = np.arange(len(self.state.log_vols))
         ax[1].plot(logX_its, self.state.log_vols, label="log X")
         ax[1].set_ylabel(r"$\log X$")
@@ -1054,7 +1082,7 @@ def plot_state(self, filename=None):
             handles + handles_time, labels + labels_time, frameon=False
         )
 
-        ax[3].plot(it, self.history["logZ"], label="logZ")
+        ax[3].plot(it, sanitise_array(self.history["logZ"]), label="logZ")
         ax[3].set_ylabel(r"$\log Z$")
         ax[3].legend(frameon=False)