diff --git a/src/pyrecest/filters/_linear_gaussian.py b/src/pyrecest/filters/_linear_gaussian.py
index 66c758c12..c87edf72a 100644
--- a/src/pyrecest/filters/_linear_gaussian.py
+++ b/src/pyrecest/filters/_linear_gaussian.py
@@ -56,6 +56,15 @@ def _contains_complex_value(x):
     return False
 
 
+def _ensure_finite(x, name):
+    try:
+        values = np.asarray(to_numpy(x), dtype=float)
+    except (TypeError, ValueError, OverflowError) as exc:
+        raise ValueError(f"{name} must contain only finite values") from exc
+    if not np.all(np.isfinite(values)):
+        raise ValueError(f"{name} must contain only finite values")
+
+
 def _as_vector(x, name):
     if _contains_boolean_value(x):
         raise ValueError(f"{name} must contain numeric values, not booleans")
@@ -64,6 +73,7 @@ def _as_vector(x, name):
     x = atleast_1d(asarray(x, dtype=float64))
     if len(x.shape) != 1:
         raise ValueError(f"{name} must be one-dimensional after coercion")
+    _ensure_finite(x, name)
     return x
 
 
@@ -75,6 +85,7 @@ def _as_matrix(x, name):
     x = atleast_2d(asarray(x, dtype=float64))
     if len(x.shape) != 2:
         raise ValueError(f"{name} must be two-dimensional after coercion")
+    _ensure_finite(x, name)
     return x
 
 
diff --git a/tests/filters/test_linear_gaussian_nonfinite_inputs.py b/tests/filters/test_linear_gaussian_nonfinite_inputs.py
new file mode 100644
index 000000000..0a4df34eb
--- /dev/null
+++ b/tests/filters/test_linear_gaussian_nonfinite_inputs.py
@@ -0,0 +1,105 @@
+import unittest
+
+import numpy as np
+import numpy.testing as npt
+from pyrecest.backend import array, to_numpy
+from pyrecest.filters import KalmanFilter
+from pyrecest.filters._linear_gaussian import (
+    linear_gaussian_predict,
+    linear_gaussian_update,
+)
+
+
+class LinearGaussianNonfiniteInputsTest(unittest.TestCase):
+    @staticmethod
+    def _state_snapshot(kf):
+        state = kf.filter_state
+        return (
+            np.asarray(to_numpy(state.mu)).copy(),
+            np.asarray(to_numpy(state.C)).copy(),
+        )
+
+    def test_predict_rejects_nonfinite_inputs(self):
+        base = {
+            "mean": array([0.0]),
+            "covariance": array([[1.0]]),
+            "system_matrix": array([[1.0]]),
+            "sys_noise_cov": array([[0.1]]),
+        }
+        cases = (
+            ("mean", array([float("nan")])),
+            ("covariance", array([[float("inf")]])),
+            ("system_matrix", array([[float("-inf")]])),
+            ("sys_noise_cov", array([[float("nan")]])),
+            ("sys_input", array([float("inf")])),
+        )
+
+        for name, value in cases:
+            kwargs = dict(base)
+            kwargs[name] = value
+            with self.subTest(name=name), self.assertRaisesRegex(
+                ValueError,
+                f"{name} must contain only finite values",
+            ):
+                linear_gaussian_predict(**kwargs)
+
+    def test_update_rejects_nonfinite_inputs(self):
+        base = {
+            "mean": array([0.0]),
+            "covariance": array([[1.0]]),
+            "measurement": array([0.0]),
+            "measurement_matrix": array([[1.0]]),
+            "meas_noise": array([[0.1]]),
+        }
+        cases = (
+            ("mean", array([float("nan")])),
+            ("covariance", array([[float("inf")]])),
+            ("measurement", array([float("-inf")])),
+            ("measurement_matrix", array([[float("nan")]])),
+            ("meas_noise", array([[float("inf")]])),
+        )
+
+        for name, value in cases:
+            kwargs = dict(base)
+            kwargs[name] = value
+            with self.subTest(name=name), self.assertRaisesRegex(
+                ValueError,
+                f"{name} must contain only finite values",
+            ):
+                linear_gaussian_update(**kwargs)
+
+    def test_failed_predict_does_not_poison_kalman_state(self):
+        kf = KalmanFilter((array([0.0]), array([[1.0]])))
+        mean_before, covariance_before = self._state_snapshot(kf)
+
+        with self.assertRaisesRegex(
+            ValueError,
+            "sys_noise_cov must contain only finite values",
+        ):
+            kf.predict_linear(array([[1.0]]), array([[float("nan")]]))
+
+        mean_after, covariance_after = self._state_snapshot(kf)
+        npt.assert_allclose(mean_after, mean_before)
+        npt.assert_allclose(covariance_after, covariance_before)
+
+    def test_failed_update_does_not_poison_kalman_state(self):
+        kf = KalmanFilter((array([0.0]), array([[1.0]])))
+        mean_before, covariance_before = self._state_snapshot(kf)
+
+        with self.assertRaisesRegex(
+            ValueError,
+            "measurement must contain only finite values",
+        ):
+            kf.update_linear(
+                array([float("inf")]),
+                array([[1.0]]),
+                array([[0.1]]),
+            )
+
+        mean_after, covariance_after = self._state_snapshot(kf)
+        npt.assert_allclose(mean_after, mean_before)
+        npt.assert_allclose(covariance_after, covariance_before)
+
+
+if __name__ == "__main__":
+    unittest.main()