diff --git a/src/relaxed/__init__.py b/src/relaxed/__init__.py
index 4d754a4..f309aa4 100644
--- a/src/relaxed/__init__.py
+++ b/src/relaxed/__init__.py
@@ -1,6 +1,14 @@
 from relaxed._version import version as __version__
 
-__all__ = ("__version__", "hist", "cramer_rao_uncert", "fisher_info", "mle", "infer")
+__all__ = (
+    "__version__",
+    "hist",
+    "cramer_rao_uncert",
+    "fisher_info",
+    "mle",
+    "infer",
+    "gaussianity",
+)
 
 from relaxed import infer, mle
-from relaxed.ops import cramer_rao_uncert, fisher_info, hist
+from relaxed.ops import cramer_rao_uncert, fisher_info, gaussianity, hist
diff --git a/src/relaxed/infer/hypothesis_test.py b/src/relaxed/infer/hypothesis_test.py
index a713741..f721296 100644
--- a/src/relaxed/infer/hypothesis_test.py
+++ b/src/relaxed/infer/hypothesis_test.py
@@ -5,15 +5,15 @@
 
 from functools import partial
 
-import jax
 import jax.numpy as jnp
 import pyhf
 from chex import Array
+from jax import jit
 
 from ..mle import fit, fixed_poi_fit
 
 
-@partial(jax.jit, static_argnames=["model", "return_mle_pars"])  # forward pass
+@partial(jit, static_argnames=["model", "return_mle_pars"])  # forward pass
 def hypotest(
     test_poi: float,
     data: Array,
diff --git a/src/relaxed/mle/constrained_fit.py b/src/relaxed/mle/constrained_fit.py
index a70096a..e87b756 100644
--- a/src/relaxed/mle/constrained_fit.py
+++ b/src/relaxed/mle/constrained_fit.py
@@ -5,9 +5,9 @@
 from functools import partial
 from typing import TYPE_CHECKING, Callable, cast
 
-import jax
 import jax.numpy as jnp
 from chex import Array
+from jax import jit
 
 if TYPE_CHECKING:
     import pyhf
@@ -33,7 +33,7 @@ def fit_objective(
     return fit_objective
 
 
-@partial(jax.jit, static_argnames=["model"])  # forward pass
+@partial(jit, static_argnames=["model"])  # forward pass
 def fixed_poi_fit(
     data: Array,
     model: pyhf.Model,
diff --git a/src/relaxed/mle/global_fit.py b/src/relaxed/mle/global_fit.py
index a5bef5e..f3ad618 100644
--- a/src/relaxed/mle/global_fit.py
+++ b/src/relaxed/mle/global_fit.py
@@ -5,8 +5,8 @@
 from functools import partial
 from typing import TYPE_CHECKING, Callable, cast
 
-import jax
 from chex import Array
+from jax import jit
 
 if TYPE_CHECKING:
     import pyhf
@@ -24,7 +24,7 @@ def fit_objective(lhood_pars_to_optimize: Array) -> float:  # NLL
     return fit_objective
 
 
-@partial(jax.jit, static_argnames=["model"])
+@partial(jit, static_argnames=["model"])
 def fit(
     data: Array,
     model: pyhf.Model,
diff --git a/src/relaxed/mle/minimize.py b/src/relaxed/mle/minimize.py
index 4082890..3052454 100644
--- a/src/relaxed/mle/minimize.py
+++ b/src/relaxed/mle/minimize.py
@@ -10,10 +10,11 @@
 import jaxopt
 import optax
 from chex import Array
+from jax import jit
 
 
 # try wrapping obj with closure_convert
-@partial(jax.jit, static_argnames=["objective_fn"])  # forward pass
+@partial(jit, static_argnames=["objective_fn"])  # forward pass
 def _minimize(
     objective_fn: Callable[..., float], init_pars: Array, lr: float, *obj_args: Any
 ) -> Array:
diff --git a/src/relaxed/ops/__init__.py b/src/relaxed/ops/__init__.py
index 2b86a52..bedf249 100644
--- a/src/relaxed/ops/__init__.py
+++ b/src/relaxed/ops/__init__.py
@@ -2,7 +2,9 @@
     "hist",
     "fisher_info",
     "cramer_rao_uncert",
+    "gaussianity",
 )
 
 from relaxed.ops.fisher_information import cramer_rao_uncert, fisher_info
 from relaxed.ops.histograms import hist
+from relaxed.ops.likelihood_gaussianity import gaussianity
diff --git a/src/relaxed/ops/histograms.py b/src/relaxed/ops/histograms.py
index 0e57b6b..ccee602 100644
--- a/src/relaxed/ops/histograms.py
+++ b/src/relaxed/ops/histograms.py
@@ -5,13 +5,13 @@
 
 from functools import partial
 
-import jax
 import jax.numpy as jnp
 import jax.scipy as jsp
 from chex import Array
+from jax import jit
 
 
-@partial(jax.jit, static_argnames=["density", "reflect_infinities"])
+@partial(jit, static_argnames=["density", "reflect_infinities"])
 def hist(
     events: Array,
     bins: Array,
diff --git a/src/relaxed/ops/likelihood_gaussianity.py b/src/relaxed/ops/likelihood_gaussianity.py
new file mode 100644
index 0000000..c8356b0
--- /dev/null
+++ b/src/relaxed/ops/likelihood_gaussianity.py
@@ -0,0 +1,65 @@
+from __future__ import annotations
+
+__all__ = ("gaussianity",)
+
+from functools import partial
+from typing import TYPE_CHECKING
+
+import jax.numpy as jnp
+import jax.scipy as jsp
+from chex import Array
+from jax import jit, vmap
+from jax.random import PRNGKey, multivariate_normal
+
+if TYPE_CHECKING:
+    import pyhf
+
+
+def gaussian_logpdf(
+    bestfit_pars: Array,
+    data: Array,
+    cov: Array,
+) -> Array:
+    return jsp.stats.multivariate_normal.logpdf(data, bestfit_pars, cov).reshape(
+        1,
+    )
+
+
+@partial(jit, static_argnames=["model", "rng_key", "n_samples"])
+def gaussianity(
+    model: pyhf.Model,
+    bestfit_pars: Array,
+    cov_approx: Array,
+    observed_data: Array,
+    rng_key: PRNGKey,
+    n_samples: int = 1000,
+) -> Array:
+    # - compare the likelihood of the fitted model with a gaussian approximation
+    # that has the same MLE (fitted_pars)
+    # - do this across a number of points in parspace (sampled from the gaussian approx)
+    # and take the mean squared diff
+    # - centre the values wrt the best-fit vals to scale the differences
+    gaussian_parspace_samples = multivariate_normal(
+        key=rng_key,
+        mean=bestfit_pars,
+        cov=cov_approx,
+        shape=(n_samples,),
+    )
+
+    relative_nlls_model = vmap(
+        lambda pars, data: -(
+            model.logpdf(pars, data)[0] - model.logpdf(bestfit_pars, data)[0]
+        ),  # scale origin to bestfit pars
+        in_axes=(0, None),
+    )(gaussian_parspace_samples, observed_data)
+
+    relative_nlls_gaussian = vmap(
+        lambda pars, data: -(
+            gaussian_logpdf(pars, data, cov_approx)[0]
+            - gaussian_logpdf(bestfit_pars, data, cov_approx)[0]
+        ),  # data fixes the lhood shape
+        in_axes=(0, None),
+    )(gaussian_parspace_samples, bestfit_pars)
+
+    diffs = relative_nlls_model - relative_nlls_gaussian
+    return jnp.nanmean(diffs ** 2, axis=0)
diff --git a/tests/test_ops.py b/tests/test_ops.py
index fd207b3..2c06d07 100644
--- a/tests/test_ops.py
+++ b/tests/test_ops.py
@@ -163,3 +163,30 @@ def model(pars, data):
         return relaxed.cramer_rao_uncert(model, pars * x, data * x)
 
     jacrev(pipeline)(4.0)  # just check you can calc it w/o exception
+
+
+def test_gaussianity():
+    """Test that the gaussianity of the distribution is preserved."""
+    pyhf.set_backend("jax")
+    m = pyhf.simplemodels.uncorrelated_background([5, 5], [50, 50], [5, 5])
+    pars = jnp.asarray(m.config.suggested_init())
+    data = jnp.asarray(m.expected_data(pars))
+    cov_approx = jnp.linalg.inv(
+        relaxed.fisher_info(lambda d, p: m.logpdf(d, p)[0], pars, data)
+    )
+    relaxed.gaussianity(m, pars, cov_approx, data, PRNGKey(0))
+
+
+def test_gaussianity_grad(example_model):
+    def pipeline(x):
+        def model(pars, data):
+            return example_model.logpdf(pars, data)[0]
+
+        pars = example_model.config.suggested_init()
+        data = example_model.expected_data(pars)
+        cov_approx = jnp.linalg.inv(relaxed.fisher_info(model, pars, data))
+        return relaxed.gaussianity(
+            example_model, pars * x, cov_approx * x, data * x, PRNGKey(0)
+        )
+
+    jacrev(pipeline)(4.0)  # just check you can calc it w/o exception