diff --git a/.gitignore b/.gitignore
index 4577fd0..c9d6924 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1,3 +1,6 @@
+# temporary ignoring
+src/pymc_elicito/optimization.py
+
 # Notebooks
 *.ipynb
 
diff --git a/pyproject.toml b/pyproject.toml
index afb7437..a65448b 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -8,7 +8,10 @@ authors = [
 license = { text = "placeholder" }
 requires-python = ">=3.9"
 dependencies = [
+    "numpy>=2.0.2",
     "pandas>=2.3.3",
+    "pymc>=5.12.0",
+    "pytensor>=2.19.0",
 ]
 readme = "README.md"
 classifiers = [
@@ -41,7 +44,7 @@ Issues = "https://github.com/pymc-devs/pymc-elicito/issues"
 
 [project.optional-dependencies]
 plots = [
-    "matplotlib>=3.7.1",
+    "matplotlib>=3.9.4",
 ]
 full = [
     "pymc-elicito[plots]",
diff --git a/requirements-docs-locked.txt b/requirements-docs-locked.txt
index fc60762..e640653 100644
--- a/requirements-docs-locked.txt
+++ b/requirements-docs-locked.txt
@@ -5,6 +5,8 @@ appnope==0.1.4 ; sys_platform == 'darwin'
 argon2-cffi==23.1.0
 argon2-cffi-bindings==21.2.0
 arrow==1.3.0
+arviz==0.16.0 ; python_full_version < '3.10'
+arviz==0.22.0 ; python_full_version >= '3.10'
 asttokens==3.0.0
 async-lru==2.0.4
 attrs==25.3.0
@@ -12,12 +14,15 @@ babel==2.16.0
 backrefs==5.9
 beautifulsoup4==4.12.3
 bleach==6.2.0
+cachetools==6.2.2
 certifi==2024.12.14
 cffi==1.17.1
 charset-normalizer==3.4.1
 click==8.1.8
+cloudpickle==3.1.2
 colorama==0.4.6
 comm==0.2.2
+cons==0.4.7
 contourpy==1.3.0 ; python_full_version < '3.10'
 contourpy==1.3.2 ; python_full_version == '3.10.*'
 contourpy==1.3.3 ; python_full_version >= '3.11'
@@ -25,14 +30,20 @@ cycler==0.12.1
 debugpy==1.8.11
 decorator==5.1.1
 defusedxml==0.7.1
+etuples==0.3.10
 exceptiongroup==1.3.0 ; python_full_version < '3.11'
 executing==2.1.0
 fastjsonschema==2.21.1
+fastprogress==1.0.3 ; python_full_version < '3.10'
+filelock==3.16.1
 fonttools==4.60.1
 fqdn==1.5.1
 ghp-import==2.1.0
 griffe==1.11.0
 h11==0.14.0
+h5netcdf==1.7.3
+h5py==3.14.0 ; python_full_version < '3.10'
+h5py==3.15.1 ; python_full_version >= '3.10'
 httpcore==1.0.7
 httpx==0.28.1
 idna==3.10
@@ -62,6 +73,7 @@ jupyterlab-server==2.27.3
 jupytext==1.17.2
 kiwisolver==1.4.7 ; python_full_version < '3.10'
 kiwisolver==1.4.9 ; python_full_version >= '3.10'
+logical-unification==0.4.7
 markdown==3.7
 markdown-it-py==3.0.0
 markupsafe==3.0.2
@@ -71,6 +83,7 @@ matplotlib-inline==0.1.7
 mdit-py-plugins==0.4.2
 mdurl==0.1.2
 mergedeep==1.3.4
+minikanren==1.0.5
 mistune==3.0.2
 mkdocs==1.6.1
 mkdocs-autorefs==1.4.2
@@ -84,6 +97,7 @@ mkdocs-section-index==0.3.10
 mkdocstrings==0.30.0
 mkdocstrings-python==1.16.12
 mkdocstrings-python-xref==1.16.3
+multipledispatch==1.0.0
 nbclient==0.10.2
 nbconvert==7.16.4
 nbformat==5.10.4
@@ -110,8 +124,14 @@ ptyprocess==0.7.0 ; (python_full_version < '3.10' and sys_platform == 'emscripte
 pure-eval==0.2.3
 pycparser==2.22
 pygments==2.19.1
+pymc==5.12.0 ; python_full_version < '3.10'
+pymc==5.25.1 ; python_full_version == '3.10.*'
+pymc==5.26.1 ; python_full_version >= '3.11'
 pymdown-extensions==10.16.1
 pyparsing==3.2.5
+pytensor==2.19.0 ; python_full_version < '3.10'
+pytensor==2.31.7 ; python_full_version == '3.10.*'
+pytensor==2.35.1 ; python_full_version >= '3.11'
 python-dateutil==2.9.0.post0
 python-json-logger==3.2.1
 pytz==2025.2
@@ -124,8 +144,12 @@ referencing==0.35.1
 requests==2.32.3
 rfc3339-validator==0.1.4
 rfc3986-validator==0.1.1
+rich==13.9.4 ; python_full_version >= '3.10'
 rpds-py==0.22.3
 ruff==0.12.8
+scipy==1.13.1 ; python_full_version < '3.10'
+scipy==1.15.3 ; python_full_version == '3.10.*'
+scipy==1.16.3 ; python_full_version >= '3.11'
 send2trash==1.8.3
 setuptools==75.6.0
 six==1.17.0
@@ -133,12 +157,14 @@ sniffio==1.3.1
 soupsieve==2.6
 stack-data==0.6.3
 terminado==0.18.1
+threadpoolctl==3.6.0 ; python_full_version >= '3.10'
 tinycss2==1.4.0
 tomli==2.2.1 ; python_full_version < '3.11'
+toolz==1.1.0
 tornado==6.4.2
 traitlets==5.14.3
 types-python-dateutil==2.9.0.20241206
-typing-extensions==4.12.2 ; python_full_version < '3.13'
+typing-extensions==4.12.2
 tzdata==2025.2
 uri-template==1.3.0
 urllib3==2.3.0
@@ -147,4 +173,10 @@ wcwidth==0.2.13
 webcolors==24.11.1
 webencodings==0.5.1
 websocket-client==1.8.0
+xarray==2024.7.0 ; python_full_version < '3.10'
+xarray==2025.6.1 ; python_full_version == '3.10.*'
+xarray==2025.11.0 ; python_full_version >= '3.11'
+xarray-einstats==0.7.0 ; python_full_version < '3.10'
+xarray-einstats==0.8.0 ; python_full_version == '3.10.*'
+xarray-einstats==0.9.1 ; python_full_version >= '3.11'
 zipp==3.23.0 ; python_full_version < '3.10'
diff --git a/requirements-incl-optional-locked.txt b/requirements-incl-optional-locked.txt
index 1d6bb03..4d93be4 100644
--- a/requirements-incl-optional-locked.txt
+++ b/requirements-incl-optional-locked.txt
@@ -1,15 +1,31 @@
 # This file was autogenerated by uv via the following command:
 #    uv export -o requirements-incl-optional-locked.txt --no-hashes --no-dev --no-emit-project --all-extras
+arviz==0.16.0 ; python_full_version < '3.10'
+arviz==0.22.0 ; python_full_version >= '3.10'
+cachetools==6.2.2
+cloudpickle==3.1.2
+cons==0.4.7
 contourpy==1.3.0 ; python_full_version < '3.10'
 contourpy==1.3.2 ; python_full_version == '3.10.*'
 contourpy==1.3.3 ; python_full_version >= '3.11'
 cycler==0.12.1
+etuples==0.3.10
+fastprogress==1.0.3 ; python_full_version < '3.10'
+filelock==3.16.1
 fonttools==4.60.1
+h5netcdf==1.7.3
+h5py==3.14.0 ; python_full_version < '3.10'
+h5py==3.15.1 ; python_full_version >= '3.10'
 importlib-resources==6.5.2 ; python_full_version < '3.10'
 kiwisolver==1.4.7 ; python_full_version < '3.10'
 kiwisolver==1.4.9 ; python_full_version >= '3.10'
+logical-unification==0.4.7
+markdown-it-py==3.0.0 ; python_full_version >= '3.10'
 matplotlib==3.9.4 ; python_full_version < '3.10'
 matplotlib==3.10.7 ; python_full_version >= '3.10'
+mdurl==0.1.2 ; python_full_version >= '3.10'
+minikanren==1.0.5
+multipledispatch==1.0.0
 numpy==2.0.2 ; python_full_version < '3.10'
 numpy==2.2.6 ; python_full_version == '3.10.*'
 numpy==2.3.5 ; python_full_version >= '3.11'
@@ -17,9 +33,30 @@ packaging==24.2
 pandas==2.3.3
 pillow==11.3.0 ; python_full_version < '3.10'
 pillow==12.0.0 ; python_full_version >= '3.10'
+pygments==2.19.1 ; python_full_version >= '3.10'
+pymc==5.12.0 ; python_full_version < '3.10'
+pymc==5.25.1 ; python_full_version == '3.10.*'
+pymc==5.26.1 ; python_full_version >= '3.11'
 pyparsing==3.2.5
+pytensor==2.19.0 ; python_full_version < '3.10'
+pytensor==2.31.7 ; python_full_version == '3.10.*'
+pytensor==2.35.1 ; python_full_version >= '3.11'
 python-dateutil==2.9.0.post0
 pytz==2025.2
+rich==13.9.4 ; python_full_version >= '3.10'
+scipy==1.13.1 ; python_full_version < '3.10'
+scipy==1.15.3 ; python_full_version == '3.10.*'
+scipy==1.16.3 ; python_full_version >= '3.11'
+setuptools==75.6.0
 six==1.17.0
+threadpoolctl==3.6.0 ; python_full_version >= '3.10'
+toolz==1.1.0
+typing-extensions==4.12.2
 tzdata==2025.2
+xarray==2024.7.0 ; python_full_version < '3.10'
+xarray==2025.6.1 ; python_full_version == '3.10.*'
+xarray==2025.11.0 ; python_full_version >= '3.11'
+xarray-einstats==0.7.0 ; python_full_version < '3.10'
+xarray-einstats==0.8.0 ; python_full_version == '3.10.*'
+xarray-einstats==0.9.1 ; python_full_version >= '3.11'
 zipp==3.23.0 ; python_full_version < '3.10'
diff --git a/requirements-locked.txt b/requirements-locked.txt
index cde6122..aad98f9 100644
--- a/requirements-locked.txt
+++ b/requirements-locked.txt
@@ -1,10 +1,62 @@
 # This file was autogenerated by uv via the following command:
 #    uv export -o requirements-locked.txt --no-hashes --no-dev --no-emit-project
+arviz==0.16.0 ; python_full_version < '3.10'
+arviz==0.22.0 ; python_full_version >= '3.10'
+cachetools==6.2.2
+cloudpickle==3.1.2
+cons==0.4.7
+contourpy==1.3.0 ; python_full_version < '3.10'
+contourpy==1.3.2 ; python_full_version == '3.10.*'
+contourpy==1.3.3 ; python_full_version >= '3.11'
+cycler==0.12.1
+etuples==0.3.10
+fastprogress==1.0.3 ; python_full_version < '3.10'
+filelock==3.16.1
+fonttools==4.60.1
+h5netcdf==1.7.3
+h5py==3.14.0 ; python_full_version < '3.10'
+h5py==3.15.1 ; python_full_version >= '3.10'
+importlib-resources==6.5.2 ; python_full_version < '3.10'
+kiwisolver==1.4.7 ; python_full_version < '3.10'
+kiwisolver==1.4.9 ; python_full_version >= '3.10'
+logical-unification==0.4.7
+markdown-it-py==3.0.0 ; python_full_version >= '3.10'
+matplotlib==3.9.4 ; python_full_version < '3.10'
+matplotlib==3.10.7 ; python_full_version >= '3.10'
+mdurl==0.1.2 ; python_full_version >= '3.10'
+minikanren==1.0.5
+multipledispatch==1.0.0
 numpy==2.0.2 ; python_full_version < '3.10'
 numpy==2.2.6 ; python_full_version == '3.10.*'
 numpy==2.3.5 ; python_full_version >= '3.11'
+packaging==24.2
 pandas==2.3.3
+pillow==11.3.0 ; python_full_version < '3.10'
+pillow==12.0.0 ; python_full_version >= '3.10'
+pygments==2.19.1 ; python_full_version >= '3.10'
+pymc==5.12.0 ; python_full_version < '3.10'
+pymc==5.25.1 ; python_full_version == '3.10.*'
+pymc==5.26.1 ; python_full_version >= '3.11'
+pyparsing==3.2.5
+pytensor==2.19.0 ; python_full_version < '3.10'
+pytensor==2.31.7 ; python_full_version == '3.10.*'
+pytensor==2.35.1 ; python_full_version >= '3.11'
 python-dateutil==2.9.0.post0
 pytz==2025.2
+rich==13.9.4 ; python_full_version >= '3.10'
+scipy==1.13.1 ; python_full_version < '3.10'
+scipy==1.15.3 ; python_full_version == '3.10.*'
+scipy==1.16.3 ; python_full_version >= '3.11'
+setuptools==75.6.0
 six==1.17.0
+threadpoolctl==3.6.0 ; python_full_version >= '3.10'
+toolz==1.1.0
+typing-extensions==4.12.2
 tzdata==2025.2
+xarray==2024.7.0 ; python_full_version < '3.10'
+xarray==2025.6.1 ; python_full_version == '3.10.*'
+xarray==2025.11.0 ; python_full_version >= '3.11'
+xarray-einstats==0.7.0 ; python_full_version < '3.10'
+xarray-einstats==0.8.0 ; python_full_version == '3.10.*'
+xarray-einstats==0.9.1 ; python_full_version >= '3.11'
+zipp==3.23.0 ; python_full_version < '3.10'
diff --git a/src/pymc_elicito/losses.py b/src/pymc_elicito/losses.py
new file mode 100644
index 0000000..6fca617
--- /dev/null
+++ b/src/pymc_elicito/losses.py
@@ -0,0 +1,127 @@
+"""
+Loss functions
+
+Computing discrepancy between expert summaries and model simulations
+"""
+
+from enum import Enum
+from typing import Any, Union
+
+import pytensor.tensor as pt
+
+
+class Kernel(str, Enum):
+    """Suppoted kernel types for MMD loss"""
+
+    ENERGY = "energy"
+    GAUSSIAN = "gaussian"
+
+
+class MMD2:
+    """
+    Maximum mean discrepancy loss
+    """
+
+    def __init__(
+        self, kernel: Union[Kernel, str] = Kernel.ENERGY, **kwargs: dict[Any, Any]
+    ) -> None:
+        """
+        Compute the biased, squared maximum mean discrepancy
+
+        Parameters
+        ----------
+        kernel :
+            Kernel type used for computing the MMD.
+            Currently supported: "energy", "gaussian".
+
+        **kwargs :
+            additional keyword arguments that might be required by the
+            different individual kernels
+        """
+        # ensure that all additionally, required arguments are provided for
+        # the respective kernel
+        try:
+            self.kernel_type = Kernel(kernel)
+        except ValueError:
+            raise ValueError(  # noqa: TRY003
+                f"Invalid kernel '{kernel}'. "
+                f"Must be one of: {[k.value for k in Kernel]}"
+            )
+
+        # ensure correct kernel specification
+        if self.kernel_type == Kernel.GAUSSIAN:
+            if "sigma" not in kwargs:
+                raise ValueError(  # noqa: TRY003
+                    "You need to pass a 'sigma' argument when using a "
+                    "Gaussian kernel."
+                )
+            self.sigma = kwargs["sigma"]
+
+        self.kernel_name = kernel
+
+    def __call__(
+        self,
+        x: pt.TensorVariable,  # shape: [B, num_stats]
+        y: pt.TensorVariable,  # shape: [B, num_stats]
+    ) -> pt.TensorVariable:  # shape: []
+        """
+        Compute the biased, squared maximum mean discrepancy of two samples
+
+        Returns
+        -------
+        MMD2_mean :
+            Average biased, squared maximum mean discrepancy.
+        """
+        # treat samples as column vectors [B, num_stats, 1]
+        # equivalent to tf.expand_dims(x, -1)
+        x = x.dimshuffle(0, 1, "x")
+        y = y.dimshuffle(0, 1, "x")
+
+        # compute dot product between samples
+        # equivalent to tf.matmul(x, x, transpose_b=True)
+        # Input x is (B, N, 1), x.transpose is (B, 1, N) -> Result (B, N, N)
+        xx = pt.batched_dot(x, x.transpose(0, 2, 1))
+        xy = pt.batched_dot(x, y.transpose(0, 2, 1))
+        yy = pt.batched_dot(y, y.transpose(0, 2, 1))
+
+        # compute squared difference
+        # We need to reshape diagonals for broadcasting:
+        # diag returns (B, N). dimshuffle(0, 1, 'x') makes it (B, N, 1)
+        # dimshuffle(0, 'x', 1) makes it (B, 1, N)
+        diag_xx = pt.diagonal(xx, axis1=1, axis2=2)
+        diag_yy = pt.diagonal(yy, axis1=1, axis2=2)
+
+        u_xx = diag_xx.dimshuffle(0, 1, "x") - 2 * xx + diag_xx.dimshuffle(0, "x", 1)
+        u_xy = diag_xx.dimshuffle(0, 1, "x") - 2 * xy + diag_yy.dimshuffle(0, "x", 1)
+        u_yy = diag_yy.dimshuffle(0, 1, "x") - 2 * yy + diag_yy.dimshuffle(0, "x", 1)
+
+        # apply kernel function to squared difference
+        XX = self.kernel_fun(u_xx)
+        XY = self.kernel_fun(u_xy)
+        YY = self.kernel_fun(u_yy)
+
+        # compute biased, squared MMD
+        # reduce_mean over axes 1 and 2
+        mmd2_val = pt.mean(XX, axis=(1, 2))
+        mmd2_val -= 2 * pt.mean(XY, axis=(1, 2))
+        mmd2_val += pt.mean(YY, axis=(1, 2))
+
+        # Final mean over the batch
+        MMD2_mean = pt.mean(mmd2_val)
+
+        return MMD2_mean
+
+    def kernel_fun(self, u: pt.TensorVariable) -> pt.TensorVariable:
+        """
+        Kernel used in MMD to compute discrepancy between samples.
+        """
+        if self.kernel_type == Kernel.ENERGY:
+            # clipping for numerical stability reasons
+            d = -pt.sqrt(pt.clip(u, 1e-8, int(1e10)))
+        elif self.kernel_type == Kernel.GAUSSIAN:
+            d = pt.exp(-0.5 * (u / self.sigma))
+        else:
+            # Fallback (should be caught in init)
+            d = u
+
+        return d
diff --git a/src/pymc_elicito/optimization.py b/src/pymc_elicito/optimization.py
new file mode 100644
index 0000000..c5a638c
--- /dev/null
+++ b/src/pymc_elicito/optimization.py
@@ -0,0 +1,3 @@
+"""
+Optimization module
+"""
diff --git a/tests/unit/test_losses.py b/tests/unit/test_losses.py
new file mode 100644
index 0000000..49a6ed0
--- /dev/null
+++ b/tests/unit/test_losses.py
@@ -0,0 +1,47 @@
+"""
+Unittests for losses module
+"""
+
+import numpy as np
+import pytensor.tensor as pt
+import pytest
+from numpy import testing as npt
+
+from pymc_elicito.losses import MMD2
+
+# assuming k(x, y) = -||x - y||^2
+# mean XX: 1/N^2 sum_i sum_j k(x_i, x_j)
+# mean YY: 1/N^2 sum_i sum_j k(y_i, y_j)
+# mean XY: -2/N^2 sum_i sum_j k(x_i, y_j)
+
+
+@pytest.mark.parametrize(
+    "x, y, expected",
+    [
+        (
+            np.array([[1.0, 2.0], [3.0, 4.0]]),  # x
+            np.array([[1.5, 2.5], [3.5, 4.5]]),  # y
+            0.5,  # expected MMD^2 value (approx)
+        ),
+        (
+            np.array([[1.0, 2.0], [3.0, 4.0]]),
+            np.array([[1.0, 2.0], [3.0, 4.0]]),
+            0.0,
+        ),
+        (
+            np.array([[0, 4]]),  # mean(XX) = 1/4 (0 - 4 - 4 + 0) = -2
+            np.array([[1, 3]]),  # mean(YY) -> 1/4(0 - 2 - 2 + 0) = -1
+            # mean(XY) = 1/4(-1 - 3 - 3 - 1) = -2
+            1.0,  # xx -2xy + yy = (-2) -2*(-2) + (-1) = 1
+        ),
+    ],
+)
+def test_mmd2_call(x, y, expected):
+    x_sample = pt.as_tensor_variable(x)
+    y_sample = pt.as_tensor_variable(y)
+
+    mmd2 = MMD2(kernel="energy")
+    result = mmd2(x_sample, y_sample)
+
+    assert isinstance(result, pt.TensorVariable)
+    npt.assert_allclose(result.eval(), expected, atol=1e-4)
diff --git a/tests/unit/test_optimization.py b/tests/unit/test_optimization.py
new file mode 100644
index 0000000..1696ca2
--- /dev/null
+++ b/tests/unit/test_optimization.py
@@ -0,0 +1,10 @@
+"""
+Unittests for optimization module
+"""
+
+from pymc_elicito.optimization import run_optimizer
+
+
+def test_run_optimizer():
+    # Just ensure the function runs without errors
+    run_optimizer()
diff --git a/uv.lock b/uv.lock
index 99c4e92..8a15b3f 100644
--- a/uv.lock
+++ b/uv.lock
@@ -82,6 +82,64 @@ wheels = [
     { url = "https://files.pythonhosted.org/packages/f8/ed/e97229a566617f2ae958a6b13e7cc0f585470eac730a73e9e82c32a3cdd2/arrow-1.3.0-py3-none-any.whl", hash = "sha256:c728b120ebc00eb84e01882a6f5e7927a53960aa990ce7dd2b10f39005a67f80", size = 66419, upload-time = "2023-09-30T22:11:16.072Z" },
 ]
 
+[[package]]
+name = "arviz"
+version = "0.16.0"
+source = { registry = "https://pypi.org/simple" }
+resolution-markers = [
+    "python_full_version < '3.10' and sys_platform == 'win32'",
+    "python_full_version < '3.10' and sys_platform != 'win32'",
+]
+dependencies = [
+    { name = "h5netcdf", marker = "python_full_version < '3.10'" },
+    { name = "matplotlib", version = "3.9.4", source = { registry = "https://pypi.org/simple" }, marker = "python_full_version < '3.10'" },
+    { name = "numpy", version = "2.0.2", source = { registry = "https://pypi.org/simple" }, marker = "python_full_version < '3.10'" },
+    { name = "packaging", marker = "python_full_version < '3.10'" },
+    { name = "pandas", marker = "python_full_version < '3.10'" },
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