diff --git a/.github/workflows/tests.yml b/.github/workflows/tests.yml
index e5ee774ad8..8f932e0747 100644
--- a/.github/workflows/tests.yml
+++ b/.github/workflows/tests.yml
@@ -110,6 +110,7 @@ jobs:
             tests/logprob/test_composite_logprob.py
             tests/logprob/test_cumsum.py
             tests/logprob/test_mixture.py
+            tests/logprob/test_order.py
             tests/logprob/test_rewriting.py
             tests/logprob/test_scan.py
             tests/logprob/test_tensor.py
diff --git a/pymc/logprob/__init__.py b/pymc/logprob/__init__.py
index 0ddea90b6f..7c3c666917 100644
--- a/pymc/logprob/__init__.py
+++ b/pymc/logprob/__init__.py
@@ -49,6 +49,7 @@
 import pymc.logprob.cumsum
 import pymc.logprob.checks
 import pymc.logprob.mixture
+import pymc.logprob.order
 import pymc.logprob.scan
 import pymc.logprob.tensor
 import pymc.logprob.transforms
diff --git a/pymc/logprob/order.py b/pymc/logprob/order.py
new file mode 100644
index 0000000000..4033bf674c
--- /dev/null
+++ b/pymc/logprob/order.py
@@ -0,0 +1,127 @@
+#   Copyright 2023 The PyMC Developers
+#
+#   Licensed under the Apache License, Version 2.0 (the "License");
+#   you may not use this file except in compliance with the License.
+#   You may obtain a copy of the License at
+#
+#       http://www.apache.org/licenses/LICENSE-2.0
+#
+#   Unless required by applicable law or agreed to in writing, software
+#   distributed under the License is distributed on an "AS IS" BASIS,
+#   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+#   See the License for the specific language governing permissions and
+#   limitations under the License.
+#
+#   MIT License
+#
+#   Copyright (c) 2021-2022 aesara-devs
+#
+#   Permission is hereby granted, free of charge, to any person obtaining a copy
+#   of this software and associated documentation files (the "Software"), to deal
+#   in the Software without restriction, including without limitation the rights
+#   to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+#   copies of the Software, and to permit persons to whom the Software is
+#   furnished to do so, subject to the following conditions:
+#
+#   The above copyright notice and this permission notice shall be included in all
+#   copies or substantial portions of the Software.
+#
+#   THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+#   IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+#   FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+#   AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+#   LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+#   OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+#   SOFTWARE.
+
+from typing import List, Optional
+
+import pytensor.tensor as pt
+
+from pytensor.graph.basic import Node
+from pytensor.graph.fg import FunctionGraph
+from pytensor.graph.rewriting.basic import node_rewriter
+from pytensor.tensor.math import Max
+from pytensor.tensor.random.op import RandomVariable
+from pytensor.tensor.var import TensorVariable
+
+from pymc.logprob.abstract import (
+    MeasurableVariable,
+    _logcdf_helper,
+    _logprob,
+    _logprob_helper,
+)
+from pymc.logprob.rewriting import measurable_ir_rewrites_db
+
+
+class MeasurableMax(Max):
+    """A placeholder used to specify a log-likelihood for a max sub-graph."""
+
+
+MeasurableVariable.register(MeasurableMax)
+
+
+@node_rewriter([Max])
+def find_measurable_max(fgraph: FunctionGraph, node: Node) -> Optional[List[TensorVariable]]:
+    rv_map_feature = getattr(fgraph, "preserve_rv_mappings", None)
+    if rv_map_feature is None:
+        return None  # pragma: no cover
+
+    if isinstance(node.op, MeasurableMax):
+        return None  # pragma: no cover
+
+    base_var = node.inputs[0]
+
+    if base_var.owner is None:
+        return None
+
+    if not rv_map_feature.request_measurable(node.inputs):
+        return None
+
+    # Non-univariate distributions and non-RVs must be rejected
+    if not (isinstance(base_var.owner.op, RandomVariable) and base_var.owner.op.ndim_supp == 0):
+        return None
+
+    # TODO: We are currently only supporting continuous rvs
+    if isinstance(base_var.owner.op, RandomVariable) and base_var.owner.op.dtype.startswith("int"):
+        return None
+
+    # univariate i.i.d. test which also rules out other distributions
+    for params in base_var.owner.inputs[3:]:
+        if params.type.ndim != 0:
+            return None
+
+    # Check whether axis covers all dimensions
+    axis = set(node.op.axis)
+    base_var_dims = set(range(base_var.ndim))
+    if axis != base_var_dims:
+        return None
+
+    measurable_max = MeasurableMax(list(axis))
+    max_rv_node = measurable_max.make_node(base_var)
+    max_rv = max_rv_node.outputs
+
+    return max_rv
+
+
+measurable_ir_rewrites_db.register(
+    "find_measurable_max",
+    find_measurable_max,
+    "basic",
+    "max",
+)
+
+
+@_logprob.register(MeasurableMax)
+def max_logprob(op, values, base_rv, **kwargs):
+    r"""Compute the log-likelihood graph for the `Max` operation."""
+    (value,) = values
+
+    logprob = _logprob_helper(base_rv, value)
+    logcdf = _logcdf_helper(base_rv, value)
+
+    n = base_rv.size
+
+    logprob = (n - 1) * logcdf + logprob + pt.math.log(n)
+
+    return logprob
diff --git a/pymc/logprob/rewriting.py b/pymc/logprob/rewriting.py
index d2dc6630fd..a2b4c3bfa2 100644
--- a/pymc/logprob/rewriting.py
+++ b/pymc/logprob/rewriting.py
@@ -57,6 +57,7 @@
     EquilibriumGraphRewriter,
     GraphRewriter,
     node_rewriter,
+    out2in,
 )
 from pytensor.graph.rewriting.db import (
     LocalGroupDB,
@@ -70,6 +71,7 @@
 from pytensor.tensor.random.rewriting import local_subtensor_rv_lift
 from pytensor.tensor.rewriting.basic import register_canonicalize
 from pytensor.tensor.rewriting.shape import ShapeFeature
+from pytensor.tensor.rewriting.uncanonicalize import local_max_and_argmax
 from pytensor.tensor.subtensor import (
     AdvancedIncSubtensor,
     AdvancedIncSubtensor1,
@@ -358,6 +360,7 @@ def incsubtensor_rv_replace(fgraph, node):
 logprob_rewrites_db = SequenceDB()
 logprob_rewrites_db.name = "logprob_rewrites_db"
 logprob_rewrites_db.register("pre-canonicalize", optdb.query("+canonicalize"), "basic")
+logprob_rewrites_db.register("local_max_and_argmax", out2in(local_max_and_argmax), "basic")
 
 # These rewrites convert un-measurable variables into their measurable forms,
 # but they need to be reapplied, because some of the measurable forms require
diff --git a/scripts/run_mypy.py b/scripts/run_mypy.py
index 72f7013007..0dc73ef2a0 100644
--- a/scripts/run_mypy.py
+++ b/scripts/run_mypy.py
@@ -33,6 +33,7 @@
 pymc/logprob/censoring.py
 pymc/logprob/basic.py
 pymc/logprob/mixture.py
+pymc/logprob/order.py
 pymc/logprob/rewriting.py
 pymc/logprob/scan.py
 pymc/logprob/tensor.py
diff --git a/tests/logprob/test_order.py b/tests/logprob/test_order.py
new file mode 100644
index 0000000000..5a3818716d
--- /dev/null
+++ b/tests/logprob/test_order.py
@@ -0,0 +1,149 @@
+#   Copyright 2023 The PyMC Developers
+#
+#   Licensed under the Apache License, Version 2.0 (the "License");
+#   you may not use this file except in compliance with the License.
+#   You may obtain a copy of the License at
+#
+#       http://www.apache.org/licenses/LICENSE-2.0
+#
+#   Unless required by applicable law or agreed to in writing, software
+#   distributed under the License is distributed on an "AS IS" BASIS,
+#   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+#   See the License for the specific language governing permissions and
+#   limitations under the License.
+#
+#   MIT License
+#
+#   Copyright (c) 2021-2022 aesara-devs
+#
+#   Permission is hereby granted, free of charge, to any person obtaining a copy
+#   of this software and associated documentation files (the "Software"), to deal
+#   in the Software without restriction, including without limitation the rights
+#   to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+#   copies of the Software, and to permit persons to whom the Software is
+#   furnished to do so, subject to the following conditions:
+#
+#   The above copyright notice and this permission notice shall be included in all
+#   copies or substantial portions of the Software.
+#
+#   THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+#   IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+#   FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+#   AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+#   LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+#   OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+#   SOFTWARE.
+
+import re
+
+import numpy as np
+import pytensor.tensor as pt
+import pytest
+
+import pymc as pm
+
+from pymc import logp
+from pymc.logprob import conditional_logp
+from pymc.testing import assert_no_rvs
+
+
+def test_argmax():
+    """Test whether the logprob for ```pt.argmax``` is correctly rejected"""
+    x = pt.random.normal(0, 1, size=(3,))
+    x.name = "x"
+    x_max = pt.argmax(x, axis=-1)
+    x_max_value = pt.vector("x_max_value")
+
+    with pytest.raises(RuntimeError, match=re.escape("Logprob method not implemented for Argmax")):
+        x_max_logprob = logp(x_max, x_max_value)
+
+
+def test_max_non_iid_fails():
+    """Test whether the logprob for ```pt.max``` for non i.i.d is correctly rejected"""
+    x = pm.Normal.dist([0, 1, 2, 3, 4], 1, shape=(5,))
+    x.name = "x"
+    x_max = pt.max(x, axis=-1)
+    x_max_value = pt.vector("x_max_value")
+    with pytest.raises(RuntimeError, match=re.escape("Logprob method not implemented")):
+        x_max_logprob = logp(x_max, x_max_value)
+
+
+def test_max_non_rv_fails():
+    """Test whether the logprob for ```pt.max``` for non-RVs is correctly rejected"""
+    x = pt.exp(pt.random.beta(0, 1, size=(3,)))
+    x.name = "x"
+    x_max = pt.max(x, axis=-1)
+    x_max_value = pt.vector("x_max_value")
+    with pytest.raises(RuntimeError, match=re.escape("Logprob method not implemented")):
+        x_max_logprob = logp(x_max, x_max_value)
+
+
+def test_max_multivariate_rv_fails():
+    _alpha = pt.scalar()
+    _k = pt.iscalar()
+    x = pm.StickBreakingWeights.dist(_alpha, _k)
+    x.name = "x"
+    x_max = pt.max(x, axis=-1)
+    x_max_value = pt.vector("x_max_value")
+    with pytest.raises(RuntimeError, match=re.escape("Logprob method not implemented")):
+        x_max_logprob = logp(x_max, x_max_value)
+
+
+def test_max_categorical():
+    """Test whether the logprob for ```pt.max``` for unsupported distributions is correctly rejected"""
+    x = pm.Categorical.dist([1, 1, 1, 1], shape=(5,))
+    x.name = "x"
+    x_max = pt.max(x, axis=-1)
+    x_max_value = pt.vector("x_max_value")
+    with pytest.raises(RuntimeError, match=re.escape("Logprob method not implemented")):
+        x_max_logprob = logp(x_max, x_max_value)
+
+
+def test_non_supp_axis_max():
+    """Test whether the logprob for ```pt.max``` for unsupported axis is correctly rejected"""
+    x = pt.random.normal(0, 1, size=(3, 3))
+    x.name = "x"
+    x_max = pt.max(x, axis=-1)
+    x_max_value = pt.vector("x_max_value")
+    with pytest.raises(RuntimeError, match=re.escape("Logprob method not implemented")):
+        x_max_logprob = logp(x_max, x_max_value)
+
+
+@pytest.mark.parametrize(
+    "shape, value, axis",
+    [
+        (3, 0.85, -1),
+        (3, 0.01, 0),
+        (2, 0.2, None),
+        (4, 0.5, 0),
+        ((3, 4), 0.9, None),
+        ((3, 4), 0.75, (1, 0)),
+    ],
+)
+def test_max_logprob(shape, value, axis):
+    """Test whether the logprob for ```pt.max``` produces the corrected
+
+    The fact that order statistics of i.i.d. uniform RVs ~ Beta is used here:
+        U_1, \\dots, U_n \\stackrel{\text{i.i.d.}}{\\sim} \text{Uniform}(0, 1) \\Rightarrow U_{(k)} \\sim \text{Beta}(k, n + 1- k)
+    for all 1<=k<=n
+    """
+    x = pt.random.uniform(0, 1, size=shape)
+    x.name = "x"
+    x_max = pt.max(x, axis=axis)
+    x_max_value = pt.scalar("x_max_value")
+    x_max_logprob = logp(x_max, x_max_value)
+
+    assert_no_rvs(x_max_logprob)
+
+    test_value = value
+
+    n = np.prod(shape)
+    beta_rv = pt.random.beta(n, 1, name="beta")
+    beta_vv = beta_rv.clone()
+    beta_rv_logprob = logp(beta_rv, beta_vv)
+
+    np.testing.assert_allclose(
+        beta_rv_logprob.eval({beta_vv: test_value}),
+        (x_max_logprob.eval({x_max_value: test_value})),
+        rtol=1e-06,
+    )