Logprob derivation of Max for Discrete IID distributions (#6790)

Author: Dhruvanshu-Joshi

pymc/logprob/order.py

@@ -56,6 +56,7 @@
     _logprob_helper,
 )
 from pymc.logprob.rewriting import measurable_ir_rewrites_db
+from pymc.math import logdiffexp
 from pymc.pytensorf import constant_fold
 
 
@@ -66,6 +67,13 @@ class MeasurableMax(Max):
 MeasurableVariable.register(MeasurableMax)
 
 
+class MeasurableMaxDiscrete(Max):
+    """A placeholder used to specify a log-likelihood for sub-graphs of maxima of discrete variables"""
+
+
+MeasurableVariable.register(MeasurableMaxDiscrete)
+
+
 @node_rewriter([Max])
 def find_measurable_max(fgraph: FunctionGraph, node: Node) -> Optional[List[TensorVariable]]:
     rv_map_feature = getattr(fgraph, "preserve_rv_mappings", None)
@@ -87,10 +95,6 @@ def find_measurable_max(fgraph: FunctionGraph, node: Node) -> Optional[List[Tens
     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:
@@ -102,7 +106,12 @@ def find_measurable_max(fgraph: FunctionGraph, node: Node) -> Optional[List[Tens
     if axis != base_var_dims:
         return None
 
-    measurable_max = MeasurableMax(list(axis))
+    # distinguish measurable discrete and continuous (because logprob is different)
+    if base_var.owner.op.dtype.startswith("int"):
+        measurable_max = MeasurableMaxDiscrete(list(axis))
+    else:
+        measurable_max = MeasurableMax(list(axis))
+
     max_rv_node = measurable_max.make_node(base_var)
     max_rv = max_rv_node.outputs
 
@@ -131,6 +140,26 @@ def max_logprob(op, values, base_rv, **kwargs):
     return logprob
 
 
+@_logprob.register(MeasurableMaxDiscrete)
+def max_logprob_discrete(op, values, base_rv, **kwargs):
+    r"""Compute the log-likelihood graph for the `Max` operation.
+
+    The formula that we use here is :
+    .. math::
+        \ln(P_{(n)}(x)) = \ln(F(x)^n - F(x-1)^n)
+    where $P_{(n)}(x)$ represents the p.m.f of the maximum statistic and $F(x)$ represents the c.d.f of the i.i.d. variables.
+    """
+    (value,) = values
+    logcdf = _logcdf_helper(base_rv, value)
+    logcdf_prev = _logcdf_helper(base_rv, value - 1)
+
+    [n] = constant_fold([base_rv.size])
+
+    logprob = logdiffexp(n * logcdf, n * logcdf_prev)
+
+    return logprob
+
+
 class MeasurableMaxNeg(Max):
     """A placeholder used to specify a log-likelihood for a max(neg(x)) sub-graph.
     This shows up in the graph of min, which is (neg(max(neg(x)))."""

tests/logprob/test_order.py

@@ -39,6 +39,7 @@
 import numpy as np
 import pytensor.tensor as pt
 import pytest
+import scipy.stats as sp
 
 import pymc as pm
 
@@ -230,3 +231,26 @@ def test_min_non_mul_elemwise_fails():
     x_min_value = pt.vector("x_min_value")
     with pytest.raises(RuntimeError, match=re.escape("Logprob method not implemented")):
         x_min_logprob = logp(x_min, x_min_value)
+
+
+@pytest.mark.parametrize(
+    "mu, size, value, axis",
+    [(2, 3, 1, -1), (2, 3, 1, 0), (1, 2, 2, None), (0, 4, 0, 0)],
+)
+def test_max_discrete(mu, size, value, axis):
+    x = pm.Poisson.dist(name="x", mu=mu, size=(size))
+    x_max = pt.max(x, axis=axis)
+    x_max_value = pt.scalar("x_max_value")
+    x_max_logprob = logp(x_max, x_max_value)
+
+    test_value = value
+
+    n = size
+    exp_rv = sp.poisson(mu).cdf(test_value) ** n
+    exp_rv_prev = sp.poisson(mu).cdf(test_value - 1) ** n
+
+    np.testing.assert_allclose(
+        np.log(exp_rv - exp_rv_prev),
+        (x_max_logprob.eval({x_max_value: test_value})),
+        rtol=1e-06,
+    )