diff --git a/pymc/logprob/utils.py b/pymc/logprob/utils.py
index f41c63c978..3b3fc4737c 100644
--- a/pymc/logprob/utils.py
+++ b/pymc/logprob/utils.py
@@ -48,7 +48,7 @@
 from pytensor.graph.op import HasInnerGraph
 from pytensor.raise_op import CheckAndRaise
 from pytensor.scalar.basic import Mul
-from pytensor.tensor.basic import get_underlying_scalar_constant_value
+from pytensor.tensor.basic import AllocEmpty, get_underlying_scalar_constant_value
 from pytensor.tensor.elemwise import Elemwise
 from pytensor.tensor.exceptions import NotScalarConstantError
 from pytensor.tensor.random.op import RandomVariable
@@ -244,7 +244,7 @@ class DiracDelta(MeasurableOp, Op):
 
     __props__ = ("rtol", "atol")
 
-    def __init__(self, rtol=1e-5, atol=1e-8):
+    def __init__(self, rtol, atol):
         self.rtol = rtol
         self.atol = atol
 
@@ -267,15 +267,25 @@ def infer_shape(self, fgraph, node, input_shapes):
         return input_shapes
 
 
-dirac_delta = DiracDelta()
+def dirac_delta(x, rtol=1e-5, atol=1e-8):
+    return DiracDelta(rtol, atol)(x)
 
 
 @_logprob.register(DiracDelta)
-def diracdelta_logprob(op, values, *inputs, **kwargs):
-    (values,) = values
-    (const_value,) = inputs
-    values, const_value = pt.broadcast_arrays(values, const_value)
-    return pt.switch(pt.isclose(values, const_value, rtol=op.rtol, atol=op.atol), 0.0, -np.inf)
+def diracdelta_logprob(op, values, const_value, **kwargs):
+    [value] = values
+
+    if const_value.owner and isinstance(const_value.owner.op, AllocEmpty):
+        # Any value is considered valid for an AllocEmpty array
+        return pt.zeros_like(value)
+
+    if op.rtol == 0 and op.atol == 0:
+        # Strict equality, cheaper logp
+        match = pt.eq(value, const_value)
+    else:
+        # Loose equality, more expensive logp
+        match = pt.isclose(values, const_value, rtol=op.rtol, atol=op.atol)
+    return pt.switch(match, 0.0, -np.inf)
 
 
 def find_negated_var(var):