Add the transformation between the inverse gamma and the exponential

aemcmc/transforms.py

@@ -70,3 +70,54 @@ def location_scale_transform(in_expr, out_expr):
         eq(out_expr, noncentered_et),
         location_scale_family(distribution_lv),
     )
+
+
+def invgamma_exponential(invgamma_expr, invexponential_expr):
+    r"""Produce a goal that represents the relation between the inverse gamma distribution
+    and the inverse of an exponential distribution.
+
+    .. math::
+
+        \begin{equation*}
+            \frac{
+                X \sim \operatorname{Gamma^{-1}}\left(1, c\right)
+            }{
+                Y = 1 / X, \quad
+                Y \sim \operatorname{Exp}\left(c\right)
+            }
+        \end{equation*}
+
+    TODO: This is a particular case of a more general relation between the inverse gamma
+    and the gamma distribution (of which the exponential distribution is a special case).
+    We should implement this more general relation, and the special case separately in the
+    future.
+
+    Parameters
+    ----------
+    invgamma_expr
+        An expression that represents a random variable with an inverse gamma
+        distribution with a shape parameter equal to 1.
+    invexponential_expr
+        An expression that represents the inverse of a random variable with an
+        exponential distribution.
+
+    """
+    c_lv = var()
+    rng_lv, size_lv, dtype_lv = var(), var(), var()
+
+    invgamma_et = etuple(
+        etuplize(at.random.invgamma), rng_lv, size_lv, dtype_lv, at.as_tensor(1.0), c_lv
+    )
+
+    exponential_et = etuple(
+        etuplize(at.random.exponential),
+        c_lv,
+        rng=rng_lv,
+        size=size_lv,
+        dtype=dtype_lv,
+    )
+    invexponential_et = etuple(at.true_div, at.as_tensor(1.0), exponential_et)
+
+    return lall(
+        eq(invgamma_expr, invgamma_et), eq(invexponential_expr, invexponential_et)
+    )

tests/test_transforms.py

@@ -3,7 +3,7 @@
 from aesara.graph.fg import FunctionGraph
 from aesara.graph.kanren import KanrenRelationSub
 
-from aemcmc.transforms import location_scale_transform
+from aemcmc.transforms import invgamma_exponential, location_scale_transform
 
 
 def test_normal_scale_loc_transform_lift():
@@ -45,3 +45,35 @@ def test_normal_scale_loc_transform_sink():
     )[0]
 
     assert isinstance(res.owner.op, type(at.random.normal))
+
+
+def test_invgamma_to_exp():
+
+    srng = at.random.RandomStream(0)
+    c_at = at.scalar()
+    X_rv = srng.invgamma(1.0, c_at)
+
+    fgraph = FunctionGraph(outputs=[X_rv], clone=False)
+    res = KanrenRelationSub(invgamma_exponential).transform(
+        fgraph, fgraph.outputs[0].owner
+    )[0]
+
+    assert isinstance(res.owner.op, type(at.true_div))
+    assert isinstance(res.owner.inputs[1].owner.op, type(at.random.exponential))
+
+
+@pytest.mark.xfail(
+    reason="Op.__call__ does not dispatch to Op.make_node for some RandomVariable and etuple evaluation returns an error"
+)
+def test_invgamma_from_exp():
+
+    srng = at.random.RandomStream(0)
+    c_at = at.scalar()
+    X_rv = 1.0 / srng.exponential(c_at)
+
+    fgraph = FunctionGraph(outputs=[X_rv], clone=False)
+    res = KanrenRelationSub(lambda x, y: invgamma_exponential(y, x)).transform(
+        fgraph, fgraph.outputs[0].owner
+    )[0]
+
+    assert isinstance(res.owner.op, type(at.random.inversegamma))