Implement Hyp2F1 and gradients

Co-authored-by: ColtAllen <[email protected]>

Author: ricardoV94

pytensor/scalar/math.py

@@ -1481,3 +1481,170 @@ def c_code(self, *args, **kwargs):
 
 
 betainc_der = BetaIncDer(upgrade_to_float_no_complex, name="betainc_der")
+
+
+class Hyp2F1(ScalarOp):
+    """
+    Gaussian hypergeometric function ``2F1(a, b; c; z)``.
+
+    """
+
+    nin = 4
+    nfunc_spec = ("scipy.special.hyp2f1", 4, 1)
+
+    @staticmethod
+    def st_impl(a, b, c, z):
+        return scipy.special.hyp2f1(a, b, c, z)
+
+    def impl(self, a, b, c, z):
+        return Hyp2F1.st_impl(a, b, c, z)
+
+    def grad(self, inputs, grads):
+        a, b, c, z = inputs
+        (gz,) = grads
+        return [
+            gz * hyp2f1_der(a, b, c, z, wrt=0),
+            gz * hyp2f1_der(a, b, c, z, wrt=1),
+            gz * hyp2f1_der(a, b, c, z, wrt=2),
+            gz * ((a * b) / c) * hyp2f1(a + 1, b + 1, c + 1, z),
+        ]
+
+    def c_code(self, *args, **kwargs):
+        raise NotImplementedError()
+
+
+hyp2f1 = Hyp2F1(upgrade_to_float, name="hyp2f1")
+
+
+class Hyp2F1Der(ScalarOp):
+    """
+    Derivatives of the Gaussian hypergeometric function ``2F1(a, b; c; z)`` with respect to one of the first 3 inputs.
+
+    """
+
+    nin = 5
+
+    def impl(self, a, b, c, z, wrt):
+        """Adapted from https://github.com/stan-dev/math/blob/develop/stan/math/prim/fun/grad_2F1.hpp"""
+
+        def check_2f1_converges(a, b, c, z) -> bool:
+            num_terms = 0
+            is_polynomial = False
+
+            def is_nonpositive_integer(x):
+                return x <= 0 and x.is_integer()
+
+            if is_nonpositive_integer(a) and abs(a) >= num_terms:
+                is_polynomial = True
+                num_terms = int(np.floor(abs(a)))
+            if is_nonpositive_integer(b) and abs(b) >= num_terms:
+                is_polynomial = True
+                num_terms = int(np.floor(abs(b)))
+
+            is_undefined = is_nonpositive_integer(c) and abs(c) <= num_terms
+
+            return not is_undefined and (
+                is_polynomial or np.abs(z) < 1 or (np.abs(z) == 1 and c > (a + b))
+            )
+
+        def compute_grad_2f1(a, b, c, z, wrt):
+            """
+            Notes
+            -----
+            The algorithm can be derived by looking at the ratio of two successive terms in the series
+            β_{k+1}/β_{k} = A(k)/B(k)
+            β_{k+1} = A(k)/B(k) * β_{k}
+            d[β_{k+1}] = d[A(k)/B(k)] * β_{k} + A(k)/B(k) * d[β_{k}] via the product rule
+
+            In the 2F1, A(k)/B(k) corresponds to (((a + k) * (b + k) / ((c + k) (1 + k))) * z
+
+            The partial d[A(k)/B(k)] with respect to the 3 first inputs can be obtained from the ratio A(k)/B(k),
+            by dropping the respective term
+            d/da[A(k)/B(k)] = A(k)/B(k) / (a + k)
+            d/db[A(k)/B(k)] = A(k)/B(k) / (b + k)
+            d/dc[A(k)/B(k)] = A(k)/B(k) * (c + k)
+
+            The algorithm is implemented in the log scale, which adds the complexity of working with absolute terms and
+            tracking the sign of the terms involved.
+            """
+
+            wrt_a = wrt_b = False
+            if wrt == 0:
+                wrt_a = True
+            elif wrt == 1:
+                wrt_b = True
+            elif wrt != 2:
+                raise ValueError(f"wrt must be 0, 1, or 2, got {wrt}")
+
+            min_steps = 10  # https://github.com/stan-dev/math/issues/2857
+            max_steps = int(1e6)
+            precision = 1e-14
+
+            res = 0
+
+            if z == 0:
+                return res
+
+            log_g_old = -np.inf
+            log_t_old = 0.0
+            log_t_new = 0.0
+            sign_z = np.sign(z)
+            log_z = np.log(np.abs(z))
+
+            log_g_old_sign = 1
+            log_t_old_sign = 1
+            log_t_new_sign = 1
+            sign_zk = sign_z
+
+            for k in range(max_steps):
+                p = (a + k) * (b + k) / ((c + k) * (k + 1))
+                if p == 0:
+                    return res
+                log_t_new += np.log(np.abs(p)) + log_z
+                log_t_new_sign = np.sign(p) * log_t_new_sign
+
+                term = log_g_old_sign * log_t_old_sign * np.exp(log_g_old - log_t_old)
+                if wrt_a:
+                    term += np.reciprocal(a + k)
+                elif wrt_b:
+                    term += np.reciprocal(b + k)
+                else:
+                    term -= np.reciprocal(c + k)
+
+                log_g_old = log_t_new + np.log(np.abs(term))
+                log_g_old_sign = np.sign(term) * log_t_new_sign
+                g_current = log_g_old_sign * np.exp(log_g_old) * sign_zk
+                res += g_current
+
+                log_t_old = log_t_new
+                log_t_old_sign = log_t_new_sign
+                sign_zk *= sign_z
+
+                if k >= min_steps and np.abs(g_current) <= precision:
+                    return res
+
+            warnings.warn(
+                f"hyp2f1_der did not converge after {k} iterations",
+                RuntimeWarning,
+            )
+            return np.nan
+
+        # TODO: We could implement the Euler transform to expand supported domain, as Stan does
+        if not check_2f1_converges(a, b, c, z):
+            warnings.warn(
+                f"Hyp2F1 does not meet convergence conditions with given arguments a={a}, b={b}, c={c}, z={z}",
+                RuntimeWarning,
+            )
+            return np.nan
+
+        return compute_grad_2f1(a, b, c, z, wrt=wrt)
+
+    def __call__(self, a, b, c, z, wrt):
+        # This allows wrt to be a keyword argument
+        return super().__call__(a, b, c, z, wrt)
+
+    def c_code(self, *args, **kwargs):
+        raise NotImplementedError()
+
+
+hyp2f1_der = Hyp2F1Der(upgrade_to_float, name="hyp2f1_der")

pytensor/tensor/inplace.py

@@ -392,6 +392,11 @@ def conj_inplace(a):
     """elementwise conjugate (inplace on `a`)"""
 
 
+@scalar_elemwise
+def hyp2f1_inplace(a, b, c, z):
+    """gaussian hypergeometric function"""
+
+
 pprint.assign(add_inplace, printing.OperatorPrinter("+=", -2, "either"))
 pprint.assign(mul_inplace, printing.OperatorPrinter("*=", -1, "either"))
 pprint.assign(sub_inplace, printing.OperatorPrinter("-=", -2, "left"))

pytensor/tensor/math.py

@@ -1384,6 +1384,11 @@ def gammal(k, x):
     """Lower incomplete gamma function."""
 
 
+@scalar_elemwise
+def hyp2f1(a, b, c, z):
+    """Gaussian hypergeometric function."""
+
+
 @scalar_elemwise
 def j0(x):
     """Bessel function of the first kind of order 0."""
@@ -3132,4 +3137,5 @@ def matmul(x1: "ArrayLike", x2: "ArrayLike", dtype: Optional["DTypeLike"] = None
     "power",
     "logaddexp",
     "logsumexp",
+    "hyp2f1",
 ]