Draft of switch StudentT cdf to use tfp's betainc (#1475)

* Switch `StudentT` `cdf` to use tfp's `betainc`

Jax's `betainc` doesn't have gradients defined for all parameters while tfp's does

* Cleanup in line with PR review

* Remove unneeded import as directed by linter

Author: colehaus

numpyro/distributions/continuous.py

@@ -1909,18 +1909,26 @@ def variance(self):
         return jnp.broadcast_to(var, self.batch_shape)
 
     def cdf(self, value):
+        try:
+            from tensorflow_probability.substrates.jax.math import betainc as betainc_fn
+        except ImportError:
+            from jax.scipy.special import betainc as betainc_fn
+
         # Ref: https://en.wikipedia.org/wiki/Student's_t-distribution#Related_distributions
         # X^2 ~ F(1, df) -> df / (df + X^2) ~ Beta(df/2, 0.5)
         scaled = (value - self.loc) / self.scale
         scaled_squared = scaled * scaled
         beta_value = self.df / (self.df + scaled_squared)
+
         # when scaled < 0, returns 0.5 * Beta(df/2, 0.5).cdf(beta_value)
         # when scaled > 0, returns 1 - 0.5 * Beta(df/2, 0.5).cdf(beta_value)
         scaled_sign_half = 0.5 * jnp.sign(scaled)
         return (
             0.5
             + scaled_sign_half
-            - 0.5 * jnp.sign(scaled) * betainc(0.5 * self.df, 0.5, beta_value)
+            - 0.5
+            * jnp.sign(scaled)
+            * betainc_fn(0.5 * jnp.asarray(self.df), 0.5, jnp.asarray(beta_value))
         )
 
     def icdf(self, q):

setup.py

@@ -64,7 +64,7 @@
             "jaxns==1.0.0",
             "optax>=0.0.6",
             "pyyaml",  # flax dependency
-            "tensorflow_probability>=0.15.0",
+            "tensorflow_probability>=0.17.0",
         ],
         "examples": [
             "arviz",