Mixture model (Zero-inflated) gives nan log-probablity at 0

To reproduce the result:

import tensorflow_probability as tfp
tfd = tfp.distributions
# Zero inflated log-normal
ziln = tfd.Mixture(
    cat=tfd.Categorical(probs=[0.9, 0.1]),
    components=[
    tfd.Deterministic(loc=0), 
    tfd.LogNormal(loc=6, scale=1),
])
print(ziln.log_prob(0))
# Expected output: log(0.9)
# Output: nan
print(ziln.log_prob(1))
# Expected output: -21.221523
# Output: -21.221523

I think the log_prob should be
if x == 0:
log(0.9)
if x >0:
log(0.1 * lognormal.pdf(x))

Source code of mixture model log_prob:

  def _log_prob(self, x):
    x = tf.convert_to_tensor(x, name='x')
    distribution_log_probs = [d.log_prob(x) for d in self.components]
    cat_log_probs = self._cat_probs(log_probs=True)
    final_log_probs = [
        cat_lp + d_lp
        for (cat_lp, d_lp) in zip(cat_log_probs, distribution_log_probs)
    ]
    concat_log_probs = tf.stack(final_log_probs, 0)
    log_sum_exp = tf.reduce_logsumexp(concat_log_probs, axis=[0])
    return log_sum_exp

Now the mixture model assumes the same x is always in the support of all the distribution components, hence having a non-nan log_prob. It cannot handle cases like zero-inflated gamma / log-normal where the continuous distribution has support (0, inf).

Start with the Mixture._log_prob implementation shown in the issue and run the zero-inflated LogNormal reproduction. Check how out-of-support component log probabilities affect the combined result; done means the two reported inputs return the expected finite log probabilities without breaking ordinary mixture cases.