def _log_cdf(self, y):
    lower_cutoff = self._lower_cutoff
    upper_cutoff = self._upper_cutoff

    # Recall the promise:
    # cdf(y) := P[Y <= y]
    #         = 1, if y >= upper_cutoff,
    #         = 0, if y < lower_cutoff,
    #         = P[X <= y], otherwise.

    # P[Y <= j] = P[floor(Y) <= j] since mass is only at integers, not in
    # between.
    j = math_ops.floor(y)

    result_so_far = self.distribution.log_cdf(j)

    # Broadcast, because it's possible that this is a single distribution being
    # evaluated on a number of samples, or something like that.
    j += array_ops.zeros_like(result_so_far)

    # Re-define values at the cutoffs.
    if lower_cutoff is not None:
      neg_inf = -np.inf * array_ops.ones_like(result_so_far)
      result_so_far = math_ops.select(j < lower_cutoff, neg_inf, result_so_far)
    if upper_cutoff is not None:
      result_so_far = math_ops.select(j >= upper_cutoff,
                                      array_ops.zeros_like(result_so_far),
                                      result_so_far)

    return result_so_far

  def pdf(self, x, name="pdf"):
    """The PDF of observations in `x` under these Uniform distribution(s).

    Args:
      x: tensor of dtype `dtype`, must be broadcastable with `a` and `b`.
      name: The name to give this op.

    Returns:
      pdf: tensor of dtype `dtype`, the pdf values of `x`. If `x` is `nan`, will
          return `nan`.
    """
    with ops.name_scope(self.name):
      with ops.op_scope([self.a, self.b, x], name):
        x = ops.convert_to_tensor(x, name="x")
        if x.dtype != self.dtype:
          raise TypeError("Input x dtype does not match dtype: %s vs. %s" %
                          (x.dtype, self.dtype))

        broadcasted_x = x * self._ones()
        return math_ops.select(
            math_ops.is_nan(broadcasted_x), broadcasted_x, math_ops.select(
                math_ops.logical_or(broadcasted_x < self.a,
                                    broadcasted_x > self.b),
                array_ops.zeros_like(broadcasted_x),
                (1.0 / self.range()) * array_ops.ones_like(broadcasted_x)))

  def _survival_function(self, y):
    lower_cutoff = self._lower_cutoff
    upper_cutoff = self._upper_cutoff

    # Recall the promise:
    # survival_function(y) := P[Y > y]
    #                       = 0, if y >= upper_cutoff,
    #                       = 1, if y < lower_cutoff,
    #                       = P[X > y], otherwise.

    # P[Y > j] = P[ceiling(Y) > j] since mass is only at integers, not in
    # between.
    j = math_ops.ceil(y)

    # P[X > j], used when lower_cutoff < X < upper_cutoff.
    result_so_far = self.distribution.survival_function(j)

    # Broadcast, because it's possible that this is a single distribution being
    # evaluated on a number of samples, or something like that.
    j += array_ops.zeros_like(result_so_far)

    # Re-define values at the cutoffs.
    if lower_cutoff is not None:
      result_so_far = math_ops.select(j < lower_cutoff,
                                      array_ops.ones_like(result_so_far),
                                      result_so_far)
    if upper_cutoff is not None:
      result_so_far = math_ops.select(j >= upper_cutoff,
                                      array_ops.zeros_like(result_so_far),
                                      result_so_far)

    return result_so_far

 def copy_through():
   # Use broadcasting select to determine which values should get
   # the previous state & zero output, and which values should get
   # a calculated state & output.
   copy_cond = (time >= sequence_length)
   return (math_ops.select(copy_cond, zero_output, output),
           math_ops.select(copy_cond, state, new_state))

    def _variance(self):
        var = self._ones() * math_ops.square(self.sigma) * self.df / (self.df - 2)
        # When 1 < df <= 2, variance is infinite.
        inf = np.array(np.inf, dtype=self.dtype.as_numpy_dtype())
        result_where_defined = math_ops.select(
            math_ops.greater(self.df, array_ops.fill(self.batch_shape(), 2.0)),
            var,
            array_ops.fill(self.batch_shape(), inf, name="inf"),
        )

        if self.allow_nan_stats:
            nan = np.array(np.nan, dtype=self.dtype.as_numpy_dtype())
            return math_ops.select(
                math_ops.greater(self.df, self._ones()),
                result_where_defined,
                array_ops.fill(self.batch_shape(), nan, name="nan"),
            )
        else:
            return control_flow_ops.with_dependencies(
                [
                    check_ops.assert_less(
                        array_ops.ones((), dtype=self.dtype),
                        self.df,
                        message="variance not defined for components of df <= 1",
                    )
                ],
                result_where_defined,
            )

 def _cdf(self, x):
   broadcasted_x = x * array_ops.ones(self.batch_shape())
   zeros = array_ops.zeros_like(x + self.a + self.b, dtype=self.dtype)
   ones = array_ops.ones_like(x + self.a + self.b, dtype=self.dtype)
   result_if_not_big = math_ops.select(
       x < self.a, zeros, (broadcasted_x - self.a) / self.range())
   return math_ops.select(x >= self.b, ones, result_if_not_big)

  def cdf(self, x, name="cdf"):
    """CDF of observations in `x` under these Uniform distribution(s).

    Args:
      x: tensor of dtype `dtype`, must be broadcastable with `a` and `b`.
      name: The name to give this op.

    Returns:
      cdf: tensor of dtype `dtype`, the CDFs of `x`. If `x` is `nan`, will
          return `nan`.
    """
    with ops.op_scope([self.a, self.b, x], self.name):
      with ops.name_scope(name):
        x = ops.convert_to_tensor(x, name="x")
        if x.dtype != self.dtype:
          raise TypeError("Input x dtype does not match dtype: %s vs. %s" %
                          (x.dtype, self.dtype))

    broadcasted_x = x * self._ones()
    return math_ops.select(broadcasted_x < self.a,
                           array_ops.zeros_like(broadcasted_x),
                           math_ops.select(broadcasted_x >= self.b,
                                           array_ops.ones_like(broadcasted_x),
                                           (broadcasted_x - self.a) /
                                           self.range))

 def _copy_some_through(new_output, new_state):
     # Use broadcasting select to determine which values should get
     # the previous state & zero output, and which values should get
     # a calculated state & output.
     copy_cond = time >= sequence_length
     return [math_ops.select(copy_cond, zero_output, new_output)] + [
         math_ops.select(copy_cond, old_s, new_s) for (old_s, new_s) in zip(state, new_state)
     ]

 def _prob(self, x):
   broadcasted_x = x * array_ops.ones(self.batch_shape())
   return math_ops.select(
       math_ops.is_nan(broadcasted_x),
       broadcasted_x,
       math_ops.select(
           math_ops.logical_or(broadcasted_x < self.a,
                               broadcasted_x > self.b),
           array_ops.zeros_like(broadcasted_x),
           (1. / self.range()) * array_ops.ones_like(broadcasted_x)))

def pick_vector(cond,
                true_vector,
                false_vector,
                name="pick_vector"):
  """Picks possibly different length row `Tensor`s based on condition.

  Value `Tensor`s should have exactly one dimension.

  If `cond` is a python Boolean or `tf.constant` then either `true_vector` or
  `false_vector` is immediately returned. I.e., no graph nodes are created and
  no validation happens.

  Args:
    cond: `Tensor`. Must have `dtype=tf.bool` and be scalar.
    true_vector: `Tensor` of one dimension. Returned when cond is `True`.
    false_vector: `Tensor` of one dimension. Returned when cond is `False`.
    name: `String`. The name to give this op.

  Example:

  ```python
  pick_vector(tf.less(0, 5), tf.range(10, 12), tf.range(15, 18))
  # result is tensor: [10, 11].
  pick_vector(tf.less(5, 0), tf.range(10, 12), tf.range(15, 18))
  # result is tensor: [15, 16, 17].
  ```

  Returns:
    true_or_false_vector: `Tensor`.

  Raises:
    TypeError: if `cond.dtype != tf.bool`
    TypeError: if `cond` is not a constant and
      `true_vector.dtype != false_vector.dtype`
  """
  with ops.op_scope((cond, true_vector, false_vector), name):
    cond = ops.convert_to_tensor(cond, name="cond")
    if cond.dtype != dtypes.bool:
      raise TypeError("%s.dtype=%s which is not %s" %
                      (cond.name, cond.dtype, dtypes.bool))
    cond_value_static = tensor_util.constant_value(cond)
    if cond_value_static is not None:
      return true_vector if cond_value_static else false_vector
    true_vector = ops.convert_to_tensor(true_vector, name="true_vector")
    false_vector = ops.convert_to_tensor(false_vector, name="false_vector")
    if true_vector.dtype != false_vector.dtype:
      raise TypeError(
          "%s.dtype=%s does not match %s.dtype=%s"
          % (true_vector.name, true_vector.dtype,
             false_vector.name, false_vector.dtype))
    n = array_ops.shape(true_vector)[0]
    return array_ops.slice(array_ops.concat(0, (true_vector, false_vector)),
                           [math_ops.select(cond, 0, n)],
                           [math_ops.select(cond, n, -1)])

def _ndtr(x):
  """Implements ndtr core logic."""
  half_sqrt_2 = constant_op.constant(
      0.5 * math.sqrt(2.), dtype=x.dtype, name="half_sqrt_2")
  w = x * half_sqrt_2
  z = math_ops.abs(w)
  y = math_ops.select(math_ops.less(z, half_sqrt_2),
                      1. + math_ops.erf(w),
                      math_ops.select(math_ops.greater(w, 0.),
                                      2. - math_ops.erfc(z),
                                      math_ops.erfc(z)))
  return 0.5 * y

 def _mode(self):
     mode = (self.a - 1.0) / (self.a_b_sum - 2.0)
     if self.allow_nan_stats:
         nan = np.array(np.nan, dtype=self.dtype.as_numpy_dtype())
         return math_ops.select(
             math_ops.logical_and(math_ops.greater(self.a, 1.0), math_ops.greater(self.b, 1.0)),
             mode,
             array_ops.fill(self.batch_shape(), nan, name="nan"),
         )
     else:
         return control_flow_ops.with_dependencies(
             [
                 check_ops.assert_less(
                     array_ops.ones((), dtype=self.dtype),
                     self.a,
                     message="Mode not defined for components of a <= 1.",
                 ),
                 check_ops.assert_less(
                     array_ops.ones((), dtype=self.dtype),
                     self.b,
                     message="Mode not defined for components of b <= 1.",
                 ),
             ],
             mode,
         )

  def mean(self, name="mean"):
    """Mean of the distribution.

    The mean of Student's T equals `mu` if `df > 1`, otherwise it is `NaN`.  If
    `self.allow_nan_stats=False`, then an exception will be raised rather than
    returning `NaN`.

    Args:
      name:  A name to give this op.

    Returns:
      The mean for every batch member, a `Tensor` with same `dtype` as self.
    """
    with ops.name_scope(self.name):
      with ops.name_scope(name, values=[self._mu]):
        result_if_defined = self._mu * self._ones()
        if self.allow_nan_stats:
          df_gt_1 = self._df > self._ones()
          nan = np.nan + self._zeros()
          return math_ops.select(df_gt_1, result_if_defined, nan)
        else:
          one = constant_op.constant(1.0, dtype=self.dtype)
          return control_flow_ops.with_dependencies(
              [check_ops.assert_less(
                  one, self._df,
                  message="mean not defined for components of df <= 1"
              )], result_if_defined)

  def mode(self, name="mode"):
    """Mode of the distribution.

    Note that the mode for the Beta distribution is only defined
    when `a > 1`, `b > 1`. This returns the mode when `a > 1` and `b > 1`,
    and NaN otherwise. If `self.allow_nan_stats` is `False`, an exception
    will be raised rather than returning `NaN`.

    Args:
      name: The name for this op.

    Returns:
      Mode of the Beta distribution.
    """
    with ops.name_scope(self.name):
      with ops.op_scope([self._a, self._b, self._a_b_sum], name):
        a = self._a
        b = self._b
        a_b_sum = self._a_b_sum
        one = constant_op.constant(1, self.dtype)
        mode = (a - 1)/ (a_b_sum - 2)

        if self.allow_nan_stats:
          return math_ops.select(
              math_ops.logical_and(
                  math_ops.greater(a, 1), math_ops.greater(b, 1)),
              mode,
              (constant_op.constant(float("NaN"), dtype=self.dtype) *
               array_ops.ones_like(a_b_sum, dtype=self.dtype)))
        else:
          return control_flow_ops.with_dependencies([
              check_ops.assert_less(one, a),
              check_ops.assert_less(one, b)], mode)

  def gradient_clipping(grads_and_vars):
    """Internal function for adaptive clipping."""
    grads, variables = zip(*grads_and_vars)

    norm = clip_ops.global_norm(grads)

    max_norm, log_mean = _adaptive_max_norm(
        norm, std_factor, decay, global_step, epsilon, name)

    # reports the max gradient norm for debugging
    if report_summary:
      summary.scalar("global_norm/adaptive_max_gradient_norm", max_norm)

    # factor will be 1. if norm is smaller than max_norm
    factor = math_ops.select(norm < max_norm,
                             array_ops.ones_like(norm),
                             math_ops.exp(log_mean) / norm)

    if static_max_norm is not None:
      factor = math_ops.minimum(static_max_norm / norm, factor)

    # apply factor
    clipped_grads = []
    for grad in grads:
      if grad is None:
        clipped_grads.append(None)
      elif isinstance(grad, ops.IndexedSlices):
        clipped_grads.append(ops.IndexedSlices(
            grad.values * factor, grad.indices, grad.dense_shape))
      else:
        clipped_grads.append(grad * factor)

    return list(zip(clipped_grads, variables))

  def mode(self, name="mode"):
    """Mode of each batch member.

    The mode of a gamma distribution is `(alpha - 1) / beta` when `alpha > 1`,
    and `NaN` otherwise.  If `self.strict_statistics` is `True`, an exception
    will be raised rather than returning `NaN`.

    Args:
      name:  A name to give this op.

    Returns:
      The mode for every batch member, a `Tensor` with same `dtype` as self.
    """
    alpha = self._alpha
    beta = self._beta
    with ops.name_scope(self.name):
      with ops.op_scope([alpha, beta], name):
        mode_if_defined = (alpha - 1.0) / beta
        if self.strict_statistics:
          one = ops.convert_to_tensor(1.0, dtype=self.dtype)
          return control_flow_ops.with_dependencies(
              [check_ops.assert_less(one, alpha)], mode_if_defined)
        else:
          alpha_ge_1 = alpha >= 1.0
          nan = np.nan * self._ones()
          return math_ops.select(alpha_ge_1, mode_if_defined, nan)

  def variance(self, name="variance"):
    """Variance of each batch member.

    Variance for inverse gamma is defined only for `alpha > 2`. If
    `self.allow_nan_stats` is `False`, an exception will be raised rather
    than returning `NaN`.

    Args:
      name: A name to give this op.

    Returns:
      The variance for every batch member, a `Tensor` with same `dtype` as self.
    """
    alpha = self._alpha
    beta = self._beta
    with ops.name_scope(self.name):
      with ops.op_scope([alpha, beta], name):
        var_if_defined = (math_ops.square(self._beta) /
                          (math_ops.square(self._alpha - 1.0) *
                           (self._alpha - 2.0)))
        if self.allow_nan_stats:
          alpha_gt_2 = alpha > 2.0
          nan = np.nan * self._ones()
          return math_ops.select(alpha_gt_2, var_if_defined, nan)
        else:
          two = constant_op.constant(2.0, dtype=self.dtype)
          return control_flow_ops.with_dependencies(
              [check_ops.assert_less(
                  two, alpha,
                  message="variance not defined for components of alpha <= 2")],
              var_if_defined)

  def mean(self, name="mean"):
    """Mean of each batch member.

    The mean of an inverse gamma distribution is `beta / (alpha - 1)`,
    when `alpha > 1`, and `NaN` otherwise.  If `self.allow_nan_stats` is
    `False`, an exception will be raised rather than returning `NaN`

    Args:
      name: A name to give this op.

    Returns:
      The mean for every batch member, a `Tensor` with same `dtype` as self.
    """
    alpha = self._alpha
    beta = self._beta
    with ops.name_scope(self.name):
      with ops.op_scope([alpha, beta], name):
        mean_if_defined = beta / (alpha - 1.0)
        if self.allow_nan_stats:
          alpha_gt_1 = alpha > 1.0
          nan = np.nan * self._ones()
          return math_ops.select(alpha_gt_1, mean_if_defined, nan)
        else:
          one = constant_op.constant(1.0, dtype=self.dtype)
          return control_flow_ops.with_dependencies(
              [check_ops.assert_less(
                  one, alpha,
                  message="mean not defined for components of alpha <= 1")],
              mean_if_defined)

    def variance(self, name="variance"):
        """Variance of each batch member.

    Variance for inverse gamma is defined only for `alpha > 2`. If
    `self.strict_statistics` is `True`, an exception will be raised rather
    than returning `NaN`.

    Args:
      name: A name to give this op.

    Returns:
      The variance for every batch member, a `Tensor` with same `dtype` as self.
    """
        alpha = self._alpha
        beta = self._beta
        with ops.name_scope(self.name):
            with ops.op_scope([alpha, beta], name):
                var_if_defined = math_ops.square(self._beta) / (
                    math_ops.square(self._alpha - 1.0) * (self._alpha - 2.0)
                )
                if self.strict_statistics:
                    two = ops.convert_to_tensor(2.0, dtype=self.dtype)
                    return control_flow_ops.with_dependencies([check_ops.assert_less(two, alpha)], var_if_defined)
                else:
                    alpha_gt_2 = alpha > 2.0
                    nan = np.nan * self._ones()
                    return math_ops.select(alpha_gt_2, var_if_defined, nan)

  def mode(self, name="mode"):
    """Mode of the distribution.

    Note that the mode for the Beta distribution is only defined
    when `alpha > 1`. This returns the mode when `alpha > 1`,
    and NaN otherwise. If `self.allow_nan_stats` is `False`, an exception
    will be raised rather than returning `NaN`.

    Args:
      name: The name for this op.

    Returns:
      Mode of the Dirichlet distribution.
    """
    with ops.name_scope(self.name):
      with ops.op_scope([self._alpha, self._alpha_0], name):
        one = constant_op.constant(1, self.dtype)
        mode = (self._alpha - 1)/ (
            array_ops.expand_dims(self._alpha_0, -1) - math_ops.cast(
                self.event_shape()[0], self.dtype))

        if self.allow_nan_stats:
          return math_ops.select(
              math_ops.greater(self._alpha, 1),
              mode,
              (constant_op.constant(float("NaN"), dtype=self.dtype) *
               array_ops.ones_like(self._alpha, dtype=self.dtype)))
        else:
          return control_flow_ops.with_dependencies([
              check_ops.assert_less(
                  one, self._alpha,
                  message="mode not defined for components of alpha <= 1")
          ], mode)