def scale_gaussian_prior(name, z, logscale_factor=3.0, trainable=True):
  """Returns N(s^i * z^i, std^i) where s^i and std^i are pre-component.

  s^i is a learnable parameter with identity initialization.
  std^i is optionally learnable with identity initialization.

  Args:
    name: variable scope.
    z: input_tensor
    logscale_factor: equivalent to scaling up the learning_rate by a factor
                     of logscale_factor.
    trainable: Whether or not std^i is learnt.
  """
  with tf.variable_scope(name, reuse=tf.AUTO_REUSE):
    z_shape = common_layers.shape_list(z)
    latent_multiplier = tf.get_variable(
        "latent_multiplier", shape=z_shape, dtype=tf.float32,
        initializer=tf.ones_initializer())
    log_scale = tf.get_variable(
        "log_scale_latent", shape=z_shape, dtype=tf.float32,
        initializer=tf.zeros_initializer(), trainable=trainable)
    log_scale = log_scale * logscale_factor
    return tfp.distributions.Normal(
        loc=latent_multiplier * z, scale=tf.exp(log_scale))