code: def sample_halton_sequence(
      dim,
      num_results=None,
      sequence_indices=None,
      dtype=tf.float32,
      randomized=True,
      seed=None,
      name=None,
  ):
      r"""Returns a sample from the `dim` dimensional Halton sequence.
  
      Warning: The sequence elements take values only between 0 and 1. Care must be
      taken to appropriately transform the domain of a function if it differs from
      the unit cube before evaluating integrals using Halton samples. It is also
      important to remember that quasi-random numbers without randomization are not
      a replacement for pseudo-random numbers in every context. Quasi random numbers
      are completely deterministic and typically have significant negative
      autocorrelation unless randomization is used.
  
      Computes the members of the low discrepancy Halton sequence in dimension
      `dim`. The `dim`-dimensional sequence takes values in the unit hypercube in
      `dim` dimensions. Currently, only dimensions up to 10000 are supported. The
      prime base for the k-th axes is the k-th prime starting from 2. For example,
      if `dim` = 3, then the bases will be [2, 3, 5] respectively and the first
      element of the non-randomized sequence will be: [0.5, 0.333, 0.2]. For a more
      complete description of the Halton sequences see
      [here](https://en.wikipedia.org/wiki/Halton_sequence). For low discrepancy
      sequences and their applications see
      [here](https://en.wikipedia.org/wiki/Low-discrepancy_sequence).
  
      If `randomized` is true, this function produces a scrambled version of the
      Halton sequence introduced by [Owen (2017)][1]. For the advantages of
      randomization of low discrepancy sequences see [here](
      https://en.wikipedia.org/wiki/Quasi-Monte_Carlo_method#Randomization_of_quasi-Monte_Carlo).
  
      The number of samples produced is controlled by the `num_results` and
      `sequence_indices` parameters. The user must supply either `num_results` or
      `sequence_indices` but not both.
      The former is the number of samples to produce starting from the first
      element. If `sequence_indices` is given instead, the specified elements of
      the sequence are generated. For example, sequence_indices=tf.range(10) is
      equivalent to specifying n=10.
  
      #### Examples
  
      ```python
      import tensorflow as tf
      import tensorflow_probability as tfp
  
      # Produce the first 1000 members of the Halton sequence in 3 dimensions.
      num_results = 1000
      dim = 3
      sample = tfp.mcmc.sample_halton_sequence(
        dim,
        num_results=num_results,
        seed=127)
  
      # Evaluate the integral of x_1 * x_2^2 * x_3^3  over the three dimensional
      # hypercube.
      powers = tf.range(1.0, limit=dim + 1)
      integral = tf.reduce_mean(tf.reduce_prod(sample ** powers, axis=-1))
      true_value = 1.0 / tf.reduce_prod(powers + 1.0)
      with tf.Session() as session:
        values = session.run((integral, true_value))
  
      # Produces a relative absolute error of 1.7%.
      print ("Estimated: %f, True Value: %f" % values)
  
      # Now skip the first 1000 samples and recompute the integral with the next
      # thousand samples. The sequence_indices argument can be used to do this.
  
  
      sequence_indices = tf.range(start=1000, limit=1000 + num_results,
                                  dtype=tf.int32)
      sample_leaped = tfp.mcmc.sample_halton_sequence(
          dim,
          sequence_indices=sequence_indices,
          seed=111217)
  
      integral_leaped = tf.reduce_mean(tf.reduce_prod(sample_leaped ** powers,
                                                      axis=-1))
      with tf.Session() as session:
        values = session.run((integral_leaped, true_value))
      # Now produces a relative absolute error of 0.05%.
      print ("Leaped Estimated: %f, True Value: %f" % values)
      ```
  
      Args:
        dim: Positive Python `int` representing each sample's `event_size.` Must
          not be greater than 10000.
        num_results: (Optional) Positive scalar `Tensor` of dtype int32. The number
          of samples to generate. Either this parameter or sequence_indices must
          be specified but not both. If this parameter is None, then the behaviour
          is determined by the `sequence_indices`.
          Default value: `None`.
        sequence_indices: (Optional) `Tensor` of dtype int32 and rank 1. The
          elements of the sequence to compute specified by their position in the
          sequence. The entries index into the Halton sequence starting with 0 and
          hence, must be whole numbers. For example, sequence_indices=[0, 5, 6] will
          produce the first, sixth and seventh elements of the sequence. If this
          parameter is None, then the `num_results` parameter must be specified
          which gives the number of desired samples starting from the first sample.
          Default value: `None`.
        dtype: (Optional) The dtype of the sample. One of: `float16`, `float32` or
          `float64`.
          Default value: `tf.float32`.
        randomized: (Optional) bool indicating whether to produce a randomized
          Halton sequence. If True, applies the randomization described in
          [Owen (2017)][1].
          Default value: `True`.
        seed: PRNG seed; see `tfp.random.sanitize_seed` for details. Only used if
          `randomized` is True. If not supplied and `randomized` is True, no seed is
          set.
          Default value: `None`.
        name:  (Optional) Python `str` describing ops managed by this function. If
          not supplied the name of this function is used.
          Default value: "sample_halton_sequence".
  
      Returns:
        halton_elements: Elements of the Halton sequence. `Tensor` of supplied dtype
          and `shape` `[num_results, dim]` if `num_results` was specified or shape
          `[s, dim]` where s is the size of `sequence_indices` if `sequence_indices`
          were specified.
  
      Raises:
        ValueError: if both `sequence_indices` and `num_results` were specified or
          if dimension `dim` is less than 1 or greater than 10000.
  
      #### References
  
      [1]: Art B. Owen. A randomized Halton algorithm in R. _arXiv preprint
           arXiv:1706.02808_, 2017. https://arxiv.org/abs/1706.02808
      """
      if dim < 1 or dim > _MAX_DIMENSION:
          raise ValueError(
              "Dimension must be between 1 and {}. Supplied {}".format(
                  _MAX_DIMENSION, dim
              )
          )
      if (num_results is None) == (sequence_indices is None):
          raise ValueError(
              "Either `num_results` or `sequence_indices` must be"
              " specified but not both."
          )
  
      if not dtype_util.is_floating(dtype):
          raise ValueError("dtype must be of `float`-type")
  
      with tf.name_scope(name or "sample"):
          # Here and in the following, the shape layout is as follows:
          # [sample dimension, event dimension, coefficient dimension].
          # The coefficient dimension is an intermediate axes which will hold the
          # weights of the starting integer when expressed in the (prime) base for
          # an event dimension.
          if sequence_indices is not None:
              sequence_indices = tf.convert_to_tensor(sequence_indices)
          indices = _get_indices(num_results, sequence_indices, dtype)
          if num_results is None:
              num_results = ps.reduce_max(indices)
          radixes = _PRIMES[0:dim][..., np.newaxis]
          max_sizes_by_axes = _base_expansion_size(
              num_results, radixes, dtype
          )
          max_size = ps.reduce_max(max_sizes_by_axes)
  
          # The powers of the radixes that we will need. Note that there is a bit
          # of an excess here. Suppose we need the place value coefficients of 7
          # in base 2 and 3. For 2, we will have 3 digits but we only need 2 digits
          # for base 3. However, we can only create rectangular tensors so we
          # store both expansions in a [2, 3] tensor. This leads to the problem that
          # we might end up attempting to raise large numbers to large powers. For
          # example, base 2 expansion of 1024 has 10 digits. If we were in 10
          # dimensions, then the 10th prime (29) we will end up computing 29^10 even
          # though we don't need it. We avoid this by setting the exponents for each
          # axes to 0 beyond the maximum value needed for that dimension.
          exponents_by_axes = tf.tile(
              [tf.range(max_size, dtype=dtype)], [dim, 1]
          )
  
          # The mask is true for those coefficients that are irrelevant.
          weight_mask = exponents_by_axes < max_sizes_by_axes
          capped_exponents = tf.where(
              weight_mask,
              exponents_by_axes,
              dtype_util.as_numpy_dtype(dtype)(0.0),
          )
          weights = tf.cast(radixes**capped_exponents, dtype=dtype)
          # The following computes the base b expansion of the indices. Suppose,
          # x = a0 + a1*b + a2*b^2 + ... Then, performing a floor div of x with
          # the vector (1, b, b^2, b^3, ...) will produce
          # (a0 + s1 * b, a1 + s2 * b, ...) where s_i are coefficients we don't care
          # about. Noting that all a_i < b by definition of place value expansion,
          # we see that taking the elements mod b of the above vector produces the
          # place value expansion coefficients.
          coeffs = tf.math.floordiv(indices, weights)
          coeffs *= tf.cast(weight_mask, dtype)
          coeffs %= radixes
          if not randomized:
              coeffs /= radixes
              return tf.reduce_sum(coeffs / weights, axis=-1)
  
          shuffle_seed, zero_correction_seed = samplers.split_seed(
              seed, salt="MCMCSampleHaltonSequence"
          )
  
          coeffs = _randomize(coeffs, radixes, seed=shuffle_seed)
          # Remove the contribution from randomizing the trailing zero for the
          # axes where max_size_by_axes < max_size. This will be accounted
          # for separately below (using zero_correction).
          coeffs *= tf.cast(weight_mask, dtype)
          coeffs /= radixes
          base_values = tf.reduce_sum(coeffs / weights, axis=-1)
  
          # The randomization used in Owen (2017) does not leave 0 invariant. While
          # we have accounted for the randomization of the first `max_size_by_axes`
          # coefficients, we still need to correct for the trailing zeros. Luckily,
          # this is equivalent to adding a uniform random value scaled so the first
          # `max_size_by_axes` coefficients are zero. The following statements perform
          # this correction.
          zero_correction = samplers.uniform(
              [dim, 1], seed=zero_correction_seed, dtype=dtype
          )
          zero_correction /= tf.cast(
              radixes**max_sizes_by_axes, dtype
          )
          return base_values + tf.reshape(zero_correction, [-1])
  
file_path: probability/tensorflow_probability/python/mcmc/sample_halton_sequence_lib.py
imports: import numpy
  import tensorflow.compat.v2
  from tensorflow_probability.python.internal import dtype_util
  from tensorflow_probability.python.internal import prefer_static
  from tensorflow_probability.python.internal import samplers
label_desc: Use image node_id 1 for calling a global function with example usage: sample_halton_sequence(dim, num_results, sequence_indices, dtype, randomized, seed, name) and returns: unknown, tf