Returns a sample from the dim dimensional Halton sequence.

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 1000 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. For low discrepancy sequences and their applications see here.

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.

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.


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(

# 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 =, 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,
sample_leaped = tfp.mcmc.sample_halton_sequence(

integral_leaped = tf.reduce_mean(tf.reduce_prod(sample_leaped ** powers,
with tf.Session() as session:
  values =, true_value))
# Now produces a relative absolute error of 0.05%.
print ("Leaped Estimated: %f, True Value: %f" % values)


  • dim: Positive Python int representing each sample's event_size. Must not be greater than 1000.
  • 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: (Optional) Python integer to seed the random number generator. 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".


  • 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.


  • ValueError: if both sequence_indices and num_results were specified or if dimension dim is less than 1 or greater than 1000.


[1]: Art B. Owen. A randomized Halton algorithm in R. arXiv preprint arXiv:1706.02808, 2017.