consistent_sampler

Routine sampler for providing 'consistent sampling' --- sampling that is consistent across subsets (as explained below).

Here we call the elements to be sampled "ids", although they may be arbitrary python objects (strings, tuples, whatever). We assume that ids are distinct.

Consistent sampling works by associating a random "ticket number" with each id; the desired sample is found by taking the subset of the desired sample size containing those elements with the smallest associated random numbers.

The random ticket numbers are computed using a given "seed"; this seed may be an arbitrary python object, typically a large integer or long string.

The sampling is consistent since it consistently favors elements with small ticket numbers; if two sets S and T have substantial overlap, then their samples of a given size will also have substantial overlap (for the same random seed).

This routine takes as input a finite collection of distinct object ids, a random seed, and some other parameters. The sampling may be "with replacement" or "without replacement". One of the additional parameters to the routine is "take" -- the size of the desired sample.

It provides as output a "sampling order" --- an ordered list of object ids that determine the sample. Each object id as associated with a random value (its "ticket number") that depends on the id and the seed; ids are output in order of increasing ticket number. For efficiency and portability, the ticket number is represented as a decimal fraction 0.dddd...dddd between 0 and 1.

For sampling without replacement, the output can not be longer than the input, as no id may appear in the sample more than once.

For sampling with replacement, the output may be infinite in length, as an id may appear in the sample an arbitrarily large (even infinite) number of times. When an id is sampled and then replaced in the set of ids being sampled, it is given a new random ticket number drawn uniformly from the set of numbers in (0, 1) larger than its previous ticket number.

The output of sampler is always a python generator, capable of producing an infinitely long stream of ids.

Example 1.

As a small example of sampling without replacement:

g = sampler(['A-1', 'A-2', 'A-3', 'B-1', 'B-2', 'B-3'], 
            with_replacement=False, take=4, seed=314159, output='id')

yields a generator g whose output can be printed:

print(list(id for id in g))

which produces:

['B-2', 'B-3', 'A-3', 'A-2']

Example 2.

Here is an example where sampling is with replacement from a set of 6 ids, and the output gives a triple (tuple) for each selected id, giving 1. the associated random ticket number, 2. the selected id, and 3. the "generation" (number of times the id has been selected so far).

>>> for t in sampler(['a1', 'b2', 'c3', 'd4', 'e5', 'f6'],
                     with_replacement=True, seed=19283746, take=10):
...     print(t)
('0.303241347', 'e5', 1)
('0.432145156', 'b2', 1)
('0.487135586', 'c3', 1)
('0.581779914', 'b2', 2)
('0.680782907', 'b2', 3)
('0.700258702', 'c3', 2)
('0.816686725', 'b2', 4)
('0.841870265', 'a1', 1)
('0.857737141', 'a1', 2)
('0.866227993', 'f6', 1)

Discussion

This routine is designed for use in election audits, where the ids being sampled are ballot ids, but this routine is suitable for general use.

For a similar election audit sampling method, see Stark's election audit tools: https://www.stat.berkeley.edu/~stark/Vote/auditTools.htm

Consistent sampling is not a new idea, see for example https://arxiv.org/abs/1612.01041 and the references to consistent sampling therein. The routine here may (or may not) be novel in that it extends consistent sampling to sampling with replacement.

For our applications, one big advantage of consistent sampling is the following. If each county collects cast ballots separately, then they can order their own ballots for sampling and interpretation independently of what other counties are doing. An overall sample order can be constructed from the individual county sample order, by merging the list of triples each produces into an overall sorted order.

Usage

Further documentation and examples are: 1. in the code consistent_sampler.py 2. obtainable by running the program demo_consistent_sampler.py 3. described in the file USAGE_EXAMPLES.md 4. in the file docs/consistent_sampling_with_replacement.pdf (This file will soon appear on arXiv as well.)

This code has been packaged and uploaded to PyPI. From python3 you can say

from consistent_sampler import sampler

and then say help(sampler) for more documentation, or run sampler as in the above examples.

Efficiency

The bulk of the work is in computing the SHA256 hash function, which takes about one microsecond per call on a typical laptop. Thus, the running time of sampler is proportional to the length if id_list (to set up the priority queue) plus, if sampling is done with replacement, the value of drop+take, where the constant of proportionality is about 1 microsecond.