Discrete Gaussian Samplers over the Integers¶
This class realizes oracles which returns integers proportionally to
\(\exp((xc)^2/(2σ^2))\). All oracles are implemented using rejection sampling.
See DiscreteGaussianDistributionIntegerSampler.__init__()
for which algorithms are
available.
AUTHORS:
 Martin Albrecht (20140628): initial version
EXAMPLES:
We construct a sampler for the distribution \(D_{3,c}\) with width \(σ=3\) and center \(c=0\):
sage: from sage.stats.distributions.discrete_gaussian_integer import DiscreteGaussianDistributionIntegerSampler
sage: sigma = 3.0
sage: D = DiscreteGaussianDistributionIntegerSampler(sigma=sigma)
We ask for 100000 samples:
sage: from six.moves import range
sage: n=100000; l = [D() for _ in range(n)]
These are sampled with a probability proportional to \(\exp(x^2/18)\). More precisely we have to normalise by dividing by the overall probability over all integers. We use the fact that hitting anything more than 6 standard deviations away is very unlikely and compute:
sage: bound = (6*sigma).floor()
sage: norm_factor = sum([exp(x^2/(2*sigma^2)) for x in range(bound,bound+1)])
sage: norm_factor
7.519...
With this normalisation factor, we can now test if our samples follow the expected distribution:
sage: x=0; l.count(x), ZZ(round(n*exp(x^2/(2*sigma^2))/norm_factor))
(13355, 13298)
sage: x=4; l.count(x), ZZ(round(n*exp(x^2/(2*sigma^2))/norm_factor))
(5479, 5467)
sage: x=10; l.count(x), ZZ(round(n*exp(x^2/(2*sigma^2))/norm_factor))
(53, 51)
We construct an instance with a larger width:
sage: from sage.stats.distributions.discrete_gaussian_integer import DiscreteGaussianDistributionIntegerSampler
sage: sigma = 127
sage: D = DiscreteGaussianDistributionIntegerSampler(sigma=sigma, algorithm='uniform+online')
ask for 100000 samples:
sage: from six.moves import range
sage: n=100000; l = [D() for _ in range(n)] # long time
and check if the proportions fit:
sage: x=0; y=1; float(l.count(x))/l.count(y), exp(x^2/(2*sigma^2))/exp(y^2/(2*sigma^2)).n() # long time
(1.0, 1.00...)
sage: x=0; y=100; float(l.count(x))/l.count(y), exp(x^2/(2*sigma^2))/exp(y^2/(2*sigma^2)).n() # long time
(1.32..., 1.36...)
We construct a sampler with \(c\%1 != 0\):
sage: from sage.stats.distributions.discrete_gaussian_integer import DiscreteGaussianDistributionIntegerSampler
sage: sigma = 3
sage: D = DiscreteGaussianDistributionIntegerSampler(sigma=sigma, c=1/2)
sage: from six.moves import range
sage: n=100000; l = [D() for _ in range(n)] # long time
sage: mean(l).n() # long time
0.486650000000000
REFERENCES:

class
sage.stats.distributions.discrete_gaussian_integer.
DiscreteGaussianDistributionIntegerSampler
¶ Bases:
sage.structure.sage_object.SageObject
A Discrete Gaussian Sampler using rejection sampling.

__init__
(sigma, c=0, tau=6, algorithm=None, precision='mp')¶ Construct a new sampler for a discrete Gaussian distribution.
INPUT:
sigma
 samples \(x\) are accepted with probability proportional to \(\exp((xc)²/(2σ²))\)c
 the mean of the distribution. The value ofc
does not have to be an integer. However, some algorithms only support integervaluedc
(default:0
)tau
 samples outside the range \((⌊c⌉⌈στ⌉,...,⌊c⌉+⌈στ⌉)\) are considered to have probability zero. This bound applies to algorithms which sample from the uniform distribution (default:6
)algorithm
 see list below (default:"uniform+table"
for \(σt\) bounded by
DiscreteGaussianDistributionIntegerSampler.table_cutoff
and"uniform+online"
for bigger \(στ\))
precision
 either"mp"
for multiprecision where the actual precision used is taken from sigma or"dp"
for double precision. In the latter case results are not reproducible. (default:"mp"
)
ALGORITHMS:
"uniform+table"
 classical rejection sampling, sampling from the uniform distribution and accepted with probability proportional to \(\exp((xc)²/(2σ²))\) where \(\exp((xc)²/(2σ²))\) is precomputed and stored in a table. Any realvalued \(c\) is supported."uniform+logtable"
 samples are drawn from a uniform distribution and accepted with probability proportional to \(\exp((xc)²/(2σ²))\) where \(\exp((xc)²/(2σ²))\) is computed using logarithmically many calls to Bernoulli distributions. See [DDLL2013] for details. Only integervalued \(c\) are supported."uniform+online"
 samples are drawn from a uniform distribution and accepted with probability proportional to \(\exp((xc)²/(2σ²))\) where \(\exp((xc)²/(2σ²))\) is computed in each invocation. Typically this is very slow. See [DDLL2013] for details. Any realvalued \(c\) is accepted."sigma2+logtable"
 samples are drawn from an easily samplable distribution with \(σ = k·σ_2\) with \(σ_2 = \sqrt{1/(2\log 2)}\) and accepted with probability proportional to \(\exp((xc)²/(2σ²))\) where \(\exp((xc)²/(2σ²))\) is computed using logarithmically many calls to Bernoulli distributions (but no calls to \(\exp\)). See [DDLL2013] for details. Note that this sampler adjusts \(σ\) to match \(k·σ_2\) for some integer \(k\). Only integervalued \(c\) are supported.
EXAMPLES:
sage: from sage.stats.distributions.discrete_gaussian_integer import DiscreteGaussianDistributionIntegerSampler sage: DiscreteGaussianDistributionIntegerSampler(3.0, algorithm="uniform+online") Discrete Gaussian sampler over the Integers with sigma = 3.000000 and c = 0 sage: DiscreteGaussianDistributionIntegerSampler(3.0, algorithm="uniform+table") Discrete Gaussian sampler over the Integers with sigma = 3.000000 and c = 0 sage: DiscreteGaussianDistributionIntegerSampler(3.0, algorithm="uniform+logtable") Discrete Gaussian sampler over the Integers with sigma = 3.000000 and c = 0
Note that
"sigma2+logtable"
adjusts \(σ\):sage: DiscreteGaussianDistributionIntegerSampler(3.0, algorithm="sigma2+logtable") Discrete Gaussian sampler over the Integers with sigma = 3.397287 and c = 0

__call__
()¶ Return a new sample.
EXAMPLES:
sage: from sage.stats.distributions.discrete_gaussian_integer import DiscreteGaussianDistributionIntegerSampler sage: DiscreteGaussianDistributionIntegerSampler(3.0, algorithm="uniform+online")() 3 sage: DiscreteGaussianDistributionIntegerSampler(3.0, algorithm="uniform+table")() 3

algorithm
¶

c
¶

sigma
¶

tau
¶
