distributions#
Probability functions for distributions commonly used in differential privacy.
Functions#
Returns the pmf for a double-sided geometric distribution at k. |
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Returns the cmf for a double-sided geometric distribution at k. |
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Returns exact value of the cmf for a double-sided geometric distribution at k. |
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Returns the inverse cmf of a double-sided geometric distribution at p. |
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Return exact value of inverse cmf of double-sided geometric distribution at p. |
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Returns the pmf for a discrete gaussian distribution at k. |
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Returns the cmf for a discrete gaussian distribution at k. |
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Returns the inverse cmf for a discrete gaussian distribution at p. |
- double_sided_geometric_pmf(k: int, alpha: float) float#
- double_sided_geometric_pmf(k: numpy.ndarray, alpha: float) numpy.ndarray
- double_sided_geometric_pmf(k: int, alpha: numpy.ndarray) numpy.ndarray
Returns the pmf for a double-sided geometric distribution at k.
For \(k \in \mathbb{Z}\)
\[f(k)= \frac {e^{1 / \alpha} - 1} {e^{1 / \alpha} + 1} \cdot e^{\frac{-\mid k \mid}{\alpha}}\]A double sided geometric distribution is the difference between two geometric distributions (It can be sampled by sampling two values from a geometric distribution, and taking their difference).
See section 4.1 in [BV18] or scipy.stats.geom for more information (Note that the parameter \(p\) used in scipy.stats.geom is related to \(\alpha\) through \(p = 1 - e^{-1 / \alpha}\)).
- Parameters
k – The values to calculate the pmf for.
alpha – The scale of the geometric distribution.
- double_sided_geometric_cmf(k: int, alpha: float) float#
- double_sided_geometric_cmf(k: numpy.ndarray, alpha: float) numpy.ndarray
Returns the cmf for a double-sided geometric distribution at k.
For \(k \in \mathbb{Z}\)
\[\begin{split}F(k) = \begin{cases} \frac{e^{1 / \alpha}}{e^{1 / \alpha} + 1} \cdot e^\frac{k}{\alpha} & \text{if k} \le 0 \\ 1 - \frac{1}{e^{1 / \alpha} + 1}\cdot e^{\frac{-k}{\alpha}} & \text{otherwise} \\ \end{cases}\end{split}\]See
double_sided_geometric_pmf()for more information.- Parameters
k – The values to calculate the pmf for.
alpha – The scale of the geometric distribution.
- double_sided_geometric_cmf_exact(k, alpha)#
Returns exact value of the cmf for a double-sided geometric distribution at k.
See
double_sided_geometric_cmf()for more information.- Parameters
k (tmlt.core.utils.exact_number.ExactNumberInput) – The values to calculate the pmf for.
alpha (tmlt.core.utils.exact_number.ExactNumberInput) – The scale of the geometric distribution.
- Return type
- double_sided_geometric_inverse_cmf(p: float, alpha: float) int#
- double_sided_geometric_inverse_cmf(p: numpy.ndarray, alpha: float) numpy.ndarray
Returns the inverse cmf of a double-sided geometric distribution at p.
In other words, it returns the smallest k s.t. CMF(k) >= p.
For \(p \in [0, 1]\)
\[\begin{split}F(k) = \begin{cases} \alpha\ln(p\cdot\frac{e^{\frac{1}{\alpha}} + 1}{e^{\frac{1}{\alpha}}}) & \text{if p} \le 0.5 \\ -\alpha\ln((e^{\frac{1}{\alpha}} + 1)(1 - p)) & \text{otherwise} \\ \end{cases}\end{split}\]See
double_sided_geometric_pmf()for more information.
- double_sided_geometric_inverse_cmf_exact(p, alpha)#
Return exact value of inverse cmf of double-sided geometric distribution at p.
See
double_sided_geometric_inverse_cmf()for more information.- Parameters
p (tmlt.core.utils.exact_number.ExactNumberInput) – The values to calculate the inverse cmf for.
alpha (tmlt.core.utils.exact_number.ExactNumberInput) – The scale of the geometric distribution.
- Return type
- discrete_gaussian_pmf(k: int, sigma_squared: float) float#
- discrete_gaussian_pmf(k: numpy.ndarray, sigma_squared: float) numpy.ndarray
Returns the pmf for a discrete gaussian distribution at k.
For \(k \in \mathbb{Z}\)
(1)#\[f(k) = \frac {e^{-k^2/2\sigma^2}} { \sum_{n\in \mathbb{Z}} e^{-n^2/2\sigma^2} }\]See [CKS20] for more information. The formula above is based on Definition 1 in the paper.
- Parameters
k – The value to evaluate the pmf at.
sigma_squared – The variance of the discrete gaussian distribution.
- discrete_gaussian_cmf(k: int, sigma_squared: float) float#
- discrete_gaussian_cmf(k: numpy.ndarray, sigma_squared: float) numpy.ndarray
Returns the cmf for a discrete gaussian distribution at k.
See (1) for the probability mass function.
- Parameters
k – The value to evaluate the cmf at.
sigma_squared – The variance of the discrete gaussian distribution.
- discrete_gaussian_inverse_cmf(p: float, sigma_squared: float) int#
- discrete_gaussian_inverse_cmf(p: numpy.ndarray, sigma_squared: float) numpy.ndarray
- discrete_gaussian_inverse_cmf(p: tmlt.core.utils.arb.Arb, sigma_squared: tmlt.core.utils.arb.Arb) int
Returns the inverse cmf for a discrete gaussian distribution at p.
In other words, it returns the smallest k s.t. CMF(k) >= p.
- Parameters
p – The value to evaluate the inverse cmf at.
sigma_squared – The variance of the discrete gaussian distribution.