class

Kumaraswamy

extendsDistribution

Kumaraswamy(concentration1: Tensor | float, concentration0: Tensor | float, validate_args: bool | None = None)

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Kumaraswamy distribution on $[0, 1]$ .

Two-parameter continuous distribution on the unit interval that mimics the shapes of the lucid.distributions.Beta distribution but has a closed-form CDF and inverse CDF. This makes it a popular choice in variational autoencoders and normalising flows where cheap, exact reparameterised sampling is needed and no special functions are required.

Parameters

concentration1Tensor or float

First shape parameter

a > 0

concentration0Tensor or float

Second shape parameter

b > 0

validate_argsbool= None

If True, validate parameter constraints at construction time.

Notes

Probability density on $x \in [0, 1]$ :

p(x; a, b) = a\, b\, x^{a - 1} (1 - x^a)^{b - 1}

Cumulative distribution (closed form):

F(x; a, b) = 1 - (1 - x^a)^b

Inverse CDF (used by rsample):

F^{-1}(u; a, b) = (1 - (1 - u)^{1/b})^{1/a}

Moments are expressible in closed form via the Beta function:

\mathbb{E}[X^n] = b\, B(1 + n/a,\, b)

Special cases:

$a = 1$ → $\mathrm{Beta}(1, b)$
$b = 1$ → $\mathrm{Beta}(a, 1)$

Unlike Beta, Kumaraswamy is not a member of the exponential family and lacks an analytic normalising integral for sums. Its main advantage is the lack of any reliance on the gamma function in forward or backward passes.

Examples

>>> import lucid
>>> from lucid.distributions import Kumaraswamy
>>> d = Kumaraswamy(concentration1=2.0, concentration0=5.0)
>>> d.rsample((4,)).shape
(4,)
>>> d.log_prob(lucid.tensor(0.3))  # log(2 * 5 * 0.3 * (1 - 0.3**2)**4)
tensor(0.7214)

Used by 2

Constructors

dunder

init

→None

__init__(concentration1: Tensor | float, concentration0: Tensor | float, validate_args: bool | None = None)

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Initialise a Kumaraswamy distribution.

Parameters

concentration1Tensor | float

First shape parameter

a > 0

(controls the left tail).

concentration0Tensor | float

Second shape parameter

b > 0

(controls the right tail).

validate_argsbool | None= None

If True, validate parameter constraints at construction time.

Properties

prop

mean

→Tensor

mean: Tensor

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E[X] = b · B(1 + 1/a, b) expressed via lgamma.

prop

mode

→Tensor

mode: Tensor

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Mode of the Kumaraswamy distribution.

\left(\frac{a - 1}{a b - 1}\right)^{1/a} \quad (a, b \geq 1)

with $a$ = concentration1 and $b$ = concentration0; nan when either is below 1, as the reference framework answers. The reference evaluates a different expression and answers nan for every $b > 1$ as well; this is the closed form.

prop

variance

→Tensor

variance: Tensor

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Var[X] = E[X²] − E[X]² where E[X²] = b · B(1 + 2/a, b).

Instance methods

cdf

→Tensor

cdf(value: Tensor)

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Cumulative distribution function of the Kumaraswamy distribution.

F(x; a, b) = 1 - (1 - x^a)^b, \quad x \in [0, 1]

Evaluated as $-\operatorname{expm1}(b \log(1 - x^a))$ , which keeps the left tail's relative precision.

Returns probabilities in $[0, 1]$ ; 0 below the support and 1 above it.

Examples

>>> Kumaraswamy(1.0, 1.0).cdf(lucid.tensor(0.25))  # Uniform(0, 1)
tensor(0.25)

entropy

→Tensor

entropy()

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Entropy via the Beta-distributed auxiliary variable Y = X^a.

If X ~ Kumaraswamy(a, b) then Y = X^a ~ Beta(1, b), giving: E[log X] = (1/a)(ψ(1) − ψ(b+1)) = −(1/a)(γ + ψ(b+1)) E[log(1−X^a)] = ψ(b) − ψ(b+1) = −1/b

H = −log a − log b − (a−1)·E[log X] − (b−1)·E[log(1−X^a)] = −log a − log b + (1 − 1/a)(γ + ψ(b+1)) + (1 − 1/b)

icdf

→Tensor

icdf(value: Tensor)

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Quantile function of the Kumaraswamy distribution.

Q(p; a, b) = \left(1 - (1 - p)^{1/b}\right)^{1/a}

Returns quantiles in $[0, 1]$ .

Examples

>>> Kumaraswamy(1.0, 1.0).icdf(lucid.tensor(0.25))
tensor(0.25)

log_prob

→Tensor

log_prob(value: Tensor)

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Log-probability density of the Kumaraswamy distribution.

Parameters

valueTensor

Point(s)

x \in [0, 1]

at which to evaluate the density.

Returns

Tensor

Log-density $\log a + \log b + (a-1)\log x + (b-1)\log(1-x^a)$ .

Notes

The two weighted logarithms are lucid.xlogy and lucid.special.xlog1py, so the ends of the support score the density's limit there: at $x = 0$ it is +inf for $a < 1$ , -inf for $a > 1$ and $\log b$ for $a = 1$ ; $x = 1$ is the same in $b$ . The reference framework answers NaN at both ends.

rsample

→Tensor

rsample(sample_shape: tuple[int, ...] = ())

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Reparameterised sample via the icdf: x = (1 − (1 − U)^(1/b))^(1/a).

>>> import lucid >>> from lucid.distributions import Kumaraswamy >>> d = Kumaraswamy(concentration1=2.0, concentration0=5.0) >>> d.rsample((4,)).shape (4,) >>> d.log_prob(lucid.tensor(0.3)) # log(2 * 5 * 0.3 * (1 - 0.3**2)**4) tensor(0.7214)