feat(distributions): add Kumaraswamy distribution#330
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Adds distrax.Kumaraswamy(concentration0, concentration1) — the
Kumaraswamy distribution on (0, 1) with parameters a > 0 and b > 0.
Unlike the Beta distribution, Kumaraswamy has a closed-form CDF and
inverse CDF, making it well-suited for reparameterized sampling in
variational inference and normalizing flows.
Implements:
- log_prob(x) = log(a) + log(b) + (a-1)log(x) + (b-1)log(1-x^a)
- log_cdf(x) = log(1 - (1-x^a)^b)
- mean() = b · B(1+1/a, b) via log-gammas (numerically stable)
- variance() = b · B(1+2/a, b) − mean²
- mode() = ((a-1)/(ab-1))^(1/a) for a,b≥1; boundary modes at 0/1
- entropy() = (1-1/b) − log(ab) + (1-1/a)·(γ + ψ(b+1))
(exact closed form derived from the digamma identity)
- sample() via the closed-form quantile F^{-1}(u) = (1-(1-u)^{1/b})^{1/a}
Adds distrax.Kumaraswamy to the public __init__.py and 20 tests
covering log_prob, log_cdf, mean, variance, mode, entropy (all
validated against TFP), sampling shape/range/moments, and JIT.
Reference: Kumaraswamy (1980), Journal of Hydrology, 46(1–2), 79–88.
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What does this PR do?
Adds
distrax.Kumaraswamy(concentration0, concentration1)— the Kumaraswamy distribution on the open interval (0, 1).The Kumaraswamy distribution (Kumaraswamy 1980) has PDF
f(x; a, b) = ab·x^{a-1}·(1−x^a)^{b-1}and the key property that its CDF and quantile function have closed forms, unlike the Beta distribution. This makes it well-suited for:Implemented
log_prob(x)log(ab) + (a-1)log x + (b-1)log(1-x^a)log_cdf(x)log(1 − (1−x^a)^b)mean()b · B(1+1/a, b)via log-gammasvariance()b · B(1+2/a, b) − mean²mode()((a-1)/(ab-1))^{1/a}for interior; 0/1 at boundariesentropy()(1−1/b) − log(ab) + (1−1/a)·(γ + ψ(b+1))(exact)sample()(1 − (1−u)^{1/b})^{1/a}, u ~ UniformTests
20 tests in
kumaraswamy_test.py— log_prob / log_cdf / mean / variance / mode / entropy all validated against TFP; sampling shape, range, and moments; JIT compatibility.Reference
Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1–2), 79–88.