pgjax¶

On-device Pólya-Gamma sampling for JAX, as an XLA FFI custom call.

pgjax.pg_sample(h, z, key) draws omega ~ PG(h, z) from inside a JIT’d / lax.scan-ed / pmap-ed program with no host round-trip — the native replacement for jax.pure_callback(random_polyagamma, ...), which pays a device↔host round-trip on every Pólya-Gamma Gibbs sweep and serializes under jax.pmap.

import jax, pgjax
jax.config.update("jax_enable_x64", True)

@jax.jit
def omega_update(h, z, key):
    return pgjax.pg_sample(h, z, key)

Method¶

Small integer h (incl. Bernoulli/logit h = 1): the exact Devroye sampler for PG(1, z) (Polson, Scott & Windle 2013), summed h times.
Real h (e.g. Negative-Binomial h = y + alpha): the exact infinite-sum (Gamma) representation truncated at K terms with a moment-matched Gamma tail correction for the remainder. The tail mean/variance have closed forms (Σ 1/d_k = (π/2a)tanh(πa), and its a-derivative), so the correction cancels the truncation bias in the mean — it is precisely that bias (truncation with no tail term) that collapses alpha in NB models when omitted.
Large h: the saddlepoint rejection sampler (Windle, Polson & Scott 2014, arXiv:1405.0506). A two-piece bounding-kernel envelope proposes, and the saddlepoint density approximation accepts, giving O(1) per accepted draw versus the O(K) Gamma sum. Used automatically when h >= 8 or (h > 4 and |z| <= 4) — the hybrid dispatch thresholds validated in zoj613/polyagamma.

Validated against random_polyagamma over h ∈ [0.5, 40] × z ∈ [-8, 8]: KS ≤ 0.009, relative mean error ≤ 0.34%. The saddlepoint path is validated to KS < 0.02 and relative mean error < 2% over h ∈ {8,…,40} × z ∈ {0,1,-4,5}.

The RNG is a per-call xoshiro256** seeded (via splitmix64) from the JAX PRNG key, so draws are reproducible and each call owns its stream — safe under the concurrent per-device calls jax.pmap issues.

Status¶

✅ Exact PG(1, z) / integer h (Devroye), real h (tail-corrected Gamma sum), and large h (saddlepoint) — covers the logit (h=1), Negative-Binomial (h=y+alpha), and large-shape regimes.
Hybrid dispatch is automatic: the appropriate sampler is selected from (h, z) with no API change.

CPU-only (the sampler is cheap scalar work; the point is to keep it on-device next to the rest of a JIT’d Gibbs sweep).