Randomized quasi-Monte Carlo

Scientific question¶

How can we integrate in moderate or high dimension when structured grids are too expensive, while retaining reproducible randomized uncertainty evidence?

Geometric picture¶

Sobol points fill the unit cube more evenly than independent random points. Digital scrambling produces independent randomized replicates without discarding the low-discrepancy structure inside each replicate.

Derivation¶

A Sobol coordinate is a base-two digital construction:

x_{n,j} = \sum_{k=1}^{B}b_k(n)v_{j,k}\pmod 2,

(1)

where (v_{j,k}) are direction numbers. With (R) scrambled replicates,

\widehat{I} =\frac{1}{R}\sum_{r=1}^{R}\widehat{I}_r, \qquad s_R^2=\frac{1}{R-1}\sum_{r=1}^{R} (\widehat{I}_r-\widehat{I})^2.

(2)

The fixed-look Student interval has half-width

h=t_{1-\alpha/2,R-1}\frac{s_R}{\sqrt{R}}.

(3)

Sequential stopping instead uses bounded empirical-Bernstein evidence and a predeclared spending schedule satisfying

\sum_{k=1}^{K}\alpha_k\leq\alpha.

(4)

Computational cost¶

Fixed cost is (R2^m) evaluations for level (m). Sequential schedules reuse the declared maximum point and replicate capacities. Vectorized payloads can increase memory even when logical point count is unchanged.

What the estimator means¶

Fixed scrambled methods report replicate standard error or a fixed-look confidence half-width. Sequential intervals require finite declared estimate bounds. Coverage is empirically calibrated only for the frozen campaign.

JAX and differentiation¶

Keys, scramble configuration, level, and replicate schedule are static or stopped. Replay differentiates the accepted randomized formula for the supplied key; it does not differentiate the randomization policy.

Quantities and units¶

Finite quantity axes are normalized before points are mapped into the domain. Sequential estimate_bounds may be quantities compatible with the result.

Worked astrophysical example¶

A separable survey selection integral is

I=\int_{\Omega} \prod_{j=1}^{d} \operatorname{sigmoid}\!\left(\frac{c_j-x_j}{w_j}\right)d^d x.

(5)

Randomized QMC is useful when the number of survey or population coordinates makes tensor rules impractical.

Failure modes¶

Audit recipe¶

Record the root key policy, scramble, bits, level, replicate count or schedule, confidence level, estimate bounds, status, coverage definition, evaluations, and the exact evidence artifact.

Warranted claim¶

Jaxstro provides deterministic Sobol, fixed scrambled Sobol, and bounded sequential scrambled Sobol integration. Confidence claims are real-scalar and limited to their declared randomized assumptions.