Slide 17
Slide 17 text
Introduction Fast Bayesian Transforms Numerical Examples Gaussian Process Diagnostics Summary References
Gaussian Probability
(a,b)
exp −1
2
tTΣ−1t
(2π)d det(Σ)
dt =
[0,1]d−1
f(x) dx by Genz’s transformation
ε = 1 × 10−4, d = 5, Σ = 0.4I + 0.611T, a = (−∞, . . . , −∞), b ∼
√
dU[0, 1]d
Method MC Lattice Sobol’ BayesLat BayesSobol
Absolute Error 2.00 × 10−5 5.00 × 10−6 4.10 × 10−6 9.20 × 10−6 3.40 × 10−6
Tolerance Met 100% 100% 100% 100% 100%
n 62 000 000 4100 4100 2000 4100
Time (seconds) 17.0000 0.0110 0.0097 0.0880 0.0950
Algorithms are implemented in GAIL and soon QMCPy
Choi, S.-C. T., Ding, Y., H., F. J., Jiang, L., Jiménez Rugama, L. A., Li, D., Jagadeeswaran, R., Tong, X., Zhang, K., et al.
GAIL: Guaranteed Automatic Integration Library (Versions 1.0–2.2). MATLAB software. 2013–2017.
http://gailgithub.github.io/GAIL_Dev/.
Choi, S.-C. T., H., F. J., McCourt, M. & Sorokin, A. QMCPy: A quasi-Monte Carlo Python Library. 2020+.
https://github.com/QMCSoftware/QMCSoftware.
Genz, A. Comparison of Methods for the Computation of Multivariate Normal Probabilities. Computing Science and Statistics
25, 400–405 (1993). 9/15