How (Much) to Sample to Estimate a Mean Fred J. Hickernell Illinois Institute of Technology Dept. Applied Math Ctr. Interdisc. Scientific Comput. Office of Research hickernell@iit.edu sites.google.com/iit.edu/fred-j-hickernell Thank you for your invitation and hospitality, Slides at speakerdeck.com/fjhickernell/how-much-to-sample-to-estimate-the-mean Jupyter notebook with figures at (H. 2023) Purdue Statistics Colloquium, Friday, March 3, 2023

Intro IID Low Discrepancy Software Summary References Fred Is Like Dennis Lin multicultural multilingual multicultural multilingual 2/23

Intro IID Low Discrepancy Software Summary References Fred Is (Not) Like Dennis Lin Colloquium Talk multicultural multilingual multicultural multilingual 2/23

Intro IID Low Discrepancy Software Summary References Computing the Mean, µ = E(Y) Probability µ = y ϱ(y) dy, Y ∼ known ϱ 3/23

Intro IID Low Discrepancy Software Summary References Computing the Mean, µ = E(Y) Probability µ = y ϱ(y) dy, Y ∼ known ϱ Statistics µ ∈ ^ µ − 2.58^ σ √ n , ^ µ + 2.58^ σ √ n with high probability ^ µ = 1 n n i=1 Yi, ^ σ2 = 1 n − 1 n i=1 (Yi − ^ µ)2, Y1, Y2, . . . IID ∼ unknown ϱ 3/23

Intro IID Low Discrepancy Software Summary References Computing the Mean, µ = E(Y) Probability µ = y ϱ(y) dy, Y ∼ known ϱ Computer Experiments Y = f(X), X ∼ U[0, 1]d µ = [0,1]d f(x) dx ∈ [^ µ − ε, ^ µ + ε], ε specified, fixed width ^ µ = 1 n n i=1 f(Xi), X1, X2, mimic ∼ U[0, 1]d Statistics µ ∈ ^ µ − 2.58^ σ √ n , ^ µ + 2.58^ σ √ n with high probability ^ µ = 1 n n i=1 Yi, ^ σ2 = 1 n − 1 n i=1 (Yi − ^ µ)2, Y1, Y2, . . . IID ∼ unknown ϱ 3/23

Intro IID Low Discrepancy Software Summary References Computing the Mean, µ = E(Y) Computer Experiments Y = f(X), X ∼ U[0, 1]d option payoff flow velocity through rock with random porosity Bayesian posterior density times parameter µ = [0,1]d f(x) dx ∈ [^ µ − ε, ^ µ + ε], ε specified, fixed width ^ µ = 1 n n i=1 f(Xi), X1, X2, mimic ∼ U[0, 1]d option price mean flow velocity through rock with random porosity Bayesian posterior mean 3/23

Intro IID Low Discrepancy Software Summary References Computing the Mean, µ = E(Y) Computer Experiments Y = f(X), X ∼ U[0, 1]d µ = [0,1]d f(x) dx ∈ [^ µ − ε, ^ µ + ε], ε specified, fixed width, n ^ µ = 1 n n i=1 f(Xi), X1, X2, mimic ∼ U[0, 1]d 3/23

Intro IID Low Discrepancy Software Summary References Fixed-Width Central Limit Theorem (CLT) Confidence Intervals Given random number generator Y1, Y2, . . . IID ∼ unknown ϱ parameter nσ Input error tolerance ε Compute ^ µσ = 1 nσ nσ i=1 Yi, ^ σ2 = 1 nσ − 1 nσ i=1 (Yi − ^ µσ)2 n = 2.58^ σ ε 2 Output ^ µ = 1 n nσ+n i=nσ+1 Yi, µ ∈ [^ µ − ε, ^ µ + ε] with high probability 4/23

Intro IID Low Discrepancy Software Summary References Fixed-Width Central Limit Theorem (CLT) Confidence Intervals Given random number generator Y1, Y2, . . . IID ∼ unknown ϱ parameter nσ Input error tolerance ε Compute ^ µσ = 1 nσ nσ i=1 Yi, ^ σ2 = 1 nσ − 1 nσ i=1 (Yi − ^ µσ)2 n = 2.58^ σ ε 2 Output ^ µ = 1 n nσ+n i=nσ+1 Yi, µ ∈ [^ µ − ε, ^ µ + ε] with high probability What might go wrong Do not know how well ^ σ2 bounds var(Y) CLT is an asymptotic (not finite sample) result 4/23

Intro IID Low Discrepancy Software Summary References Fixed-Width Confidence Intervals Assuming Bounded Kurtosis Given Y1, Y2, . . . IID ∼ unknown ϱ, kurt(Y) ⩽ κmax parameters nσ, κmax Input error tolerance ε Compute ^ µσ = 1 nσ nσ i=1 Yi, ^ σ2 = 1 nσ − 1 nσ i=1 (Yi − ^ µσ)2, n = CBE Cσ(κmax , nσ)^ σ ε , κmax Output ^ µ = 1 n nσ+n i=nσ+1 Yi, µ ∈ [^ µ − ε, ^ µ + ε] with 99% probability The fix (H., Jiang, Liu, and Owen 2013) var(Y) ⩽ C2 σ (κmax , nσ)^ σ2 with probability √ 99% Berry-Esseen inequality gives finite sample √ 99% confidence interval 5/23

Intro IID Low Discrepancy Software Summary References Uncertainty in a Cantilevered Beam (Parno and Seelinger 2022) u(x) = g(Z, x) = beam deflection = solution of a differential equation boundary value problem Z ∼ U[1, 1.2]3 defines uncertainty in Young’s modulus x = position µ(x) = E[u(x)] = [0,1]3 g(z, x) dz ≈ 1 n n i=1 g(Zi, x) µ(end) = 1037 figs. at (H. 2023) 101 102 Tolerance, ε sec min hr Time (s) IID (ε−2) 101 102 Tolerance, ε 102 103 104 105 106 n IID (ε−2) 6/23

Intro IID Low Discrepancy Software Summary References Cones Are Better than Balls for Adaptive Algorithms Fixed Width Confidence Intervals (using IID sampling) Y = Y : E[(Y − µ)4] var2(Y) ⩽ κmax Y = Y : var(Y) ⩽ σ2 max Use Berry-Esseen inequality Use Chebyshev inequality Y is not too heavy tailed Y is not too big Y ∈ Y =⇒ aY ∈ Y Y ∈ Y ̸ =⇒ aY ∈ Y Adaptive sample size Fixed sample size 7/23

Intro IID Low Discrepancy Software Summary References Cones Are Better than Balls for Adaptive Algorithms Fixed Width Confidence Intervals (using IID sampling) Y = Y : E[(Y − µ)4] var2(Y) ⩽ κmax Y = Y : var(Y) ⩽ σ2 max Use Berry-Esseen inequality Use Chebyshev inequality Y is not too heavy tailed Y is not too big Y ∈ Y =⇒ aY ∈ Y Y ∈ Y ̸ =⇒ aY ∈ Y Adaptive sample size Fixed sample size Choice of inflation factors depends on or implies κmax Non-convex cones may permit successful adaptive algorithms (Bahadur and Savage 1956; Bakhvalov 1970) 7/23

Intro IID Low Discrepancy Software Summary References Low Discrepancy Samples Fill Space Better 0 1 xi,1 0 1 xi,2 n = 64 0 1 xi,1 0 1 xi,2 n = 128 0 1 xi,1 0 1 xi,2 n = 256 0 1 xi,1 0 1 xi,2 n = 512 IID Points 0 1 xi, 1 0 1 xi, 2 n = 64 0 1 xi, 1 0 1 xi, 2 n = 128 0 1 xi, 1 0 1 xi, 2 n = 256 0 1 xi, 1 0 1 xi, 2 n = 512 Lattice Points 8/23

Intro IID Low Discrepancy Software Summary References Low Discrepancy Samples Fill Space Better 0 1 xi,1 0 1 xi,2 n = 64 0 1 xi,1 0 1 xi,2 n = 128 0 1 xi,1 0 1 xi,2 n = 256 0 1 xi,1 0 1 xi,2 n = 512 IID Points 0 1 xi, 1 0 1 xi, 2 n = 64 0 1 xi, 1 0 1 xi, 2 n = 128 0 1 xi, 1 0 1 xi, 2 n = 256 0 1 xi, 1 0 1 xi, 2 n = 512 Sobol' Points 8/23

Intro IID Low Discrepancy Software Summary References Low Discrepancy Samples Approximate Integrals Better Discrepancy measures how far the empirical distribution of {x1, . . . , xn} is from the uniform distribution (H. 1999; Niederreiter 1992) µ [0,1]d f(x) dx − 1 n n i=1 f(xi) ⩽ sample quality discrepancy(x1, . . . , xn) × function roughness semi-norm(f) 0 1 xi, 1 0 1 xi, 2 n = 64 0 1 xi, 1 0 1 xi, 2 n = 128 0 1 xi, 1 0 1 xi, 2 n = 256 0 1 xi, 1 0 1 xi, 2 n = 512 Lattice Points 9/23

Intro IID Low Discrepancy Software Summary References Low Discrepancy Samples Approximate Integrals Better Discrepancy measures how far the empirical distribution of {x1, . . . , xn} is from the uniform distribution (H. 1999; Niederreiter 1992) µ [0,1]d f(x) dx − 1 n n i=1 f(xi) ⩽ sample quality discrepancy(x1, . . . , xn) expensive to compute × function roughness semi-norm(f) no way to bound 0 1 xi, 1 0 1 xi, 2 n = 64 0 1 xi, 1 0 1 xi, 2 n = 128 0 1 xi, 1 0 1 xi, 2 n = 256 0 1 xi, 1 0 1 xi, 2 n = 512 Lattice Points 9/23

Intro IID Low Discrepancy Software Summary References Fixed-Width Confidence Intervals via Lattice Sampling f(x) = k∈Zd f(k) exp(2π √ −1k′x), f(k) = [0,1]d f(x) exp(−2π √ −1k′x) dx Fourier series µ [0,1]d f(x) dx f(0) − 1 n n i=1 f(xi) = k∈Zd k′xi∈Z ∀i f(k) ⩽ k∈Zd k′xi∈Z ∀i f(k) depends on terms aliased w/ constant ⩽ k∈Zd k′xi∈Z ∀i 1 |weight(k)|2 discrepancy(x1,...,xn) × 0̸=k∈Zd f(k)weight(k) 2 semi-norm(f) 10/23

Intro IID Low Discrepancy Software Summary References Fixed-Width Confidence Intervals via Lattice Sampling f(x) = k∈Zd f(k) exp(2π √ −1k′x), f(k) = [0,1]d f(x) exp(−2π √ −1k′x) dx Fourier series µ [0,1]d f(x) dx f(0) − 1 n n i=1 f(xi) ⩽ k∈Zd k′xi∈Z ∀i f(k) depends on terms aliased w/ constant Adaptive stopping rule for lattices (H. and Jiménez Rugama 2016) Define = {f whose Fourier coefficients decay nicely} Bound k∈Zd k′xi∈Z ∀i f(k) in terms of sums of f(k) = 1 n n i=1 f(xi) exp(−2π √ −1k′x) fast Fourier transform for f ∈ Increase n until error bound ⩽ ε 10/23

Intro IID Low Discrepancy Software Summary References IID vs. Low Discrepancy for the Cantilevered Beam u(x) = g(Z, x) = beam deflection Z ∼ U[1, 1.2]3 defines uncertainty x = position µ(x) = E[u(x)] ≈ 1037 10−2 10−1 100 101 102 Tolerance, ε sec min hr Time (s) Lattice (ε−1) IID (ε−2) 10−2 10−1 100 101 102 Tolerance, ε 102 103 104 105 106 n Lattice (ε−1) IID (ε−2) 11/23

Intro IID Low Discrepancy Software Summary References There Is More Adaptive stopping criteria also available for ▶ Sobol’ points (Jiménez Rugama and H. 2016), but not Halton ▶ Relative and hybrid error criteria (H., Jiménez Rugama, and Li 2018) ▶ Multiple answers based on a number of integrals, such as Sobol’ indices (Sorokin and Jagadeeswaran 2022+) 12/23

Intro IID Low Discrepancy Software Summary References There Is More Adaptive stopping criteria also available for ▶ Sobol’ points (Jiménez Rugama and H. 2016), but not Halton ▶ Relative and hybrid error criteria (H., Jiménez Rugama, and Li 2018) ▶ Multiple answers based on a number of integrals, such as Sobol’ indices (Sorokin and Jagadeeswaran 2022+) Bayesian adaptive stopping criteria (Jagadeeswaran and H. 2019, 2022) where ▶ f is an instance of a Gaussian process ▶ Credible interval gives the error bound ▶ Covariance kernel has hyperparameters that are tuned by empirical Bayes or cross-validation ▶ Choosing kernels that partner with lattice or Sobol’ makes computation efficient (O(n log n) not O(n3)) 12/23

Intro IID Low Discrepancy Software Summary References Where to Find Low Discrepancy Sequences BRODA (Kucherenko 2022) Dakota (Adams, Bohnhoff, et al. 2022) Julia (SciML QuasiMonteCarlo.jl 2022) MATLAB (The MathWorks, Inc. 2022) NAG (The Numerical Algorithms Group 2021) PyTorch (Paszke, Gross, et al. 2019) R (Hofert and Lemieux 2017) SAS (SAS Institute 2023) SciPy (Virtanen, Gommers, et al. 2020) TensorFlow (TF Quant Finance Contributors 2021), but beware of some limited, outdated, and incorrect implementations 13/23

Intro IID Low Discrepancy Software Summary References QMCPy, a Python quasi-Monte Carlo Library Quasi-Monte Carlo methods use low discrepancy sampling Open source project (Choi, H., et al. 2022) A blog qmcpy.org Low discrepancy generators Variable transformations Stopping criteria Links with models via UM-Bridge (Davis, Parno, Reinarz, and Seelinger 2022) 14/23

Intro IID Low Discrepancy Software Summary References The Whole Cantilevered Beam u(x) = g(Z, x) = beam deflection Z ∼ U[1, 1.2]3 defines uncertainty x = position 0 10 20 30 x 0 250 500 750 1000 u(x) Cantilevered Beam 10−2 10−1 100 Tolerance, ε sec min Time (s) Lattice (ε−1) Lattice Parallel (ε−1) 10−2 10−1 100 Tolerance, ε 101 102 103 104 105 106 n Lattice (ε−1) Lattice Parallel (ε−1) 15/23

Intro IID Low Discrepancy Software Summary References Take Aways How to sample? ▶ Low discrepancy sequences (aka quasi-Monte Carlo methods) for high efficiency How much to sample? ▶ Data-based, theoretically justified adaptive stopping criteria Traditional theory impractical Based on cones of random variables or integrands 16/23

Intro IID Low Discrepancy Software Summary References Take Aways How to sample? ▶ Low discrepancy sequences (aka quasi-Monte Carlo methods) for high efficiency ▶ Better variable transforms needed for peaky f How much to sample? ▶ Data-based, theoretically justified adaptive stopping criteria Traditional theory impractical Based on cones of random variables or integrands ▶ Criteria and theory needed for Multilevel methods Densities and quantiles 16/23

Intro IID Low Discrepancy Software Summary References Take Aways How to sample? ▶ Low discrepancy sequences (aka quasi-Monte Carlo methods) for high efficiency ▶ Better variable transforms needed for peaky f How much to sample? ▶ Data-based, theoretically justified adaptive stopping criteria Traditional theory impractical Based on cones of random variables or integrands ▶ Criteria and theory needed for Multilevel methods Densities and quantiles Use and develop good software, make research reproducible (H. 2023) 16/23

Thank you These slides are available at speakerdeck.com/fjhickernell/how-much-to-sample-to-estimate-the-mean Jupyter notebook with figures at (H. 2023)

Intro IID Low Discrepancy Software Summary References References I H., F. J. (1999). “Goodness-of-Fit Statistics, Discrepancies and Robust Designs”. In: Statist. Probab. Lett. 44, pp. 73–78. doi: 10.1016/S0167-7152(98)00293-4. — (2023). Jupyter Notebook for Computations and Figures for the 2023 March 3 Statistics Colloquium at Purdue University. https://github.com/QMCSoftware/ QMCSoftware/blob/Purdue2023Talk/demos/Purdue_Talk_Figures.ipynb. H., F. J. and Ll. A. Jiménez Rugama (2016). “Reliable Adaptive Cubature Using Digital Sequences”. In: Monte Carlo and Quasi-Monte Carlo Methods: MCQMC, Leuven, Belgium, April 2014. Ed. by R. Cools and D. Nuyens. Vol. 163. Springer Proceedings in Mathematics and Statistics. arXiv:1410.8615 [math.NA]. Springer-Verlag, Berlin, pp. 367–383. 18/23

Intro IID Low Discrepancy Software Summary References References II H., F. J., Ll. A. Jiménez Rugama, and D. Li (2018). “Adaptive Quasi-Monte Carlo Methods for Cubature”. In: Contemporary Computational Mathematics — a celebration of the 80th birthday of Ian Sloan. Ed. by J. Dick, F. Y. Kuo, and H. Woźniakowski. Springer-Verlag, pp. 597–619. doi: 10.1007/978-3-319-72456-0. H., F. J. et al. (2013). “Guaranteed Conservative Fixed Width Confidence Intervals Via Monte Carlo Sampling”. In: Monte Carlo and Quasi-Monte Carlo Methods 2012. Ed. by J. Dick et al. Vol. 65. Springer Proceedings in Mathematics and Statistics. Springer-Verlag, Berlin, pp. 105–128. doi: 10.1007/978-3-642-41095-6. Adams, Brian M. et al. (May 2022). Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.16 User’s Manual. Sandia National Laboratories. 19/23

Intro IID Low Discrepancy Software Summary References References III Bahadur, R. R. and L. J. Savage (1956). “The Nonexistence of Certain Statistical Procedures in Nonparametric Problems”. In: Ann. Math. Stat. 27, pp. 1115–1122. Bakhvalov, N. S. (1970). “On the optimality of linear methods for operator approximation in convex classes of functions (in Russian)”. In: Zh. Vychisl. Mat. i Mat. Fiz. 10. English transl.: USSR Comput. Math. Math. Phys. 11 (1971) 244–249., pp. 555–568. Choi, S.-C. T. et al. (2022). QMCPy: A quasi-Monte Carlo Python Library. doi: 10.5281/zenodo.3964489. url: https://qmcsoftware.github.io/QMCSoftware/. Cools, R. and D. Nuyens, eds. (2016). Monte Carlo and Quasi-Monte Carlo Methods: MCQMC, Leuven, Belgium, April 2014. Vol. 163. Springer Proceedings in Mathematics and Statistics. Springer-Verlag, Berlin. Davis, A. et al. (2022). UQ and Model Bridge (UM-Bridge). url: https://um-bridge-benchmarks.readthedocs.io/en/docs/. 20/23

Intro IID Low Discrepancy Software Summary References References IV Hofert, M. and C. Lemieux (2017). qrng R package. url: https://cran.r-project.org/web/packages/qrng/qrng.pdf (visited on 2017). Jagadeeswaran, R. and F. J. H. (2019). “Fast Automatic Bayesian Cubature Using Lattice Sampling”. In: Stat. Comput. 29, pp. 1215–1229. doi: 10.1007/s11222-019-09895-9. — (2022). “Fast Automatic Bayesian Cubature Using Sobol’ Sampling”. In: Advances in Modeling and Simulation: Festschrift in Honour of Pierre L’Ecuyer. Ed. by Z. Botev et al. Springer, Cham, pp. 301–318. doi: 10.1007/978-3-031-10193-9\_15. Jiménez Rugama, Ll. A. and F. J. H. (2016). “Adaptive Multidimensional Integration Based on Rank-1 Lattices”. In: Monte Carlo and Quasi-Monte Carlo Methods: MCQMC, Leuven, Belgium, April 2014. Ed. by R. Cools and D. Nuyens. Vol. 163. Springer Proceedings in Mathematics and Statistics. arXiv:1411.1966. Springer-Verlag, Berlin, pp. 407–422. 21/23

Intro IID Low Discrepancy Software Summary References References V Kucherenko, S. (2022). BRODA. url: https://www.broda.co.uk/index.html. Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia: SIAM. Parno, M. and L. Seelinger (2022). Uncertainty propagation of material properties of a cantilevered beam. url: https://um-bridge-benchmarks.readthedocs.io/en/docs/forward- benchmarks/muq-beam-propagation.html. Paszke, Adam et al. (2019). “PyTorch: An imperative style, high-performance deep learning library”. In: Advances in neural information processing systems 32, pp. 8026–8037. SAS Institute (2023). SAS Documentation for the MODEL Procedure. url: https://documentation.sas.com/doc/en/pgmsascdc/9.4_3.4/etsug/etsug_model_ sect197.htm. 22/23

Intro IID Low Discrepancy Software Summary References References VI SciML QuasiMonteCarlo.jl (2022). url: https://github.com/SciML/QuasiMonteCarlo.jl. Sorokin, A. G. and R. Jagadeeswaran (2022+). “Monte Carlo for Vector Functions of Integrals”. In: Monte Carlo and Quasi-Monte Carlo Methods: MCQMC, Linz, Austria, July 2022. Ed. by A. Hinrichs, P. Kritzer, and F. Pillichshammer. Springer Proceedings in Mathematics and Statistics. in preparation for submission for publication. Springer, Cham. TF Quant Finance Contributors (2021). Quasi Monte-Carlo Methods. url: https://github.com/google/tf-quant-finance/math/qmc. The MathWorks, Inc. (2022). MATLAB R2022b. Natick, MA. The Numerical Algorithms Group (2021). The NAG Library. Mark 27. Oxford. Virtanen, Pauli et al. (2020). “SciPy 1.0: fundamental algorithms for scientific computing in Python”. In: Nature Methods 17.3, pp. 261–272. 23/23