The document discusses methods for efficiently and accurately estimating integrals, including Monte Carlo simulation, low-discrepancy sampling, and Bayesian cubature. It notes that product rules for estimating high-dimensional integrals become prohibitively expensive as dimension increases. Adaptive low-discrepancy sampling is proposed as a method that uses Sobol' or lattice points and normally doubles the number of points until a tolerance is reached.

1. Simulating the Mean Efficiently
and to a Given Tolerance
Fred J. Hickernell
Department of Applied Mathematics, Illinois Institute of Technology
hickernell@iit.edu mypages.iit.edu/~hickernell
Thanks to Lan Jiang, Tony Jiménez Rugama, Jagadees Rathinavel,
and the rest of the the Guaranteed Automatic Integration Library (GAIL) team
Supported by NSF-DMS-1522687
Thanks for your kind invitation

2. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Estimating/Simulating/Computing an Integral
Gaussian probability =
ż
[a,b]
e´xT
Σ´1
x/2
(2π)d/2 |Σ|1/2
dx
2/16

3. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Estimating/Simulating/Computing an Integral
Gaussian probability =
ż
[a,b]
e´xT
Σ´1
x/2
(2π)d/2 |Σ|1/2
dx
option price =
ż
Rd
payoff(x)
e´xT
Σ´1
x/2
(2π)d/2 |Σ|1/2
looooooomooooooon
PDF of Brownian motion at d times
dx
2/16

4. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Estimating/Simulating/Computing an Integral
Gaussian probability =
ż
[a,b]
e´xT
Σ´1
x/2
(2π)d/2 |Σ|1/2
dx
option price =
ż
Rd
payoff(x)
e´xT
Σ´1
x/2
(2π)d/2 |Σ|1/2
looooooomooooooon
PDF of Brownian motion at d times
dx
Bayesian ^βj =
ż
Rd
βj prob(β|data) dβ =
ş
Rd βj prob(data|β) probprior(β) dβ
ş
Rd prob(data|β) probprior(β) dβ
2/16

5. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Estimating/Simulating/Computing an Integral
Gaussian probability =
ż
[a,b]
e´xT
Σ´1
x/2
(2π)d/2 |Σ|1/2
dx
option price =
ż
Rd
payoff(x)
e´xT
Σ´1
x/2
(2π)d/2 |Σ|1/2
looooooomooooooon
PDF of Brownian motion at d times
dx
Bayesian ^βj =
ż
Rd
βj prob(β|data) dβ =
ş
Rd βj prob(data|β) probprior(β) dβ
ş
Rd prob(data|β) probprior(β) dβ
Sobol’ indexj =
ş
[0,1]2d output(x) ´ output(xj, x1
´j) output(x1
) dx dx1
ş
[0,1]d output(x)2 dx ´
ş
[0,1]d output(x) dx
2
2/16

6. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Estimating/Simulating/Computing the Mean
Gaussian probability =
ż
[a,b]
e´xT
Σ´1
x/2
(2π)d/2 |Σ|1/2
dx
option price =
ż
Rd
payoff(x)
e´xT
Σ´1
x/2
(2π)d/2 |Σ|1/2
looooooomooooooon
PDF of Brownian motion at d times
dx
Bayesian ^βj =
ż
Rd
βj prob(β|data) dβ =
ş
Rd βj prob(data|β) probprior(β) dβ
ş
Rd prob(data|β) probprior(β) dβ
Sobol’ indexj =
ş
[0,1]2d output(x) ´ output(xj, x1
´j) output(x1
) dx dx1
ş
[0,1]d output(x)2 dx ´
ş
[0,1]d output(x) dx
2
µ =
ż
Rd
g(x) dx = E[f(X)] =
ż
Rd
f(x) ν(dx) =?, ^µn =
nÿ
i=1
wif(xi)
How to choose ν, txiun
i=1, and twiun
i=1 to make |µ ´ ^µn| small? (trio identity)
Given εa, how big must n be to guarantee |µ ´ ^µn| ď εa? (adaptive cubature) 2/16

7. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Product Rules Using Rectangular Grids
µ =
ż
Rd
f(x) ν(dx) « ^µn =
nÿ
i=1
wif(xi)
If
ż 1
0
f(x) dx ´
mÿ
i=1
wif(ti) = O(m´r
), then
ż
[0,1]d
f(x) dx
´
mÿ
i1=1
¨ ¨ ¨
mÿ
id=1
wi1
¨ ¨ ¨ wid
f(ti1
, . . . , tid
)
= O(m´r
) = O(n´r/d
)
assuming rth
derivatives in each direction exist.
3/16

8. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Product Rules Using Rectangular Grids
µ =
ż
Rd
f(x) ν(dx) « ^µn =
nÿ
i=1
wif(xi)
If
ż 1
0
f(x) dx ´
mÿ
i=1
wif(ti) = O(m´r
), then
ż
[0,1]d
f(x) dx
´
mÿ
i1=1
¨ ¨ ¨
mÿ
id=1
wi1
¨ ¨ ¨ wid
f(ti1
, . . . , tid
)
= O(m´r
) = O(n´r/d
)
assuming rth
derivatives in each direction exist.
3/16

9. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Product Rules Using Rectangular Grids
µ =
ż
Rd
f(x) ν(dx) « ^µn =
nÿ
i=1
wif(xi)
If
ż 1
0
f(x) dx ´
mÿ
i=1
wif(ti) = O(m´r
), then
ż
[0,1]d
f(x) dx
´
mÿ
i1=1
¨ ¨ ¨
mÿ
id=1
wi1
¨ ¨ ¨ wid
f(ti1
, . . . , tid
)
= O(m´r
) = O(n´r/d
)
assuming rth
derivatives in each direction exist. But the computational cost
becomes prohibitive for large dimensions, d:
d 1 2 5 10 100
n = 8d
8 64 3.3E4 1.0E9 2.0E90
Product rules are typically a bad idea unless d is small. 3/16

10. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Monte Carlo Simulation in the News
Sampling with a computer can be fast
How big is our error?
4/16

11. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
IID Monte Carlo
µ =
ż
Rd
f(x) ν(dx) « ^µn =
1
n
nÿ
i=1
f(xi), xi
IID
„ ν
µ ´ ^µn =
µ ´ ^µn
std(f(X))/
?
nlooooooomooooooon
CNF„(0,1)
ˆ
1
?
nloomoon
DSC(txiu)
ˆ std(f(X))loooomoooon
VAR(f)
trio identity (Meng, 2017+)
µ ´ ^µn =
ż
Rd
f(x) (ν ´ ^νn)(dx)
^νn =
1
n
nÿ
i=1
δxi
5/16

12. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Central Limit Theorem Stopping Rule for IID Monte Carlo
µ =
ż
Rd
f(x) ν(dx) « ^µn =
1
n
nÿ
i=1
f(xi), xi
IID
„ ν
µ ´ ^µn =
µ ´ ^µn
std(f(X))/
?
nlooooooomooooooon
CNF„(0,1)
ˆ
1
?
nloomoon
DSC(txiu)
ˆ std(f(X))loooomoooon
VAR(f)
P[|µ ´ ^µn| ď errn] « 99%
for errn =
2.58 ˆ 1.2^σ
?
n
by the Central Limit Theorem (CLT),
where ^σ2
is the sample variance.
5/16

13. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Central Limit Theorem Stopping Rule for IID Monte Carlo
µ =
ż
Rd
f(x) ν(dx) « ^µn =
1
n
nÿ
i=1
f(xi), xi
IID
„ ν
µ ´ ^µn =
µ ´ ^µn
std(f(X))/
?
nlooooooomooooooon
CNF„(0,1)
ˆ
1
?
nloomoon
DSC(txiu)
ˆ std(f(X))loooomoooon
VAR(f)
P[|µ ´ ^µn| ď errn] « 99%
for errn =
2.58 ˆ 1.2^σ
?
n
by the Central Limit Theorem (CLT),
where ^σ2
is the sample variance. But the CLT is only an asymptotic result, and
1.2^σ may be an overly optimistic upper bound on σ.
5/16

14. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Berry-Esseen Stopping Rule for IID Monte Carlo
µ =
ż
Rd
f(x) ν(dx) « ^µn =
1
n
nÿ
i=1
f(xi), xi
IID
„ ν
µ ´ ^µn =
µ ´ ^µn
std(f(X))/
?
nlooooooomooooooon
CNF„(0,1)
ˆ
1
?
nloomoon
DSC(txiu)
ˆ std(f(X))loooomoooon
VAR(f)
P[|µ ´ ^µn| ď errn] ě 99%
for Φ ´
?
n errn /(1.2^σnσ
)
+ ∆n(´
?
n errn /(1.2^σnσ
), κmax) = 0.0025
by the Berry-Esseen Inequality,
where ^σ2
nσ
is the sample variance using an independent sample from that used to
simulate the mean, and provided that kurt(f(X)) ď κmax(nσ) (H. et al., 2013;
Jiang, 2016)
5/16

15. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

16. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

17. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

18. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

19. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

20. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

21. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

22. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

23. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

24. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

25. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

26. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

27. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

28. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

29. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

30. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

31. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Normally n should be a power of 2
6/16

32. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Let tpf(k)uk denote the coefficients of the
Fourier Walsh or complex exponential
expansion of f. Let tω(k)uk be some
weights. Then
µ ´ ^µn =
´
ÿ
0‰kPdual
pf(k)
! pf(k)
ω(k)
)
k 2
tω(k)u0‰kPdual 2
loooooooooooooooooooomoooooooooooooooooooon
CNFP[´1,1]
ˆ tω(k)u0‰kPdual 2loooooooooomoooooooooon
DSC(txiun
i=1)=O(n´1+ )
ˆ
#
pf(k)
ω(k)
+
k 2looooooomooooooon
VAR(f)
6/16

33. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Adaptive Low Discrepancy Sampling Cubature
µ =
ż
[0,1]d
f(x) dx
^µn =
1
n
nÿ
i=1
f(xi), xi Sobol’ or lattice
Let tpf(k)uk denote the coefficients of the
Fourier Walsh or complex exponential
expansion of f.
Assuming that the pf(k) do not decay erratically as k Ñ ∞, the discrete
transform, rfn(k)
(
k
, may be used to bound the error reliably (H. and Jiménez
Rugama, 2016; Jiménez Rugama and H., 2016; H. et al., 2017+):
|µ ´ ^µn| ď errn := C(n)
ÿ
certaink
rfn(k)
6/16

34. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Bayesian Cubature—f Is Random
µ =
ż
Rd
f(x) ν(dx) « ^µn =
nÿ
i=1
wi f(xi)
Assume f „ GP(0, s2
Cθ) (Diaconis, 1988;
O’Hagan, 1991; Ritter, 2000; Rasmussen and
Ghahramani, 2003)
c0 =
ż
RdˆRd
Cθ(x, t) ν(dx)ν(dt)
c =
ż
Rd
Cθ(xi, t) ν(dt)
n
i=1
, C = Cθ(xi, xj)
n
i,j=1
Choosing w = wi
n
i=1
= C´1
c is optimal
µ ´ ^µn =
µ ´ ^µn
b
c0 ´ cTC´1c yTC´1y
nlooooooooooooooomooooooooooooooon
CNF„N(0,1)
ˆ
a
c0 ´ cTC´1cloooooooomoooooooon
DSC
ˆ
c
yTC´1y
nlooooomooooon
VAR(f)
where y = f(xi)
n
i=1
.
7/16

35. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Bayesian Cubature—f Is Random
µ =
ż
Rd
f(x) ν(dx) « ^µn =
nÿ
i=1
wi f(xi)
Assume f „ GP(0, s2
Cθ) (Diaconis, 1988;
O’Hagan, 1991; Ritter, 2000; Rasmussen and
Ghahramani, 2003)
c0 =
ż
RdˆRd
Cθ(x, t) ν(dx)ν(dt)
c =
ż
Rd
Cθ(xi, t) ν(dt)
n
i=1
, C = Cθ(xi, xj)
n
i,j=1
Choosing w = wi
n
i=1
= C´1
c is optimal
P[|µ ´ ^µn| ď errn] = 99% for errn = 2.58
c
c0 ´ cTC´1c
yTC´1y
n
where y = f(xi)
n
i=1
. But, θ needs to be inferred (by MLE), and C´1
typically
requires O(n3
) operations
7/16

36. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Gaussian Probability
µ =
ż
[a,b]
exp ´1
2 tT
Σ´1
t
a
(2π)d det(Σ)
dt
Genz (1993)
=
ż
[0,1]d´1
f(x) dx
For some typical choice of a, b, Σ, d = 3, εa = 0; µ « 0.6763
Worst 10% Worst 10%
εr Method % Accuracy n Time (s)
IID Monte Carlo 100% 8.1E4 1.8E´2
1E´2 Sobol’ Sampling 100% 1.0E3 5.1E´3
Bayesian Lattice 100% 1.0E3 2.8E´3
IID Monte Carlo 100% 2.0E6 3.8E´1
1E´3 Sobol’ Sampling 100% 2.0E3 7.7E´3
Bayesian Lattice 100% 1.0E3 2.8E´3
1E´4 Sobol’ Sampling 100% 1.6E4 1.8E´2
Bayesian Lattice 100% 8.2E3 1.4E´2
Bayesian lattice cubature uses covariance kernel C for which C is circulant,
and operations on C require only O(n log(n)) operations 8/16

37. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Asian Option Pricing
fair price =
ż
Rd
e´rT
max

1
d
dÿ
j=1
Sj ´ K, 0

 e´xT
Σ´1
x/2
(2π)d/2 |Σ|1/2
dx « $13.12
Sj = S0e(r´σ2
/2)jT/d+σxj
= stock price at time jT/d,
Σ = min(i, j)T/d
d
i,j=1
Worst 10% Worst 10%
εa = 1E´4 Method % Accuracy n Time (s)
Sobol’ Sampling 100% 2.1E6 4.3
Sobol’ Sampling w/ control variates 97% 1.0E6 2.1
The coefficient of the control variate for low discrepancy sampling is different than
for IID Monte Carlo (H. et al., 2005; H. et al., 2017+)
9/16

38. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Sobol’ Indices
Y = output(X), where X „ U[0, 1]d
; Sobol’ Indexj(µ) describes how much
coordinate j of input X influences output Y (Sobol’, 1990; 2001):
Sobol’ Indexj(µ) :=
µ1
µ2 ´ µ2
3
, j = 1, . . . , d
µ1 :=
ż
[0,1)2d
[output(x) ´ output(xj, x1
´j)]output(x1
) dx dx1
µ2 :=
ż
[0,1)d
output(x)2
dx, µ3 :=
ż
[0,1)d
output(x) dx
output(x) = ´x1 + x1x2 ´ x1x2x3 + ¨ ¨ ¨ + x1x2x3x4x5x6 (Bratley et al., 1992)
εa = 1E´3, εr = 0 j 1 2 3 4 5 6
n 65 536 32 768 16 384 16 384 2 048 2 048
Sobol’ Indexj 0.6529 0.1791 0.0370 0.0133 0.0015 0.0015
{Sobol’ Indexj 0.6528 0.1792 0.0363 0.0126 0.0010 0.0012
Sobol’ Indexj(pµn) 0.6492 0.1758 0.0308 0.0083 0.0018 0.0039
10/16

39. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Summary
The error in simulating the mean can be decomposed as a trio identity
(Meng, 2017+; H., 2017+)
Knowing when to stop a simulation of the mean is not trivial (H. et al., 2017+)
The Berry-Esseen inequality can tell us when to stop an IID simulation
Fourier analysis can tell us when to stop a low discrepancy simulation
Bayesian cubature can tell us when to stop a simulation if you can afford the
computational cost
All methods can be fooled by nasty functions, f
Relative error tolerances and problems involving functions of integrals can
be handled (H. et al., 2017+)
Our algorithms are implemented in the Guaranteed Automatic Integration
Library (GAIL) (Choi et al., 2013–2015), which is under continuous
development
11/16

40. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Upcoming SAMSI Quasi-Monte Carlo Program
12/16

41. Thank you
Slides available at www.slideshare.net/fjhickernell/
tulane-march-2017-talk

42. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
References I
Bratley, P., B. L. Fox, and H. Niederreiter. 1992. Implementation and tests of low-discrepancy
sequences, ACM Trans. Model. Comput. Simul. 2, 195–213.
Choi, S.-C. T., Y. Ding, F. J. H., L. Jiang, Ll. A. Jiménez Rugama, X. Tong, Y. Zhang, and X. Zhou.
2013–2015. GAIL: Guaranteed Automatic Integration Library (versions 1.0–2.1).
Cools, R. and D. Nuyens (eds.) 2016. Monte Carlo and quasi-Monte Carlo methods: MCQMC,
Leuven, Belgium, April 2014, Springer Proceedings in Mathematics and Statistics, vol. 163,
Springer-Verlag, Berlin.
Diaconis, P. 1988. Bayesian numerical analysis, Statistical decision theory and related topics IV,
Papers from the 4th Purdue symp., West Lafayette, Indiana 1986, pp. 163–175.
Genz, A. 1993. Comparison of methods for the computation of multivariate normal probabilities,
Computing Science and Statistics 25, 400–405.
H., F. J. 2017+. Error analysis of quasi-Monte Carlo methods. submitted for publication,
arXiv:1702.01487.
H., F. J., L. Jiang, Y. Liu, and A. B. Owen. 2013. Guaranteed conservative fixed width confidence
intervals via Monte Carlo sampling, Monte Carlo and quasi-Monte Carlo methods 2012, pp. 105–128.
H., F. J. and Ll. A. Jiménez Rugama. 2016. Reliable adaptive cubature using digital sequences,
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Jiang, L. 2016. Guaranteed adaptive Monte Carlo methods for estimating means of random
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44. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Maximum Likelihood Estimation of the Covariance Kernel
f „ GP(0, s2
Cθ), Cθ = Cθ(xi, xj)
n
i,j=1
y = f(xi)
n
i=1
, ^µn = cT
^θ
C´1
^θ
y
^θ = argmin
θ
yT
C´1
θ y
[det(C´1
θ )]1/n
P[|µ ´ ^µn| ď errn] = 99% for errn =
2.58
?
n
b
c0,^θ ´ cT
^θ
C´1
^θ
c^θ yTC´1
^θ
y
16/16

45. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Maximum Likelihood Estimation of the Covariance Kernel
f „ GP(0, s2
Cθ), Cθ = Cθ(xi, xj)
n
i,j=1
y = f(xi)
n
i=1
, ^µn = cT
^θ
C´1
^θ
y
^θ = argmin
θ
yT
C´1
θ y
[det(C´1
θ )]1/n
P[|µ ´ ^µn| ď errn] = 99% for errn =
2.58
?
n
b
c0,^θ ´ cT
^θ
C´1
^θ
c^θ yTC´1
^θ
y
There is a de-randomized interpretation of Bayesian cubature (H., 2017+)
f P Hilbert space w/ reproducing kernel Cθ and with best interpolant rfy
16/16

46. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Maximum Likelihood Estimation of the Covariance Kernel
f „ GP(0, s2
Cθ), Cθ = Cθ(xi, xj)
n
i,j=1
y = f(xi)
n
i=1
, ^µn = cT
^θ
C´1
^θ
y
^θ = argmin
θ
yT
C´1
θ y
[det(C´1
θ )]1/n
P[|µ ´ ^µn| ď errn] = 99% for errn =
2.58
?
n
b
c0,^θ ´ cT
^θ
C´1
^θ
c^θ yTC´1
^θ
y
There is a de-randomized interpretation of Bayesian cubature (H., 2017+)
f P Hilbert space w/ reproducing kernel Cθ and with best interpolant rfy
|µ ´ ^µn| ď
2.58
?
n
b
c0,^θ ´ cT
^θ
C´1
^θ
c^θ
loooooooooomoooooooooon
error representer ^θ
b
yTC´1
^θ
y
looooomooooon
rfy ^θ
if f ´ rfy ^θ
ď
2.58 rf ^θ?
n
16/16

47. Introduction IID Monte Carlo Low Discrepancy Sampling Bayesian Cubature Examples References
Maximum Likelihood Estimation of the Covariance Kernel
f „ GP(0, s2
Cθ), Cθ = Cθ(xi, xj)
n
i,j=1
y = f(xi)
n
i=1
, ^µn = cT
^θ
C´1
^θ
y
^θ = argmin
θ
yT
C´1
θ y
[det(C´1
θ )]1/n
P[|µ ´ ^µn| ď errn] = 99% for errn =
2.58
?
n
b
c0,^θ ´ cT
^θ
C´1
^θ
c^θ yTC´1
^θ
y
There is a de-randomized interpretation of Bayesian cubature (H., 2017+)
f P Hilbert space w/ reproducing kernel Cθ and with best interpolant rfy
^θ = argmin
θ
yT
C´1
θ y
[det(C´1
θ )]1/n
= argmin
θ
vol z P Rn
: rfz θ ď rfy θ
(
|µ ´ ^µn| ď
2.58
?
n
b
c0,^θ ´ cT
^θ
C´1
^θ
c^θ
loooooooooomoooooooooon
error representer ^θ
b
yTC´1
^θ
y
looooomooooon
rfy ^θ
if f ´ rfy ^θ
ď
2.58 rf ^θ?
n
16/16