This document describes an automatic Bayesian method for numerical integration. It begins by introducing the problem of multivariate integration and current approaches like Monte Carlo integration that have limitations. It then presents the Bayesian cubature algorithm which chooses sample points and weights to minimize the error in approximating an integral. This is done by modeling the integrand as a Gaussian process, deriving identities relating the error to properties of the covariance kernel, and estimating its hyperparameters. The kernel used is shift-invariant, allowing fast matrix computations. Simulation results show Bayesian cubature achieves high accuracy with fewer samples compared to other methods.
Optimal interval clustering: Application to Bregman clustering and statistica...Frank Nielsen
We present a generic dynamic programming method to compute the optimal clustering of n scalar elements into k pairwise disjoint intervals. This case includes 1D Euclidean k-means, k-medoids, k-medians, k-centers, etc. We extend the method to incorporate cluster size constraints and show how to choose the appropriate k by model selection. Finally, we illustrate and refine the method on two case studies: Bregman clustering and statistical mixture learning maximizing the complete likelihood.
http://arxiv.org/abs/1403.2485
We examine the effectiveness of randomized quasi Monte Carlo (RQMC) to improve the convergence rate of the mean integrated square error, compared with crude Monte Carlo (MC), when estimating the density of a random variable X defined as a function over the s-dimensional unit cube (0,1)^s. We consider histograms and kernel density estimators. We show both theoretically and empirically that RQMC estimators can achieve faster convergence rates in
some situations.
This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer.
Optimal interval clustering: Application to Bregman clustering and statistica...Frank Nielsen
We present a generic dynamic programming method to compute the optimal clustering of n scalar elements into k pairwise disjoint intervals. This case includes 1D Euclidean k-means, k-medoids, k-medians, k-centers, etc. We extend the method to incorporate cluster size constraints and show how to choose the appropriate k by model selection. Finally, we illustrate and refine the method on two case studies: Bregman clustering and statistical mixture learning maximizing the complete likelihood.
http://arxiv.org/abs/1403.2485
We examine the effectiveness of randomized quasi Monte Carlo (RQMC) to improve the convergence rate of the mean integrated square error, compared with crude Monte Carlo (MC), when estimating the density of a random variable X defined as a function over the s-dimensional unit cube (0,1)^s. We consider histograms and kernel density estimators. We show both theoretically and empirically that RQMC estimators can achieve faster convergence rates in
some situations.
This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer.
The standard Galerkin formulation of the acoustic wave propagation, governed by the Helmholtz partial differential equation (PDE), is indefinite for large wavenumbers. However, the Helmholtz PDE is in general not indefinite. The lack of coercivity (indefiniteness) is one of the major difficulties for approximation and simulation of heterogeneous media wave propagation models, including application to stochastic wave propagation Quasi Monte Carlo (QMC) analysis. We will present a new class of sign-definite continuous and discrete preconditioned FEM Helmholtz wave propagation models.
Efficient Simulations for Contamination of Groundwater Aquifers under Uncerta...Alexander Litvinenko
1. Solved time-dependent density driven flow problem with uncertain porosity and permeability in 2D and 3D
2. Computed propagation of uncertainties in porosity into the mass fraction.
3. Computed the mean, variance, exceedance probabilities, quantiles, risks.
4. Such QoIs as the number of fingers, their size, shape, propagation time can be unstable
5. For moderate perturbations, our gPCE surrogate results are similar to qMC results.
6. Used highly scalable solver on up to 800 computing nodes,
Many mathematical models use a large number of poorly-known parameters as inputs. Quantifying the influence of each of these parameters is one of the aims of sensitivity analysis. Global Sensitivity Analysis is an important paradigm for understanding model behavior, characterizing uncertainty, improving model calibration, etc. Inputs’ uncertainty is modeled by a probability distribution. There exist various measures built in that paradigm. This tutorial focuses on the so-called Sobol’ indices, based on functional variance analysis. Estimation procedures will be presented, and the choice of the designs of experiments these procedures are based on will be discussed. As Sobol’ indices have no clear interpretation in the presence of statistical dependences between inputs, it also seems promising to measure sensitivity with Shapley effects, based on the notion of Shapley value, which is a solution concept in cooperative game theory.
The standard Galerkin formulation of the acoustic wave propagation, governed by the Helmholtz partial differential equation (PDE), is indefinite for large wavenumbers. However, the Helmholtz PDE is in general not indefinite. The lack of coercivity (indefiniteness) is one of the major difficulties for approximation and simulation of heterogeneous media wave propagation models, including application to stochastic wave propagation Quasi Monte Carlo (QMC) analysis. We will present a new class of sign-definite continuous and discrete preconditioned FEM Helmholtz wave propagation models.
Efficient Simulations for Contamination of Groundwater Aquifers under Uncerta...Alexander Litvinenko
1. Solved time-dependent density driven flow problem with uncertain porosity and permeability in 2D and 3D
2. Computed propagation of uncertainties in porosity into the mass fraction.
3. Computed the mean, variance, exceedance probabilities, quantiles, risks.
4. Such QoIs as the number of fingers, their size, shape, propagation time can be unstable
5. For moderate perturbations, our gPCE surrogate results are similar to qMC results.
6. Used highly scalable solver on up to 800 computing nodes,
Many mathematical models use a large number of poorly-known parameters as inputs. Quantifying the influence of each of these parameters is one of the aims of sensitivity analysis. Global Sensitivity Analysis is an important paradigm for understanding model behavior, characterizing uncertainty, improving model calibration, etc. Inputs’ uncertainty is modeled by a probability distribution. There exist various measures built in that paradigm. This tutorial focuses on the so-called Sobol’ indices, based on functional variance analysis. Estimation procedures will be presented, and the choice of the designs of experiments these procedures are based on will be discussed. As Sobol’ indices have no clear interpretation in the presence of statistical dependences between inputs, it also seems promising to measure sensitivity with Shapley effects, based on the notion of Shapley value, which is a solution concept in cooperative game theory.
Multidimensional integrals may be approximated by weighted averages of integrand values. Quasi-Monte Carlo (QMC) methods are more accurate than simple Monte Carlo methods because they carefully choose where to evaluate the integrand. This tutorial focuses on how quickly QMC methods converge to the correct answer as the number of integrand values increases. The answer may depend on the smoothness of the integrand and the sophistication of the QMC method. QMC error analysis may assumes the integrand belongs to a reproducing kernel Hilbert space or may assume that the integrand is an instance of a stochastic process with known covariance structure. These two approaches have interesting parallels. This tutorial also explores how the computational cost of achieving a good approximation to the integral depends on the dimension of the domain of the integrand. Finally, this tutorial explores methods for determining how many integrand values are needed to satisfy the error tolerance. Relevant software is described.
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSTahia ZERIZER
In this article, we study boundary value problems of a large
class of non-linear discrete systems at two-time-scales. Algorithms are given to implement asymptotic solutions for any order of approximation.
We research behavior and sharp bounds for the zeros of infinite sequences of polynomials orthogonal with respect to a Geronimus perturbation of a positive Borel measure on the real line.
A numerical method to solve fractional Fredholm-Volterra integro-differential...OctavianPostavaru
The Goolden ratio is famous for the predictability it provides both in the microscopic world as well as in the dynamics of macroscopic structures of the universe. The extension of the Fibonacci series to the Fibonacci polynomials gives us the opportunity to use this powerful tool in the study of Fredholm-Volterra integro-differential equations. In this paper, we define a new hybrid fractional function consisting of block-pulse functions and Fibonacci polynomials (FHBPF). For this, in the Fibonacci polynomials we perform the transformation $x\to x^{\alpha}$, with $\alpha$ a real parameter. In the method developed in this paper, we propose that the unknown function $D^{\alpha}f(x)$ be written as a linear combination of FHBPF. We consider the fractional derivative $D^{\alpha}$ in the Caputo sense. Using theoretical considerations, we can write both the function $f(x)$ and other involved functions of type $D^{\beta}f(x)$ on the same basis. For this operation, we have to define an integral operator of Riemann-Liouville type associated to FHBPF, and with the help of hypergeometric functions, we can express this operator exactly. All these ingredients together with the collocation in the Newton-Cotes nodes transform the integro-differential equation into an algebraic system that we solve by applying Newton's iterative method. We conclude the paper with some examples to demonstrate that the proposed method is simple to implement and accurate. There are situations when by simply considering $\alpha\ne1$, we obtain an improvement in accuracy by 12 orders of magnitude.
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Multi-source connectivity as the driver of solar wind variability in the heli...
Automatic Bayesian method for Numerical Integration
1. Automatic Bayesian method for Numerical
Integration
Jagadeeswaran Rathinavel, Fred J. Hickernell
Department of Applied Mathematics, Illinois Institute of Technology
jrathin1@iit.edu
Thanks to the CASSC 2017 organizers
2. Introduction Bayesian Cubature Simulation results Conclusion References
Multivariate Integration
Approximate the d-dimensional integral over [0, 1)d
µ := E[f(X)] :=
ż
[0,1)d
f(x) ν(dx)
by a simple cubature rule
µ « ^µ :=
n´1ÿ
j=0
f(xj)wj =
ż
[0,1)d
f(x) ^ν(dx)
using points (xj)n´1
j=0 and associated weights wj. Then the approximation error
µ ´ ^µ =
ş
[0,1)d f(x) (ν ´ ^ν)(dx)
use extensible pointset and an algorithm that allows to add more points if needed.
2/22
3. Introduction Bayesian Cubature Simulation results Conclusion References
Motivating Example
How to measure the water volume of a creek in a given area (For Ex: 10sqft) ?
Figure: (Nuyens, 2007)
3/22
7. Introduction Bayesian Cubature Simulation results Conclusion References
What is multivariate integration?
Figure: (Nuyens, 2007)
A d-dim integral:
ż
[0,1)d
f(x) dx =
ż 1
0
ż 1
0
. . .
ż 1
0
f(x1, x2, . . . xd) dx1 dx2 . . . dxd
Grid points: Curse of dimensionality
IID Monte Carlo: Converges O(n´ 1
2 )
^µ = 1
n
řn´1
j=0 f(xj)
Typical Quasi Monte Carlo: Converges O(n´1+
)
xj chosen carefully (low-discrepancy points)
4/22
8. Introduction Bayesian Cubature Simulation results Conclusion References
What is multivariate integration?
Figure: (Nuyens, 2007)
A d-dim integral:
ż
[0,1)d
f(x) dx =
ż 1
0
ż 1
0
. . .
ż 1
0
f(x1, x2, . . . xd) dx1 dx2 . . . dxd
Grid points: Curse of dimensionality
IID Monte Carlo: Converges O(n´ 1
2 )
^µ = 1
n
řn´1
j=0 f(xj)
Typical Quasi Monte Carlo: Converges O(n´1+
)
xj chosen carefully (low-discrepancy points)
Can we do better? 4/22
9. Introduction Bayesian Cubature Simulation results Conclusion References
Algorithm
1: procedure AutoCubature(f, errtol) Ź Integrate within the error tolerance
2: n Ð 28
3: do
4: Generate (xi)n´1
i=0
5: Sample (f(xi))n´1
i=0
6: Compute errn
7: n Ð 2 ˆ n
8: while errn ą errtol Ź Iterate till error tolerance is met
9: Compute weights (wi)n´1
i=0
10: Compute integral ^µn
11: return ^µn Ź Integral estimate ^µn
12: end procedure
Problem:
How to choose (xj)n´1
j=0 and (wj)n´1
j=0 to make |µ ´ ^µ| small?
Given error tolerance errtol, how big must ‘n’ be to guarantee |µ ´ ^µ| ď errtol
5/22
10. Introduction Bayesian Cubature Simulation results Conclusion References
Algorithm
1: procedure AutoCubature(f, errtol) Ź Integrate within the error tolerance
2: n Ð 28
3: do
4: Generate (xi)n´1
i=0
5: Sample (f(xi))n´1
i=0
6: Compute errn
7: n Ð 2 ˆ n
8: while errn ą errtol Ź Iterate till error tolerance is met
9: Compute weights (wi)n´1
i=0
10: Compute integral ^µn
11: return ^µn Ź Integral estimate ^µn
12: end procedure
Problem:
How to choose (xj)n´1
j=0 and (wj)n´1
j=0 to make |µ ´ ^µ| small?
Given error tolerance errtol, how big must ‘n’ be to guarantee |µ ´ ^µ| ď errtol
5/22
11. Introduction Bayesian Cubature Simulation results Conclusion References
Bayesian Trio Identity
Random f : f „ GP(0, s2
Cθ), a Gaussian process
from the sample space F
with zero mean and covariance kernel, s2
Cθ, Cθ : X ˆ X Ñ R. Then
6/22
12. Introduction Bayesian Cubature Simulation results Conclusion References
Bayesian Trio Identity
Random f : f „ GP(0, s2
Cθ), a Gaussian process
from the sample space F
with zero mean and covariance kernel, s2
Cθ, Cθ : X ˆ X Ñ R. Then
c0 =
ż
XˆX
C(x, t) ν(dx)ν(dt), c =
ż
X
C(xi, t) ν(dt)
n´1
i=0
C = C(xi, xj)
n´1
i,j=0
, w = wi
n´1
i=0
,
µ ´ ^µ =
ş
X f(x) (ν ´ ^ν)(dx)
s
a
c0 ´ 2cTw + wTCwloooooooooooooomoooooooooooooon
ALNB
(f, ν ´ ^ν)
a
c0 ´ 2cTw + wTCwlooooooooooooomooooooooooooon
DSCB
(ν ´ ^ν)
sloomoon
VARB
(f)
6/22
13. Introduction Bayesian Cubature Simulation results Conclusion References
Bayesian Trio Identity
Random f : f „ GP(0, s2
Cθ), a Gaussian process
from the sample space F
with zero mean and covariance kernel, s2
Cθ, Cθ : X ˆ X Ñ R. Then
c0 =
ż
XˆX
C(x, t) ν(dx)ν(dt), c =
ż
X
C(xi, t) ν(dt)
n´1
i=0
C = C(xi, xj)
n´1
i,j=0
, w = wi
n´1
i=0
,
µ ´ ^µ =
ş
X f(x) (ν ´ ^ν)(dx)
s
a
c0 ´ 2cTw + wTCwloooooooooooooomoooooooooooooon
ALNB
(f, ν ´ ^ν)
a
c0 ´ 2cTw + wTCwlooooooooooooomooooooooooooon
DSCB
(ν ´ ^ν)
sloomoon
VARB
(f)
The scale parameter, s, and shape parameter, θ, should be estimated.
w = C´1
c minimizes DSCB
(ν ´ ^ν)
makes ALNB
(f, ν ´ ^ν)
ˇ
ˇ(f(xi) = yi)n´1
i=0 „ N(0, 1)
Ref: Diaconis (1988), O’Hagan (1991), Ritter (2000), Rasmussen (2003) and
others 6/22
19. Introduction Bayesian Cubature Simulation results Conclusion References
Rank-1 Lattice rules : Low discrepancy point set
Given the “generating vector” g, the construction of n - Rank-1 lattice points is
given by
"
kg
n
* n´1
k=0
then the Lattice rule approximation is
1
n
n´1ÿ
k=0
f
"
kg
n
*
with t.u the fractional part, i.e, ‘modulo 1’ operator.
9/22
22. Introduction Bayesian Cubature Simulation results Conclusion References
Bayesian Cubature
µ =
ż
X
f(x)ν(dx) « ^µn =
n´1ÿ
i=0
wif(xi)
Assume f „ GP(0, s2
Cθ)
µ ´ ^µn
ˇ
ˇ(f(xi) = yi)n´1
i=0 „ N yT
(C´1
c ´ w), s2
(c0 ´ cT
C´1
c)
10/22
23. Introduction Bayesian Cubature Simulation results Conclusion References
Bayesian Cubature
µ =
ż
X
f(x)ν(dx) « ^µn =
n´1ÿ
i=0
wif(xi)
Assume f „ GP(0, s2
Cθ)
µ ´ ^µn
ˇ
ˇ(f(xi) = yi)n´1
i=0 „ N yT
(C´1
c ´ w), s2
(c0 ´ cT
C´1
c)
Choosing w = C´1
^θ
c^θ is optimal
P[|µ ´ ^µn| ď errn] ě 99% for errn = 2.58
d
c^θ,0 ´ cT
^θ
C´1
^θ
c^θ
yTC´1
^θ
y
n
MLE ^θ = argmin
θ
yT
C´1
θ y
[det(C´1
θ )]1/n
, where y = f(xi)
n´1
i=0
.
10/22
24. Introduction Bayesian Cubature Simulation results Conclusion References
Bayesian Cubature
µ =
ż
X
f(x)ν(dx) « ^µn =
n´1ÿ
i=0
wif(xi)
Assume f „ GP(0, s2
Cθ)
µ ´ ^µn
ˇ
ˇ(f(xi) = yi)n´1
i=0 „ N yT
(C´1
c ´ w), s2
(c0 ´ cT
C´1
c)
Choosing w = C´1
^θ
c^θ is optimal
P[|µ ´ ^µn| ď errn] ě 99% for errn = 2.58
d
c^θ,0 ´ cT
^θ
C´1
^θ
c^θ
yTC´1
^θ
y
n
MLE ^θ = argmin
θ
yT
C´1
θ y
[det(C´1
θ )]1/n
, where y = f(xi)
n´1
i=0
.
C´1
typically requires O(n3
) operations.
But with covariance kernel C for which matrix C is symmetric circulant.
So operations on C require only O(n log(n)) operations.
10/22
25. Introduction Bayesian Cubature Simulation results Conclusion References
Optimal Shape parameter θ
Maximum likelihood estimate of θ by
argmin
θ
yT
C´1
θ y
[det(C´1
θ )]1/n
11/22
26. Introduction Bayesian Cubature Simulation results Conclusion References
Optimal Shape parameter θ
Maximum likelihood estimate of θ by
argmin
θ
yT
C´1
θ y
[det(C´1
θ )]1/n
simplified to (Bernstein, 2009)
argmin
θ
log
n´1ÿ
i=0
|zi|2
γi
+
1
n
n´1ÿ
i=0
log(γi)
where (γi) eigen values of Cθ
z := (zi)n´1
i=0 = DFT y , γ := (γi)n´1
i=0 = DFT C(xi, x0)
n´1
i=0
O(n log(n)) operations to estimate the ^θ
11/22
38. Introduction Bayesian Cubature Simulation results Conclusion References
Summary
Automatic Bayesian Cubature with O(n) complexity and O(n log(n)) MLE
Complexity
Having the advantages of a kernel method and the low computation cost of
Quasi Monte carlo
Scalable based on the complexity of the Integrand
i.e, Kernel order and Lattice-points can be chosen to suit the smoothness of
the integrand
More about Guaranteed Automatic Algorithms (GAIL):
http://gailgithub.github.io/GAIL_Dev/
These slides will also be available here.
19/22
39. Introduction Bayesian Cubature Simulation results Conclusion References
Future work
Dimension independent
Tighter error bound
Roundoff error
Lattice points optimization specific the the kernel
Choosing kernel order automatically as part of MLE
Can we compute n directly instead of in Loop?
Choosing Periodization transform automatically
20/22
41. Introduction Bayesian Cubature Simulation results Conclusion References
References I
Bernstein, Dennis S. 2009. Matrix mathematics, theory, facts, and formulas, Princeton University
Press.
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.
Nuyens, Dirk. 2007. Fast construction of good lattice rules, Ph.D. Thesis.
O’Hagan, A. 1991. Bayes-Hermite quadrature, J. Statist. Plann. Inference 29, 245–260.
Olver, F. W. J., D. W. Lozier, R. F. Boisvert, C. W. Clark, and A. B. O. Dalhuis. 2013. Digital library of
mathematical functions.
Rasmussen, C. E. 2003. Bayesian Monte Carlo, Advances in Neural Information Processing
Systems, pp. 489–496.
Ritter, K. 2000. Average-case analysis of numerical problems, Lecture Notes in Mathematics,
vol. 1733, Springer-Verlag, Berlin.
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