Bayesian probability updates using SamplingImportance Resampling Applications in nuclear theory W. G. Jiang and C. Forss en

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Bayesian probability updates using Sampling/Importance

Resampling: Applications in nuclear theory

W. G. Jiang and C. Forss´en

Department of Physics, Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden

Abstract

We review an established Bayesian sampling method called sampling/importance resampling

and highlight situations in nuclear theory when it can be particularly useful. To this end we both

analyse a toy problem and demonstrate realistic applications of importance resampling to infer the

posterior distribution for parameters of ∆NNLO interaction model based on chiral eﬀective ﬁeld

theory and to estimate the posterior probability distribution of target observables. The limitation

of the method is also showcased in extreme situations where importance resampling breaks.

PACS numbers: 21.30.-x

arXiv:2210.02507v1 [nucl-th] 5 Oct 2022

I. INTRODUCTION

Bayesian inference is an appealing approach for dealing with theoretical uncertainties

and has been applied in diﬀerent nuclear physics studies [

–

]. In the practice of Bayesian

analyses, a sampling procedure is usually inevitable for approximating the posterior probability

distribution of model parameters and for performing predictive computations. Various

Markov chain Monte Carlo (

MCMC

) methods [

–

] are often used for this purpose, even

for complicated models with high-dimensional parameter spaces. However,

MCMC

sampling

typically requires many likelihood evaluations, which is often a costly operation in nuclear

theory, and there is a need to explore other sampling techniques. In this paper, we review an

established method called sampling/importance resampling (

S/IR

) [

–

] and demonstrate

its use in realistic nuclear physics applications where we also perform comparisons with

MCMC sampling.

In recent years, there has been an increasing demand for precision nuclear theory.This

implies a challenge to not just achieve accurate theoretical predictions but also to quantify

accompanying uncertainties. The use of ab initio many-body methods and nuclear interaction

models based on chiral eﬀective ﬁeld theory (

EFT) has shown a potential to describe

ﬁnite nuclei and nuclear matter based on extant experimental data (e.g. nucleon-nucleon

scattering, few-body sector) with controlled approximations [

–

]. The interaction model is

parametrized in terms of low-energy constants (

LEC

s), the number of which is growing order-

by-order according to the rules of a corresponding power counting [

–

]. Very importantly,

the systematic expansion allows to quantify the truncation error and to incorporate this

knowledge in the analysis [

–

]. Indeed, Bayesian inference is an excellent framework

to incorporate diﬀerent sources of uncertainty and to propagate error bars to the model

predictions. Starting from Bayes’ theorem

pr(θ|D)∝ L(θ)pr(θ),(1)

where

(

θ|D

) is the posterior probability density function (

PDF

) for the vector

LEC

(conditional on the data

(

)

≡pr

(

D |θ

) is the likelihood and

(

) is the prior. Then

for any model prediction one needs to evaluate the expectation value of a function of interest

y(θ) (target observables) according to the posterior. This involves integrals such as

Zdθy(θ)pr(θ|D),(2)

which can not be analytically solved for realistic cases. Fortunately, integrals such as Eq.

(2)

can be approximately evaluated using a ﬁnite set of samples

{θi}N

i=1

from

(

θ|D

MCMC

sampling methods are the main computational tool for providing such samples, even for

high-dimensional parameter volumes [

]. However the use of

MCMC

in nuclear theory

typically requires massive computations to record suﬃciently many samples from the Markov

chain. There are certainly situations where

MCMC

sampling is not ideal, or even becomes

infeasible:

When the posterior is conditioned on some calibration data for which our model

evaluations are very costly. Then we might only aﬀord a limited number of full

likelihood evaluations and our MCMC sampling becomes less likely to converge.

Bayesian posterior updates in which calibration data is added in several diﬀerent stages.

This typically requires that the

MCMC

sampling must be carried out repeatedly from

scratch.

Model checking where we want to explore the sensitivity to prior assignments. This is

a second example of posterior updating.

The prediction of target observables for which our model evaluations become very

costly and the handling of a large number of MCMC samples becomes infeasible.

These are situations where one might want to use the

S/IR

method [

], which allows

posterior probability updates with a minimum amount of computation (previous results of

model evaluations remain useful). In the following sections we ﬁrst review the

S/IR

method

and then present both toy and realistic applications in which its performance is compared

with full

MCMC

sampling. Finally, we illustrate limitations of the method by considering

cases where

S/IR

fails and we highlight the importance of the so-called eﬀective number of

samples. More diﬃcult scenarios, in which the method fails without a clear warning, are left

for the concluding remarks.

II. SAMPLING/IMPORTANCE RESAMPLING

The basic idea of

S/IR

is to utilize the inherent duality between samples and the density

(probability distribution) from which they were generated [

]. This duality oﬀers an

opportunity to indirectly recreate a density (that might be hard to compute) from samples

that are easy to obtain. Here we give a brief review of the method and illustrate with a toy

problem.

Let us consider a target density

(

). In our applications this target will be the posterior

PDF pr

(

θ|D

) from Eq.

(1)

. Instead of attempting to directly collect samples from

(

), as

would be the goal in

MCMC

approaches, the

S/IR

method uses a detour. We ﬁrst obtain

samples from a simple (even analytic) density

(

). We then resample from this ﬁnite set

using a resampling algorithm to approximately recreate samples from the target density

(

). There are (at least) two diﬀerent resampling methods. In this paper we only focus on

one of them called weighted bootstrap (more details of resampling methods can be found in

Refs. [20, 21]).

Assuming we are interested in the target density

(

) =

(

)

/Rf

(

) d

, the procedure

of resampling via weighted bootstrap can be summarized as follows:

1. Generate the set {θi}n

i=1 of samples from a sampling density g(θ).

Calculate

ωi

(

θi

)

/ g

(

θi

) for the

samples and deﬁne importance weights as:

qi=ωi/Pn

j=1 ωj.

Draw

new samples

{θ∗

i}N

i=1

from the discrete distribution

{θi}n

i=1

with probability

mass qion θi.

The set of samples

{θ∗

i}N

i=1

will then be approximately distributed according to the

target density h(θ).

Intuitively, the distribution of

θ∗

should be good approximation of

(

) when

is large

enough. Here we justify this claim via the cumulative distribution function of

θ∗

(for the

one-dimensional case)

pr(θ∗≤a) =

i=1

qi·H(a−θi) =

i=1

ωi·H(a−θi)

i=1

ωi

−−−→

n→∞

Eghf(θ)

g(θ)·H(a−θi)i

Eghf(θ)

g(θ)i=Ra

−∞ f(θ)dθ

R∞

−∞ f(θ)dθ =Za

−∞

h(θ)dθ,

(3)

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BayesianprobabilityupdatesusingSampling/ImportanceResampling:ApplicationsinnucleartheoryW.G.JiangandC.ForssenDepartmentofPhysics,ChalmersUniversityofTechnology,SE-41296Goteborg,SwedenAbstractWereviewanestablishedBayesiansamplingmethodcalledsampling/importanceresamplingandhighlightsituationsinnucle...

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