Fast estimation of expected information gains for Bayesian experimental designs based on Laplace approximations

Long, Quan; Scavino, Marco; Tempone, Raúl; Wang, Suojin

doi:10.1016/j.cma.2013.02.017

Cited by 124 publications

(182 citation statements)

References 23 publications

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“…Many algorithms have been proposed to perform this optimization in general non-linear cases (see e.g. [20][21][22]). Huan and Marzouk [22], for example, rewrite the expected relative entropy for a joint analysis of D 1 and D 2 as:…”

Section: B Surprisementioning

confidence: 99%

Quantifying concordance in cosmology

Seehars

Grandis

Amara³

et al. 2016

Phys. Rev. D

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Quantifying the concordance between different cosmological experiments is important for testing the validity of theoretical models and systematics in the observations. In earlier work, we thus proposed the Surprise, a concordance measure derived from the relative entropy between posterior distributions. We revisit the properties of the Surprise and describe how it provides a general, versatile, and robust measure for the agreement between datasets. We also compare it to other measures of concordance that have been proposed for cosmology. As an application, we extend our earlier analysis and use the Surprise to quantify the agreement between WMAP 9, Planck 13 and Planck 15 constraints on the ΛCDM model. Using a principle component analysis in parameter space, we find that the large Surprise between WMAP 9 and Planck 13 (S = 17.6 bits, implying a deviation from consistency at 99.8% confidence) is due to a shift along a direction that is dominated by the amplitude of the power spectrum. The Planck 15 constraints deviate from the Planck 13 results (S = 56.3 bits), primarily due to a shift in the same direction. The Surprise between WMAP and Planck consequently disappears when moving to Planck 15 (S = −5.1 bits). This means that, unlike Planck 13, Planck 15 is not in tension with WMAP 9. These results illustrate the advantages of the relative entropy and the Surprise for quantifying the disagreement between cosmological experiments and more generally as an information metric for cosmology.

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Section: B Surprisementioning

confidence: 99%

Quantifying concordance in cosmology

Seehars

Grandis

Amara³

et al. 2016

Phys. Rev. D

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“…It is likely that the issues with importance sampling will be exacerbated in higher dimensional problems, and so, importance sampling from the Laplace approximation may be more useful in this setting. Polynomial-based sparse quadrature methods [36] may also be useful in higher dimensional model settings to perform the integration over θ in Equation (1).…”

Section: Discussionmentioning

confidence: 99%

“…Laplace approximations also suffer from the curse of dimensionality. To overcome this issue, Long et al [36] used polynomial-based sparse quadrature to integrate over the prior distribution. Furthermore, the estimated posterior distribution obtained from the Laplace approximation may not accommodate the tails of the posterior distribution, and so, we investigate alternative methods to obtain better coverage of the tails.…”

Section: Laplace Approximationmentioning

confidence: 99%

Fully Bayesian Experimental Design for Pharmacokinetic Studies

Ryan

Drovandi

Pettitt

2015

Entropy

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Utility functions in Bayesian experimental design are usually based on the posterior distribution. When the posterior is found by simulation, it must be sampled from for each future dataset drawn from the prior predictive distribution. Many thousands of posterior distributions are often required. A popular technique in the Bayesian experimental design literature, which rapidly obtains samples from the posterior, is importance sampling, using the prior as the importance distribution. However, importance sampling from the prior will tend to break down if there is a reasonable number of experimental observations. In this paper, we explore the use of Laplace approximations in the design setting to overcome this drawback. Furthermore, we consider using the Laplace approximation to form the importance distribution to obtain a more efficient importance distribution than the prior. The methodology is motivated by a pharmacokinetic study, which investigates the effect of extracorporeal membrane oxygenation on the pharmacokinetics of antibiotics in sheep. The design problem is to find 10 near optimal plasma sampling times that produce precise estimates of pharmacokinetic model parameters/measures of interest. We consider several different utility functions of interest in these studies, which involve the posterior distribution of parameter functions.

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“…The main results obtained in are as follows.Theorem Under the assumption that the smallest singular value of the Jacobian matrix is bounded from zero by a small constant and the model output g ( θ , ξ ) has continuous second derivatives, the expected information gain can be approximated by

I (bold-italicξ) = \int_{bold-italic Θ} ([], - \frac{1}{2} \log (| bold-italic Σ ({bold-italicθ}_{0}, bold-italicξ) |) - \frac{tr (bold-italic Σ ({bold-italicθ}_{0}, bold-italicξ) {bold-italicH}_{h} ({bold-italicθ}_{0}))}{2} - \frac{d}{2} - \frac{d}{2} \log (2 π)) p ({bold-italicθ}_{0}) d {bold-italicθ}_{0} + scriptO ((), \frac{1}{s}),

where H h is the Hessian of

h (bold-italicθ) = \log ([], p (bold-italicθ))

.Theorem Under the assumption that r < d and that the model output g ( θ , ξ ) has continuous second derivatives, the expected information gain can be approximated by

I (bold-italicξ) = \int_{bold-italic Θ} ([], - \frac{1}{2} \log | {bold-italic Σ}_{proj} ({bold-italicθ}_{0}, bold-italicξ) | - \log ([], \int_{T} p_{bold-italics, bold-italict} (bold-italic 0, bold-italict) d bold-italict) - \frac{r}{2} - \frac{r}{2} \log (2 π)) p ({bold-italicθ}_{0}) d {bold-italicθ}_{}

…”

Section: Expected Information Gain In a Proposed Experimentsmentioning

confidence: 99%

“…The following theorem holds. Theorem Under the assumption that the smallest singular value of the Jacobian matrix is bounded from zero by a small constant and the model output g ( θ , ξ ) has continuous second derivatives, the expected information gain can be approximated by

I (bold-italicξ) = \int_{bold-italic Θ} {\overset{D}{true}}_{KL} ({bold-italicθ}_{0}, bold-italicξ) p ({bold-italicθ}_{0}) d {bold-italicθ}_{0} + scriptO ((), \frac{1}{s}),

with

{\tilde{D}}_{KL} (θ_{0}, ξ) = \int_{normalΘ} \frac{ϕ (θ | y)}{p (\vec{θ})} ϕ (θ | y) d θ and ϕ (θ | y) = \frac{\tilde{p} (θ | y, ξ)}{\int_{normalΘ} \tilde{p} (θ | y, ξ) d θ} .

Proof The proof is very similar to that of Theorem in , and only an outline is provided here. Let

ϕ (θ) = p (\hat{θ} | y) \exp (- {(θ - \hat{θ})}^{⊤} normalΣ ()

…”

Section: Expected Information Gain In a Proposed Experimentsmentioning

confidence: 99%

Optimal Bayesian Experimental Design for Priors of Compact Support with Application to Shock‐Tube Experiments for Combustion Kinetics

Bisetti

Kim

Knio

et al. 2016

Numerical Meth Engineering

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SUMMARYThe analysis of reactive systems in combustion science and technology relies on detailed models comprising many chemical reactions that describe the conversion of fuel and oxidizer into products and the formation of pollutants. Shock-tube experiments are a convenient setting for measuring the rate parameters of individual reactions. The temperature, pressure, and concentration of reactants are chosen to maximize the sensitivity of the measured quantities to the rate parameter of the target reaction. In this study, we optimize the experimental setup computationally by optimal experimental design (OED) in a Bayesian framework. We approximate the posterior probability density functions (pdf) using truncated Gaussian distributions in order to account for the bounded domain of the uniform prior pdf of the parameters. The underlying Gaussian distribution is obtained in the spirit of the Laplace method, more precisely, the mode is chosen as the maximum a posteriori (MAP) estimate, and the covariance is chosen as the negative inverse of the Hessian of the misfit function at the MAP estimate. The model related entities are obtained from a polynomial surrogate. The optimality, quantified by the information gain measures, can be estimated efficiently by a rejection sampling algorithm against the underlying Gaussian probability distribution, rather than against the true posterior. This approach offers a significant error reduction when the magnitude of the invariants of the posterior covariance are comparable to the size of the bounded domain of the prior. We demonstrate the accuracy and superior computational efficiency of our method for shock-tube experiments aiming to measure the model parameters of a key reaction which is part of the complex kinetic network describing the hydrocarbon oxidation. In the experiments, the initial temperature and fuel concentration are optimized with respect to the expected information gain in the estimation of the parameters of the target reaction rate. We show that the expected information gain surface can change its "shape" dramatically according to the level of noise introduced into the synthetic data. The information that can be extracted from the data saturates as a logarithmic function of the number of experiments, and few experiments are needed when they are conducted at the optimal experimental design conditions. Furthermore, inversion of the legacy data indicates the validity and robustness of our designs.

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Fast estimation of expected information gains for Bayesian experimental designs based on Laplace approximations

Cited by 124 publications

References 23 publications

Quantifying concordance in cosmology

Quantifying concordance in cosmology

Fully Bayesian Experimental Design for Pharmacokinetic Studies

Optimal Bayesian Experimental Design for Priors of Compact Support with Application to Shock‐Tube Experiments for Combustion Kinetics

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