Combining Push-Forward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems

Butler, Troy; Jakeman, John Davis; Wildey, Timothy Michael

doi:10.1137/16m1087229

Cited by 37 publications

(91 citation statements)

References 24 publications

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“…Figure 10 plots the output samples of enzyme and substrate from the 493 last step of CMC for t = 1 (blue points) and t = 2 (orange points) versus 494 the contours (black lines) of the joint marginal distributions of eq. (20). 495 The distribution of paired enzyme-substrate samples illustrates that the 496 CMC output distribution closely approximates the target density, itself 497 representing dynamic evolution of the covariance between enzyme and 498 substrate measurements.…”

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confidence: 90%

“…In measure theoretic terms, the intrinsic measure implied by p(θ|Φ) is 223 known as the push forward of the measure implied by p(θ|Φ) with respect 224 to the model [20].…”

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confidence: 99%

“…The output target distribution shown by the contours corresponds to the marginal densities of each pair of enzyme-substrate measurements given by eq. (20). Figure 10 plots the output samples of enzyme and substrate from the 493 last step of CMC for t = 1 (blue points) and t = 2 (orange points) versus 494 the contours (black lines) of the joint marginal distributions of eq.…”

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confidence: 99%

“…Target marginal distributions (solid lines) along 499 with their approximations from kernel density estimation (dashed lines) 500 are also shown above and to the right of the main panel of Figure 10 and 501 largely indicate correspondence. (20). Enzyme and substrate measurements are given by the horizontal and vertical axes, respectively.…”

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confidence: 99%

“…608 This is because, rather than aiming to formulate a model that describes 609 output observations, our approach aims to recapitulate output distributions. 610 Others [20], (including us [22]), have considered similar problems before; 611 perhaps most notably by Albert Tarantola in his landmark work on inverse 612 problem theory (see, for example, [35]). In Tarantola's framework, a joint 613 input parameter and output space is considered, where prior knowledge and 614 experimental theory combine elegantly to produce a posterior distribution 615 whose marginal output distribution is a weighted "conjunction" of various 616 sources of information.…”

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confidence: 99%

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A Monte Carlo method to estimate cell population heterogeneity

Lambert

Gavaghan

Tavener

2019

Preprint

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1 Abstract 1Variation is characteristic of all living systems. Laboratory techniques 2 such as flow cytometry can probe individual cells, and, after decades of 3 experimentation, it is clear that even members of genetically identical cell 4 populations can exhibit differences. To understand whether variation is 5 biologically meaningful, it is essential to discern its source. Mathematical 6 models of biological systems are tools that can be used to investigate causes 7 of cell-to-cell variation. From mathematical analysis and simulation of these 8 models, biological hypotheses can be posed and investigated, then parameter 9 inference can determine which of these is compatible with experimental data. 10 Data from laboratory experiments often consist of "snapshots" representing 11 distributions of cellular properties at different points in time, rather than 12 individual cell trajectories. These data are not straightforward to fit using 13 hierarchical Bayesian methods, which require the number of cell population 14 clusters to be chosen a priori. Here, we introduce a computational sampling 15 method named "Contour Monte Carlo" for estimating mathematical model 16 parameters from snapshot distributions, which is straightforward to imple-17 ment and does not require cells be assigned to predefined categories. Our 18 method is appropriate for systems where observed variation is mostly due to 19 variability in cellular processes rather than experimental measurement error, 20 which may be the case for many systems due to continued improvements in 21 resolution of laboratory techniques. In this paper, we apply our method 22 to quantify cellular variation for three biological systems of interest and 23 provide Julia code enabling others to use this method. 24

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confidence: 90%

“…In measure theoretic terms, the intrinsic measure implied by p(θ|Φ) is 223 known as the push forward of the measure implied by p(θ|Φ) with respect 224 to the model [20].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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A Monte Carlo method to estimate cell population heterogeneity

Lambert

Gavaghan

Tavener

2019

Preprint

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Sequential Maximal Updated Density Parameter Estimation for Dynamical Systems With Parameter Drift

del‐Castillo‐Negrete,

Spence,

Butler

et al. 2024

Numerical Meth Engineering

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We present a novel method for generating sequential parameter estimates and quantifying epistemic uncertainty in dynamical systems within a data‐consistent (DC) framework. The DC framework differs from traditional Bayesian approaches due to the incorporation of the push‐forward of an initial density, which performs selective regularization in parameter directions not informed by the data in the resulting updated density. This extends a previous study that included the linear Gaussian theory within the DC framework and introduced the maximal updated density (MUD) estimate as an alternative to both least squares and maximum a posterior (MAP) estimates. In this work, we introduce algorithms for operational settings of MUD estimation in real‐ or near‐real time where spatio‐temporal datasets arrive in packets to provide updated estimates of parameters and identify potential parameter drift. Computational diagnostics within the DC framework prove critical for evaluating (1) the quality of the DC update and MUD estimate and (2) the detection of parameter value drift. The algorithms are applied to estimate (1) wind drag parameters in a high‐fidelity storm surge model, (2) thermal diffusivity field for a heat conductivity problem, and (3) changing infection and incubation rates of an epidemiological model.

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Solving Stochastic Inverse Problems for Structure-Property Linkages Using Data-Consistent Inversion

Tran

Wildey

2021

The Minerals, Metals &Amp; Materials Series

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Combining Push-Forward Measures and Bayes' Rule to Construct Consistent Solutions to Stochastic Inverse Problems

Cited by 37 publications

References 24 publications

A Monte Carlo method to estimate cell population heterogeneity

A Monte Carlo method to estimate cell population heterogeneity

Sequential Maximal Updated Density Parameter Estimation for Dynamical Systems With Parameter Drift

Solving Stochastic Inverse Problems for Structure-Property Linkages Using Data-Consistent Inversion

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