Measuring statistical evidence using relative belief

Evans, Michael

doi:10.1016/j.csbj.2015.12.001

Cited by 22 publications

(30 citation statements)

References 23 publications

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“…Inspecting s λ for a fixed c λ , say 0.95, is a rather subjective choice. According to [16], we may exploit a statistically meaningful interpretation of the measured data to define another kind of Bayesian region. If we suppose that r is plausibly the true value, then we say that there is evidence in favor of this supposition when its normalized posterior probability  …”

Section: Plausible Regionsmentioning

confidence: 99%

Bayesian error regions in quantum estimation I: analytical reasonings

Teo

Jeong

2018

New J. Phys.

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Results concerning the construction of quantum Bayesian error regions as a means to certify the quality of parameter point estimators have been reported in recent years. This task remains numerically formidable in practice for large dimensions and so far, no analytical expressions of the region size and credibility (probability of any given true parameter residing in the region) are known, which form the two principal region properties to be reported alongside a point estimator obtained from collected data. We first establish analytical formulas for the size and credibility that are valid for a uniform prior distribution over parameters, sufficiently large data samples and general constrained convex parameter estimation settings. These formulas provide a means to an efficient asymptotic error certification for parameters of arbitrary dimensions. Next, we demonstrate the accuracies of these analytical formulas as compared to numerically computed region quantities with simulated examples in qubit and qutrit quantum-state tomography where computations of the latter are feasible.

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Section: Plausible Regionsmentioning

confidence: 99%

Bayesian error regions in quantum estimation I: analytical reasonings

Teo

Jeong

2018

New J. Phys.

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“…Some of the parameters appear to be effectively nonidentifiable, as indicated by the lack of updating when comparing the prior to posterior distributions (see for a systematic review and discussion of measuring statistical evidence in a Bayesian setting; Evans, 2015). This lack of identifiability can also be quantified using, for example, the Kullback-Leibler divergence; however, we prefer to present comparisons graphically, following the general Bayesian data analysis philosophy of Gelman et al (2013).…”

Section: Posterior Parameter Distributionsmentioning

confidence: 99%

Incorporating Posterior‐Informed Approximation Errors Into a Hierarchical Framework to Facilitate Out‐of‐the‐Box MCMC Sampling for Geothermal Inverse Problems and Uncertainty Quantification

Maclaren

Nicholson

Bjarkason

et al. 2020

Water Resources Research

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We consider geothermal inverse problems and uncertainty quantification from a Bayesian perspective. Our main goal is to make standard, "out-of-the-box" Markov chain Monte Carlo (MCMC) sampling more feasible for complex simulation models by using suitable approximations. To do this, we first show how to pose both the inverse and prediction problems in a hierarchical Bayesian framework. We then show how to incorporate so-called posterior-informed model approximation error into this hierarchical framework, using a modified form of the Bayesian approximation error approach. This enables the use of a "coarse," approximate model in place of a finer, more expensive model, while accounting for the additional uncertainty and potential bias that this can introduce. Our method requires only simple probability modeling, a relatively small number of fine model simulations and only modifies the target posterior-any standard MCMC sampling algorithm can be used to sample the new posterior. These corrections can also be used in methods that are not based on MCMC sampling. We show that our approach can achieve significant computational speedups on two geothermal test problems. We also demonstrate the dangers of naively using coarse, approximate models in place of finer models, without accounting for the induced approximation errors. The naive approach tends to give overly confident and biased posteriors while incorporating Bayesian approximation error into our hierarchical framework corrects for this while maintaining computational efficiency and ease of use. Key Points: • We consider geothermal inverse problems and uncertainty quantification from a Bayesian perspective • We present a simple method for incorporating posterior-informed approximation errors into a hierarchical Bayesian framework • Our method makes standard out-of-the-box MCMC sampling feasible for more complex models while correcting for bias and overconfidence Supporting Information: • Supporting Information S1 , M. J. (2020). Incorporating posterior-informed approximation errors into a hierarchical framework to facilitate out-of-the-box MCMC sampling for geothermal inverse problems and uncertainty quantification. Water Resources Research, 56, e2018WR024240. https://

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“…This proved a surprisingly controversial topic and we encountered continuing debate 411 about fundamental principles and definitions of statistical evidence [35,[66][67][68][69].…”

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confidence: 99%

“…Following our conditional modelling approach, we decided to adopt the simple -yet approach is not without criticism, however (again, see [35,[66][67][68][69] for an entry point to 421 the ongoing debates).…”

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confidence: 99%

“…quite generally applicable -principle of evidence based on conditional probability: if414 we observe and ( | ) > ( ) then we have evidence for . A 'gold-standard' theory 415 of statistical evidence starting from this premise has been developed and defended 416 recently by Evans in a series of papers (summarised in[35]). Besides simplicity, a nice 417 feature of this approach, that we used below, is that it can be applied both to prior418 and posterior predictive distribution comparisons such as (y|y 0 ) ?…”

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confidence: 99%

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A hierarchical Bayesian framework for understanding the spatiotemporal dynamics of the intestinal epithelium

Maclaren

Parker

et al. 2016

Preprint

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Our work addresses two key challenges, one biological and one methodological. First, we aim to understand how proliferation and cellular migration rates in the intestinal epithelium are related under healthy, damaged (Ara-C treated) and recovering conditions, and how these relations can be used to identify mechanisms of repair and regeneration. We analyse new data, presented in more detail in a companion paper, in which BrdU/IdU cell-labelling experiments were performed under these respective conditions. Second, in considering how to more rigorously process these data and 1/42 interpret them using mathematical models, we develop a probabilistic, hierarchical framework. This framework provides a best-practice approach for systematically modelling and understanding the uncertainties that can otherwise undermine drawing reliable conclusions -uncertainties in experimental measurement and treatment, difficult-to-compare mathematical models of underlying mechanisms, and unknown or unobserved parameters. Both discrete and continuous mechanistic models are considered and related via hierarchical conditional probability assumptions. This allows the incorporation of features of both continuum tissue models and discrete cellular models. We perform model checks on both in-sample and out-of-sample datasets and use these checks to illustrate how to test possible model improvements and assess the robustness of our conclusions. This allows us to consider -and ultimately decide against -the need to retain finite-cell-size effects to explain a small misfit appearing in one set of long-time, out-of-sample predictions. Our approach leads us to conclude, for the present set of experiments, that a primarily proliferation-driven model is adequate for predictions over most time-scales. We describe each stage of our framework in detail, and hope that the present work may also serve as a guide for other applications of the hierarchical approach to problems in computational and systems biology more generally. Author SummaryThe intestinal epithelium serves as an important model system for studying the dynamics and regulation of multicellular populations. It is characterised by rapid rates of self-renewal and repair; failure of the regulation of these processes is thought to explain, in part, why many tumours occur in the intestinal and similar epithelial tissues. These features have led to a large amount of work on estimating rate parameters in the intestine. There still remain, however, large gaps between the raw data collected, the experimental interpretation of these data, and speculative mechanistic models for underlying processes. In our view hierarchical statistical modelling provides an ideal, but currently underutilised, method to begin to bridge these gaps. This approach makes essential use of the distinction between 2/42 'measurement', 'process' and 'parameter' models, giving an explicit framework for combining experimental data and mechanistic modelling in the presence of multiple sources of uncertainty. As we illustrate, the hie...

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Measuring statistical evidence using relative belief

Cited by 22 publications

References 23 publications

Bayesian error regions in quantum estimation I: analytical reasonings

Bayesian error regions in quantum estimation I: analytical reasonings

Incorporating Posterior‐Informed Approximation Errors Into a Hierarchical Framework to Facilitate Out‐of‐the‐Box MCMC Sampling for Geothermal Inverse Problems and Uncertainty Quantification

A hierarchical Bayesian framework for understanding the spatiotemporal dynamics of the intestinal epithelium

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