14,956 research outputs found
Thermodynamic assessment of probability distribution divergencies and Bayesian model comparison
Within path sampling framework, we show that probability distribution
divergences, such as the Chernoff information, can be estimated via
thermodynamic integration. The Boltzmann-Gibbs distribution pertaining to
different Hamiltonians is implemented to derive tempered transitions along the
path, linking the distributions of interest at the endpoints. Under this
perspective, a geometric approach is feasible, which prompts intuition and
facilitates tuning the error sources. Additionally, there are direct
applications in Bayesian model evaluation. Existing marginal likelihood and
Bayes factor estimators are reviewed here along with their stepping-stone
sampling analogues. New estimators are presented and the use of compound paths
is introduced
Bayesian analysis of hierarchical multi-fidelity codes
This paper deals with the Gaussian process based approximation of a code
which can be run at different levels of accuracy. This method, which is a
particular case of co-kriging, allows us to improve a surrogate model of a
complex computer code using fast approximations of it. In particular, we focus
on the case of a large number of code levels on the one hand and on a Bayesian
approach when we have two levels on the other hand. The main results of this
paper are a new approach to estimate the model parameters which provides a
closed form expression for an important parameter of the model (the scale
factor), a reduction of the numerical complexity by simplifying the covariance
matrix inversion, and a new Bayesian modelling that gives an explicit
representation of the joint distribution of the parameters and that is not
computationally expensive. A thermodynamic example is used to illustrate the
comparison between 2-level and 3-level co-kriging
An analytical approach to bayesian evidence computation
Bayesian evidence is a key tool in model selection, allowing a comparison of models with different numbers of parameters. Its use in the analysis of cosmological models has been limited by difficulties in calculating it, with current numerical algorithms requiring supercomputers. In this paper we give exact formulae for the Bayesian evidence in the case of Gaussian likelihoods with arbitrary correlations and top-hat priors, and approximate formulae for the case of likelihood distributions with leading non-Gaussianities (skewness and kurtosis). We apply these formulae to cosmological models with and without isocurvature components, and compare with results we previously obtained using numerical thermodynamic integration. We find that the results are of lower precision than the thermodynamic integration, while still being good enough to be usefu
Model selection for spectro-polarimetric inversions
Inferring magnetic and thermodynamic information from spectropolarimetric
observations relies on the assumption of a parameterized model atmosphere whose
parameters are tuned by comparison with observations. Often, the choice of the
underlying atmospheric model is based on subjective reasons. In other cases,
complex models are chosen based on objective reasons (for instance, the
necessity to explain asymmetries in the Stokes profiles) but it is not clear
what degree of complexity is needed. The lack of an objective way of comparing
models has, sometimes, led to opposing views of the solar magnetism because the
inferred physical scenarios are essentially different. We present the first
quantitative model comparison based on the computation of the Bayesian evidence
ratios for spectropolarimetric observations. Our results show that there is not
a single model appropriate for all profiles simultaneously. Data with moderate
signal-to-noise ratios favor models without gradients along the line-of-sight.
If the observations shows clear circular and linear polarization signals above
the noise level, models with gradients along the line are preferred. As a
general rule, observations with large signal-to-noise ratios favor more complex
models. We demonstrate that the evidence ratios correlate well with simple
proxies. Therefore, we propose to calculate these proxies when carrying out
standard least-squares inversions to allow for model comparison in the future.Comment: 16 pages, 2 figures, 8 tables, accepted for publication in Ap
Targeting Bayes factors with direct-path non-equilibrium thermodynamic integration
Thermodynamic integration (TI) for computing marginal likelihoods is based on an inverse annealing path from the prior to the posterior distribution. In many cases, the resulting estimator suffers from high variability, which particularly stems from the prior regime. When comparing complex models with differences in a comparatively small number of parameters, intrinsic errors from sampling fluctuations may outweigh the differences in the log marginal likelihood estimates. In the present article, we propose a thermodynamic integration scheme that directly targets the log Bayes factor. The method is based on a modified annealing path between the posterior distributions of the two models compared, which systematically avoids the high variance prior regime. We combine this scheme with the concept of non-equilibrium TI to minimise discretisation errors from numerical integration. Results obtained on Bayesian regression models applied to standard benchmark data, and a complex hierarchical model applied to biopathway inference, demonstrate a significant reduction in estimator variance over state-of-the-art TI methods
Prior-predictive value from fast growth simulations
Building on a variant of the Jarzynski equation we propose a new method to
numerically determine the prior-predictive value in a Bayesian inference
problem. The method generalizes thermodynamic integration and is not hampered
by equilibration problems. We demonstrate its operation by applying it to two
simple examples and elucidate its performance. In the case of multi-modal
posterior distributions the performance is superior to thermodynamic
integration.Comment: 8 pages, 11 figure
Bayesian model selection for exponential random graph models via adjusted pseudolikelihoods
Models with intractable likelihood functions arise in areas including network
analysis and spatial statistics, especially those involving Gibbs random
fields. Posterior parameter es timation in these settings is termed a
doubly-intractable problem because both the likelihood function and the
posterior distribution are intractable. The comparison of Bayesian models is
often based on the statistical evidence, the integral of the un-normalised
posterior distribution over the model parameters which is rarely available in
closed form. For doubly-intractable models, estimating the evidence adds
another layer of difficulty. Consequently, the selection of the model that best
describes an observed network among a collection of exponential random graph
models for network analysis is a daunting task. Pseudolikelihoods offer a
tractable approximation to the likelihood but should be treated with caution
because they can lead to an unreasonable inference. This paper specifies a
method to adjust pseudolikelihoods in order to obtain a reasonable, yet
tractable, approximation to the likelihood. This allows implementation of
widely used computational methods for evidence estimation and pursuit of
Bayesian model selection of exponential random graph models for the analysis of
social networks. Empirical comparisons to existing methods show that our
procedure yields similar evidence estimates, but at a lower computational cost.Comment: Supplementary material attached. To view attachments, please download
and extract the gzzipped source file listed under "Other formats
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