Credibility Regions Research Articles

LYαNNA: A deep learning field-level inference machine for the Lyman-α forest

The inference of astrophysical and cosmological properties from the Lyman-α forest conventionally relies on summary statistics of the transmission field that carry useful but limited information. We present a deep learning framework for inference from the Lyman-α forest at the field level. This framework consists of a 1D residual convolutional neural network (ResNet) that extracts spectral features and performs regression on thermal parameters of the intergalactic medium that characterize the power-law temperature-density relation. We trained this supervised machinery using a large set of mock absorption spectra from NYX hydrodynamic simulations at z = 2.2 with a range of thermal parameter combinations (labels). We employed Bayesian optimization to find an optimal set of hyperparameters for our network, and then employed a committee of 20 neural networks for increased statistical robustness of the network inference. In addition to the parameter point predictions, our machine also provides a self-consistent estimate of their covariance matrix with which we constructed a pipeline for inferring the posterior distribution of the parameters. We compared the results of our framework with the traditional summary based approach, namely the power spectrum and the probability density function (PDF) of transmission, in terms of the area of the 68% credibility regions as our figure of merit (FoM). In our study of the information content of perfect (noise- and systematics-free) Lyα forest spectral datasets, we find a significant tightening of the posterior constraints – factors of 10.92 and 3.30 in FoM over the power spectrum only and jointly with PDF, respectively – which is the consequence of recovering the relevant parts of information that are not carried by the classical summary statistics.

Fast and robust parameter estimation with uncertainty quantification for the cardiac function

Background and objectivesParameter estimation and uncertainty quantification are crucial in computational cardiology, as they enable the construction of digital twins that faithfully replicate the behavior of physical patients. Many model parameters regarding cardiac electromechanics and cardiovascular hemodynamics need to be robustly fitted by starting from a few, possibly non-invasive, noisy observations. Moreover, short execution times and a small amount of computational resources are required for the effective clinical translation. MethodsIn the framework of Bayesian statistics, we combine Maximum a Posteriori estimation and Hamiltonian Monte Carlo to find an approximation of model parameters and their posterior distributions. Fast simulations and minimal memory requirements are achieved by using an accurate and geometry-specific Artificial Neural Network surrogate model for the cardiac function, matrix–free methods, automatic differentiation and automatic vectorization. Furthermore, we account for the surrogate modeling error and measurement error. ResultsWe perform three different in silico test cases, ranging from the ventricular function to the entire cardiocirculatory system, involving whole-heart mechanics, arterial and venous hemodynamics. By employing a single central processing unit on a standard laptop, we attain highly accurate estimations for all model parameters in short computational times. Furthermore, we obtain posterior distributions that contain the true values inside the 90% credibility regions. ConclusionsMany model parameters regarding the entire cardiovascular system can be fastly and robustly identified with minimal hardware requirements. This can be achieved when a small amount of non-invasive data is available and when high levels of signal-to-noise ratio are present in the quantities of interest. With these features, our approach meets the requirements for clinical exploitation, while being compliant with Green Computing practices.

Stability and adaptability of elite upland rice lines using Bayesian-AMMI model

Rice is one of the world’s most important crops. The search for genotypes that are more productive and have wide adaptation to different environments is paramount. One of the major breeder’s obstacles faced is identification of superior strains is the presence of Genotype × Environment Interaction (GEI), which motivated the development of countless statistical procedures aiming to offer more efficient studies. In this work we analysed adaptability and stability of 13 upland rice lineages as part of a genetic improvement program in nine different environments, resulting from local combination and years of agriculture. The experiment was conducted in a completely randomized block design, with three replicates. The main variable is the grain storage in kg/ha. The model applied is the Bayesian Main Additive Effects and Multiplicative Interaction (Bayesian-AMMI). Our implementation implies an extra assumption of random effects from genotypes coming from a single population as opposed to previous works in the literature. Credibility regions with maximum posteriori density allowed identification of cultivars with higher average yield. Stable genotypes showed an initial evidence of adaptation to an environment in this rice breeding program. Bayesian-AMMI is flexible, and starts to be more widely used, but our suggestion is promising in making it a more powerful tool

Bayes estimates of multimodal density features using DNA and Economic Data

In several scientific fields, like bioinformatics, financial and macro-economics, important theoretical and practical issues exist that involve multimodal data distributions. We propose a Bayesian approach using mixtures distributions to approximate accurately such data distributions. Shape and other features of the mixture approximations are estimated including their uncertainty. For discrete data, we introduce a novel mixture of shifted Poisson distributions with an unknown number of components, which overcomes the equidispersion restriction in the standard Poisson which accomodates a wide range of shapes such as multimodality and long tails. Our simulation-based Bayesian inference treats the density features as random variables and highest credibility regions around features are easily obtained. For discrete data we develop an adapted version of the Reversible Jump Markov Chain Monte Carlo (RJMCMC) method, which allows for an unknown number of components instead of the more restrictive approach of choosing a particular number of mixture components using information criteria. Using simulated data, we show that our approach works successfully for three issues that one encounters during the estimation of mixtures: label switching; mixture complexity and prior information and mode membership versus component membership. The proposed method is applied to three empirical data sets: The count data method yields a novel perspective of the data on DNA tandem repeats in \cite{DNA_leiden}; the bimodal distribution of payment details of clients obtaining a loan from a financial institution in Spain in 1990 gives insight into the repayment ability of individual clients; and the distribution of the modes of real GDP growth data from the PennWorld Tables and their evolution over time explores possible world-wide economic convergence as well as group convergence between the US and European countries. The results of our descriptive analysis may be used as input for forecasting and policy analysis.

Implications for new physics in brightarrow s mu mu transitions after recent measurements by Belle and LHCb

We present a Bayesian analysis of the implications for new physics in semileptonic brightarrow s transitions after including new measurements of R_K at LHCb and new determinations of R_{K^*} and R_{K^{*+}} at Belle. We perform global fits with 1, 2, 4, and 8 input Wilson coefficients, plus one CKM nuisance parameter to take into account uncertainties that are not factorizable. We infer the 68% and 95.4% credibility regions of the marginalized posterior probability density for all scenarios and perform comparisons of models in pairs by calculating the Bayes factor given a common data set. We then proceed to analyzing a few well-known BSM models that can provide a high energy framework for the EFT analysis. These include the exchange of a heavy Z^{'} boson in models with heavy vector-like fermions and a scalar field, and a model with scalar leptoquarks. We provide predictions for the BSM couplings and expected mass values.

Adaptability of cotton (Gossypium hirsutum) genotypes analysed using a Bayesian AMMI model

Cotton (Gossypium spp.) provides ~90% of the world’s textile fibre. The aim of this study was to use the principal additive effects and multiplicative interaction (AMMI) model under the Bayesian approach to recommend cotton genotypes for the Central-West region of Brazil. Eight trials with upland cotton genotypes were conducted during the 2008–09 harvest in the State of Mato Grosso, Brazil. The experiment included a randomised block design with 16 genotypes. The genotypes were evaluated for fibre yield, length and strength. Chains were simulated via the Markov chain Monte Carlo method with 300000 iterations for the parameters of the Bayesian AMMI model. From the chains generated, the first 20000 burn-in observations were discarded and samples were taken by jumping every 20 observations (thin). Bayesian analysis provided additional results to those obtained by the frequentist approach, highlighting the credibility regions in the biplot for the genotypic and environmental scores. Bayesian AMMI model allowed identification of a genotype that can be widely recommended; this genotype has genotypic values above the overall mean for the three evaluated traits and did not contribute to the genotype × environment interactions observed in these traits. In addition, adaptability of genotypes to specific environments was observed, which makes it possible to capitalise the positive effect of the genotype × environment interaction.

Bayesian estimation of stable CARMA spot models for electricity prices

We develop a Bayesian estimation procedure for the electricity spot price model in Benth et al. (2014). This model incorporates a trend and seasonality component, a stable CARMA process for the price spikes, and an additional Lévy process for mid-range price level changes. Our MCMC algorithm has two advantages over the existing stepwise estimation procedure presented in Benth et al. (2014): First, since our algorithm produces samples from the full posterior distribution over all parameters, we can estimate the parameters much more accurately, which is shown in simulation studies. Second, we can provide accuracy measures as credibility intervals in addition to the point estimates. The approach is quite general, so that it can be adapted also to other similar pricing models. For illustration, we analyse spot and future prices from the EEX using the new Bayesian method and provide estimates for the risk premium together with credibility regions.

Performance of Cowpea Genotypes in the Brazilian Midwest Using the Bayesian Additive Main Effects and Multiplicative Interaction Model

Core Ideas We used the AMMI Bayesian model for selection of cowpea genotypes.We identified two genotypes with more expressive genotypic values.It is not possible to carry out the general recommendation of cowpea genotypes for the Brazilian Midwest. Cowpea is one of the world's most important tropical and subtropical food sources. In Brazil, it is grown principally in three regions: the Midwest, the Northeast, and the North. This makes the genotype × environment (G×E) interaction a challenge on the recommendation of the best genotypes for these regions. The aim of this study was to use the additive main effects and multiplicative interaction model under the Bayesian approach to recommend the best genotypes for the Brazilian Midwest. Six trials using cowpea genotypes were conducted in the municipalities of Aquidauana, Chapadão do Sul, Dourados, and Primavera. The experimental design included a randomized block design with 20 treatments (17 lines and three cowpea cultivars) and four replications. We identified only two genotypes (G15 and G2) with more expressive genotypic values and whose credibility regions do not include zero. From the credibility regions built for the genotypic and environmental scores, we verify that these genotypes contributed to the G×E interaction, presenting specific adaptability to the subgroup of environments (E5, E6). These results indicate that it is not possible to carry out the general recommendation of high‐yielding cowpea genotypes for the Brazilian Midwest region.

Early cosmology constrained

We investigate our knowledge of early universe cosmology by exploring how much additional energy density can be placed in different components beyond those in the ΛCDM model. To do this we use a method to separate early- and late-universe information enclosed in observational data, thus markedly reducing the model-dependency of the conclusions. We find that the 95% credibility regions for extra energy components of the early universe at recombination are: non-accelerating additional fluid density parameter ΩMR < 0.006 and extra radiation parameterised as extra effective neutrino species 2.3 < Neff < 3.2 when imposing flatness. Our constraints thus show that even when analyzing the data in this largely model-independent way, the possibility of hiding extra energy components beyond ΛCDM in the early universe is seriously constrained by current observations. We also find that the standard ruler, the sound horizon at radiation drag, can be well determined in a way that does not depend on late-time Universe assumptions, but depends strongly on early-time physics and in particular on additional components that behave like radiation. We find that the standard ruler length determined in this way is rs = 147.4 ± 0.7 Mpc if the radiation and neutrino components are standard, but the uncertainty increases by an order of magnitude when non-standard dark radiation components are allowed, to rs = 150 ± 5 Mpc.

Maximum-a-Posteriori Estimation with Bayesian Confidence Regions

Solutions to inverse problems that are ill-conditioned or ill-posed may have significant intrinsic uncertainty. Unfortunately, analyzing and quantifying this uncertainty is very challenging, particularly in high-dimensional problems. As a result, while most modern mathematical imaging methods produce impressive point estimation results, they are generally unable to quantify the uncertainty in the solutions delivered. This paper presents a new general methodology for approximating Bayesian high-posterior-density credibility regions in inverse problems that are convex and potentially very high-dimensional. The approximations are derived by using recent concentration of measure results related to information theory for log-concave random vectors. A remarkable property of the approximations is that they can be computed very efficiently, even in large-scale problems, by using standard convex optimization techniques. In particular, they are available as a by-product in problems solved by maximum-a-posteriori es...

Do dark matter halos explain lensing peaks?

We have investigated a recently proposed halo-based model, Camelus, for predicting weak-lensing peak counts, and compared its results over a collection of 162 cosmologies with those from N-body simulations. While counts from both models agree for peaks with $\mathcal{S/N}>1$ (where $\mathcal{S/N}$ is the ratio of the peak height to the r.m.s. shape noise), we find $\approx 50\%$ fewer counts for peaks near $\mathcal{S/N}=0$ and significantly higher counts in the negative $\mathcal{S/N}$ tail. Adding shape noise reduces the differences to within $20\%$ for all cosmologies. We also found larger covariances that are more sensitive to cosmological parameters. As a result, credibility regions in the $\{\Omega_m, \sigma_8\}$ are $\approx 30\%$ larger. Even though the credible contours are commensurate, each model draws its predictive power from different types of peaks. Low peaks, especially those with $2<\mathcal{S/N}<3$, convey important cosmological information in N-body data, as shown in \cite{DietrichHartlap, Kratochvil2010}, but \textsc{Camelus} constrains cosmology almost exclusively from high significance peaks $(\mathcal{S/N}>3)$. Our results confirm the importance of using a cosmology-dependent covariance with at least a 14\% improvement in parameter constraints. We identified the covariance estimation as the main driver behind differences in inference, and suggest possible ways to make Camelus even more useful as a highly accurate peak count emulator.

A Bayesian framework for infrasound location

SUMMARY We develop a framework for location of infrasound events using backazimuth and infrasonic arrival times from multiple arrays. Bayesian infrasonic source location (BISL) developed here estimates event location and associated credibility regions. BISL accounts for unknown source-to-array path or phase by formulating infrasonic group velocity as random. Differences between observed and predicted source-to-array traveltimes are partitioned into two additive Gaussian sources, measurement error and model error, the second of which accounts for the unknown influence of wind and temperature on path. By applying the technique to both synthetic tests and ground-truth events, we highlight the complementary nature of back azimuths and arrival times for estimating well-constrained event locations. BISL is an extension to methods developed earlier by Arrowsmith et al. that provided simple bounds on location using a grid-search technique.

Bayesian methods for the analysis of inequality constrained contingency tables

A Bayesian methodology for the analysis of inequality constrained models for contingency tables is presented. The problem of interest lies in obtaining the estimates of functions of cell probabilities subject to inequality constraints, testing hypotheses and selection of the best model. Constraints on conditional cell probabilities and on local, global, continuation and cumulative odds ratios are discussed. A Gibbs sampler to obtain a discrete representation of the posterior distribution of the inequality constrained parameters is used. Using this discrete representation, the credibility regions of functions of cell probabilities can be constructed. Posterior model probabilities are used for model selection and hypotheses are tested using posterior predictive checks. The Bayesian methodology proposed is illustrated in two examples.

Highest posterior density estimation from multiply censored Pareto data

In statistical practice, it is quite common that some data are unknown or disregarded for various reasons. In the present paper, on the basis of a multiply censored sample from a Pareto population, the problem of finding the highest posterior density (HPD) estimates of the inequality and precision parameters is discussed assuming a natural joint conjugate prior. HPD estimates are obtained in closed forms for complete or right censored data. In the general multiple censoring case, it is shown the existence and uniqueness of the estimates. Explicit lower and upper bounds are also provided. Due to the posterior unimodality, HPD credibility regions are simply connected sets. For illustration, two numerical examples are included.

Bayesian estimation based on trimmed samples from Pareto populations

Trimmed samples are widely employed in several areas of statistical practice, especially when some sample values at either or both extremes might have been contaminated. The problem of estimating the inequality and precision parameters of a Pareto distribution based on a trimmed sample and prior information is considered. From an inferential viewpoint, the problem of finding the highest posterior density (HPD) estimates of the Pareto parameters is discussed. The existence and uniqueness of the HPD estimates are established under mild conditions; explicit and accurate lower and upper bounds are also provided. Adopting a decision-theoretic perspective, several Bayesian estimators for standard loss functions are presented. In addition, two-sided and HPD credibility intervals for each Pareto parameter and joint HPD credibility regions for both parameters are derived, which have the corresponding frequentist confidence level in the noninformative case. Finally, an illustrative example concerning annual wage data is included.

BAYESIAN ANALYSIS OF ECONOMETRIC TIME SERIES MODELS USING HYBRID INTEGRATION RULES

ABSTRACT This paper is concerned with the study of Bayesian inference procedures to commonly used time series models. In particular, the dynamic or state-space models, the time-varying vector autoregressive model and the structural vector autoregressive model are considered in detail. Inference procedures are based on a hybrid integration scheme where state parameters are analytically integrated and hyperparameters are integrated by Markov chain Monte Carlo methods. Credibility regions for forecasts and impulse responses are then derived. The procedures are illustrated in real data sets.

Threshold model for misclassified binary responses with applications to animal breeding.

A Bayesian procedure for misclassified binary data was developed. An animal breeding simulation indicated that, when error of classification was ignored, the variance between clusters was inferred incorrectly. Data were reanalyzed assuming that the probability of misclassification was either known or unknown. In the first case, input parameter values were recovered in the analysis. When the probability was unknown, there was a slight bias; the true probability of misclassification and the true number of miscoded observations appeared within high credibility regions. An analysis of fertility in dairy cows is presented.

Relationships Among Sample Size, Model Selection and Likelihood Regions, and Scientifically Important Differences

In multidimensional models, simultaneous confidence or credibility regions for continuous parameters hold the overall frequentist long run or Bayesian posterior probability constant at some level such as 95%. This means that, as the dimensionality of the problem increases, the precision or information required about each individual parameter also rapidly increases so parameters have progressively less chance of being set to 0 in such a model selection procedure. These methods do not appropriately answer most of the inference questions that are generally encountered in applied statistics. Thus, sample size, model selection criteria and the estimation of parameter precision are intimately related. In contrast with frequentist and Bayesian procedures, direct likelihood inference, calibrating acceptable likelihood regions by criteria derived from model selection, such as the Akaike information criterion, holds the precision requirements per parameter constant as the dimensionality grows, thus allowing the series of inferences to remain compatible. Other model selection criteria, such as the Bayes information criterion, that depend on the sample size, maintain compatibility but decrease the precision per parameter as the sample increases, so, in the limit, the null model tends to be chosen (Lindley's paradox).

Confidence regions for multidimensional scaling analysis

Techniques are developed for surrounding each of the points in a multidimensional scaling solution with a region which will contain the population point with some level of confidence. Bayesian credibility regions are also discussed. A general theorem is proven which describes the asymptotic distribution of maximum likelihood estimates subject to identifiability constraints. This theorem is applied to a number of models to display asymptotic variance-covariance matrices for coordinate estimates under different rotational constraints. A technique is described for displaying Bayesian conditional credibility regions for any sample size.