Bayesian Posterior Distribution Research Articles

Sampling with prior knowledge for high-dimensional gravitational wave data analysis

Extracting knowledge from high-dimensional data has been notoriously difficult, primarily due to the so-called "curse of dimensionality" and the complex joint distributions of these dimensions. This is a particularly profound issue for high-dimensional gravitational wave data analysis where one requires to conduct Bayesian inference and estimate joint posterior distributions. In this study, we incorporate prior physical knowledge by sampling from desired interim distributions to develop the training dataset. Accordingly, the more relevant regions of the high-dimensional feature space are covered by additional data points, such that the model can learn the subtle but important details. We adapt the normalizing flow method to be more expressive and trainable, such that the information can be effectively extracted and represented by the transformation between the prior and target distributions. Once trained, our model only takes approximately 1 s on one V100 GPU to generate thousands of samples for probabilistic inference purposes. The evaluation of our approach confirms the efficacy and efficiency of gravitational wave data inferences and points to a promising direction for similar research. The source code, specifications, and detailed procedures are publicly accessible on GitHub.

BIDIMENSIONAL LINKED MATRIX FACTORIZATION FOR PAN-OMICS PAN-CANCER ANALYSIS.

Several modern applications require the integration of multiple large data matrices that have shared rows and/or columns. For example, cancer studies that integrate multiple omics platforms across multiple types of cancer, pan-omics pan-cancer analysis, have extended our knowledge of molecular heterogeneity beyond what was observed in single tumor and single platform studies. However, these studies have been limited by available statistical methodology. We propose a flexible approach to the simultaneous factorization and decomposition of variation across such bidimensionally linked matrices, BIDIFAC+. BIDIFAC+ decomposes variation into a series of low-rank components that may be shared across any number of row sets (e.g., omics platforms) or column sets (e.g., cancer types). This builds on a growing literature for the factorization and decomposition of linked matrices which has primarily focused on multiple matrices that are linked in one dimension (rows or columns) only. Our objective function extends nuclear norm penalization, is motivated by random matrix theory, gives a unique decomposition under relatively mild conditions, and can be shown to give the mode of a Bayesian posterior distribution. We apply BIDIFAC+ to pan-omics pan-cancer data from TCGA, identifying shared and specific modes of variability across four different omics platforms and 29 different cancer types.

Dimension free convergence rates for Gibbs samplers for Bayesian linear mixed models

The emergence of big data has led to a growing interest in so-called convergence complexity analysis, which is the study of how the convergence rate of a Monte Carlo Markov chain (for an intractable Bayesian posterior distribution) scales as the underlying data set grows in size. Convergence complexity analysis of practical Monte Carlo Markov chains on continuous state spaces is quite challenging, and there have been very few successful analyses of such chains. One fruitful analysis was recently presented by Qin and Hobert (2022), who studied a Gibbs sampler for a simple Bayesian random effects model. These authors showed that, under regularity conditions, the geometric convergence rate of this Gibbs sampler converges to zero as the data set grows in size. It is shown herein that similar behavior is exhibited by Gibbs samplers for more general Bayesian models that possess both random effects and traditional continuous covariates, the so-called mixed models. The analysis employs the Wasserstein-based techniques introduced by Qin and Hobert (2022).

Sex-specific lesion pattern of functional outcomes after stroke.

Stroke represents a considerable burden of disease for both men and women. However, a growing body of literature suggests clinically relevant sex differences in the underlying causes, presentations and outcomes of acute ischaemic stroke. In a recent study, we reported sex divergences in lesion topographies: specific to women, acute stroke severity was linked to lesions in the left-hemispheric posterior circulation. We here determined whether these sex-specific brain manifestations also affect long-term outcomes. We relied on 822 acute ischaemic patients [age: 64.7 (15.0) years, 39% women] originating from the multi-centre MRI-GENIE study to model unfavourable outcomes (modified Rankin Scale >2) based on acute neuroimaging data in a Bayesian hierarchical framework. Lesions encompassing bilateral subcortical nuclei and left-lateralized regions in proximity to the insula explained outcomes across men and women (area under the curve = 0.81). A pattern of left-hemispheric posterior circulation brain regions, combining left hippocampus, precuneus, fusiform and lingual gyrus, occipital pole and latero-occipital cortex, showed a substantially higher relevance in explaining functional outcomes in women compared to men [mean difference of Bayesian posterior distributions (men – women) = −0.295 (90% highest posterior density interval = −0.556 to −0.068)]. Once validated in prospective studies, our findings may motivate a sex-specific approach to clinical stroke management and hold the promise of enhancing outcomes on a population level.

Evaluation of a PEGylated Fibroblast Growth Factor 21 Variant Using Novel Preclinical Magnetic Resonance Imaging and Magnetic Resonance Elastography in a Mouse Model of Nonalcoholic Steatohepatitis.

BackgroundTreatments for nonalcoholic steatohepatitis (NASH) are urgently needed. Hepatic fat fraction and shear stiffness quantified by magnetic resonance imaging (MRI‐HFF) and magnetic resonance elastography (MRE‐SS), respectively, are biomarkers for hepatic steatosis and fibrosis.PurposeThis study assessed the longitudinal effects of fibroblast growth factor 21 variant (polyethylene glycol [PEG]‐FGF21v) on MRI‐HFF and MRE‐SS in a NASH mouse model.Study TypePreclinical.Animal ModelThis study included a choline‐deficient, amino acid‐defined, high‐fat diet (CDAHFD) model and 6‐week‐old, male C57BL/6J mice (N = 78).Field Strength/SequenceThis study was performed using: 3T: gradient‐echo two‐point Dixon and spin‐echo (SE) echo‐planar imaging elastography (200 Hz) and 7T: SE two‐point Dixon and SE elastography (200 Hz).AssessmentMRI and MRE were performed before control diet (CD) or CDAHFD (BD), before PEG‐FGF21v dosing (baseline), and after PEG‐FGF21v treatment (WK4/8). Regions of interest for MRI‐HFF and MRE‐SS were delineated by J.L. and H.T. (>5 years of experience each). Fibrosis and steatosis were measured histologically after picrosirius red and H&E staining. Alkaline phosphatase, alanine transaminase, bile acids, and triglycerides (TGs) were measured.Statistical TestsTwo‐tailed Dunnett's tests were used for statistical analysis; untreated CDAHFD or baseline was used for comparisons. Imaging and histology/biochemistry data were determined using Spearman correlations. Bayesian posterior distributions for MRE‐SS at WK8, posterior means, and 95% credible intervals were presented.ResultsCDAHFD significantly increased baseline MRI‐HFF (3T: 21.97% ± 0.29%; 7T: 40.12% ± 0.35%) and MRE‐SS (3T: 1.25 ± 0.02; 7T: 1.78 ± 0.06 kPa) vs. CD (3T: 3.45% ± 0.7%; 7T: 12.06% ± 1.4% and 3T: 1.01 ± 0.02; 7T: 0.89 ± 0.06 kPa). At 7T, PEG‐FGF21v significantly decreased MRI‐HFF (WK4: 28.97% ± 1.22%; WK8: 20.93% ± 1.15%) and MRE‐SS (WK4: 1.57 ± 0.04; WK8: 1.36 ± 0.05 kPa) vs. untreated (WK4: 36.36% ± 0.62%; WK8: 30.58% ± 0.81% and WK4: 2.03 ± 0.06; WK8: 2.01 ± 0.04 kPa); 3T trends were similar. WK8 SS posterior mean percent attenuation ratios (R DI) were −68% (−90%, −44%; 3T) and −64% (−78%, −52%; 7T). MRI‐HFF was significantly correlated with H&E (3T, r = 0.93; 7T, r = 0.94) and TGs (both, r = 0.92).Data ConclusionsMRI‐HFF and MRE‐SS showed PEG‐FGF21v effects on hepatic steatosis and fibrosis across 3 and 7T, consistent with histological and biochemical data.Level of Evidence1Technical Efficacy Stage2

Quantifying Uncertainties in Passive Microwave Remote Sensing of Soil Moisture via a Bayesian Probabilistic Inversion Method

Improving the accuracy of remotely sensed soil moisture (SM) is a challenging and popular topic. Quantifying the uncertainty in the SM inversion process and enhancing the confidence of SM retrieval are promising ways to address this challenge but have received little attention. We present a Bayesian probabilistic inversion algorithm that can simultaneously retrieve SM, surface roughness, and vegetation optical depth data and quantify the uncertainty in the inversion. The proposed algorithm is evaluated using airborne polarimetric L-band multibeam radiometer (PLMR) observations. We use three combinations, 3-angular observations at V-polarization (3CV), 3-angular observations at H-polarization (3CH) and 6-channel observations (6CA), to identify the optimal configuration for SM retrieval by taking advantage of the PLMR’s dual-polarization and multiple angles. Uncertainties are quantified by introducing multiple uncertainty quantification metrics into Bayesian posterior distributions of SM retrievals. The estimates are validated against multiscale ground-based measurements, including manual measurements and wireless sensor network (WSN) measurements, and the spatial representativeness of the ground-based reference regarding the validation of pixel-scale SM retrievals is discussed. The 6CA attempt yields the best SM estimates (correlation coefficient (R) ≥ 0.864, root mean square error (RMSE) ≤0.04 m3/m3, and unbiased RMSE (unRMSE) ≤ 0.035 m3/m3), while the 3CH attempt yields the lowest uncertainty. Additionally, dense manual measurements are more representative than sparsely distributed WSN measurements. Overall, combining dual-polarized observations yields the best SM estimates but introduces additional uncertainty. This study highlights uncertainties quantification in SM inversion and thus provides confidence in SM inversion, facilitating improved SM retrieval algorithms.

Efficient Bayesian estimation of permutation entropy with Dirichlet priors

Estimation of permutation entropy (PE) using Bayesian statistical methods is presented for systems where the ordinal pattern sampling follows an independent, multinomial distribution. The desired PE posterior distribution is demonstrated to closely approximate a standard Beta distribution whose hyperparameters can be computed directly from ordinal pattern counts. It is further shown that this Bayesian approach is a generalization of previously-published frequentist methods. Because Bayesian posterior distributions can be estimated for very short time series, this method enables PE analysis on data sets currently not compatible with existing methods due to their limited size. To demonstrate the power and flexibility of this technique, the PE of a semiconductor laser with optical feedback (SLWOF) is shown to be time-variant by visualizing how the PE posterior distribution varies over sequences of small (1000 point) partitions of the output time-series.

Bayesian Processing of Data on Bursts of Pressure Vessels

Two alternative Bayesian approaches are proposed for the prediction of fragmentation of pressure vessels triggered off by accidental explosions (bursts) of these containment structures. It is shown how to carry out this prediction with post-mortem data on fragment numbers counted after past explosion accidents. Results of the prediction are estimates of probabilities of individual fragment numbers. These estimates are expressed by means of Bayesian prior or posterior distributions. It is demonstrated how to elicit the prior distributions from relatively scarce post-mortem data on vessel fragmentations. Specifically, it is suggested to develop priors with two Bayesian models known as compound Poisson-gamma and multinomial-Dirichlet probability distributions. The available data is used to specify non-informative prior for Poisson parameter that is subsequently transformed into priors of individual fragment number probabilities. Alternatively, the data is applied to a specification of Dirichlet concentration parameters. The latter priors directly express epistemic uncertainty in the fragment number probabilities. Example calculations presented in the study demonstrate that the suggested non-informative prior distributions are responsive to updates with scarce data on vessel explosions. It is shown that priors specified with Poisson-gamma and multinomial-Dirichlet models differ tangibly; however, this difference decreases with increasing amount of new data. For the sake of brevity and concreteness, the study was limited to fire induced vessel bursts known as boiling liquid expanding vapour explosions (BLEVEs).

Real-Time Gravitational Wave Science with Neural Posterior Estimation.

We demonstrate unprecedented accuracy for rapid gravitational wave parameter estimation with deep learning. Using neural networks as surrogates for Bayesian posterior distributions, we analyze eight gravitational wave events from the first LIGO-Virgo Gravitational-Wave Transient Catalog and find very close quantitative agreement with standard inference codes, but with inference times reduced from O(day) to 20s per event. Our networks are trained using simulated data, including an estimate of the detector noise characteristics near the event. This encodes the signal and noise models within millions of neural-network parameters and enables inference for any observed data consistent with the training distribution, accounting for noise nonstationarity from event to event. Our algorithm-called "DINGO"-sets a new standard in fast and accurate inference of physical parameters of detected gravitational wave events, which should enable real-time data analysis without sacrificing accuracy.

Posterior-based Wald-type statistics for hypothesis testing

A new Wald-type statistic is proposed for hypothesis testing based on Bayesian posterior distributions under the correct model specification. The new statistic can be explained as a posterior version of the Wald statistic and has several nice properties. First, it is well-defined under improper prior distributions. Second, it avoids Jeffreys–Lindley–Bartlett’s paradox. Third, under the null hypothesis and repeated sampling, it follows a χ2 distribution asymptotically, offering an asymptotically pivotal test. Fourth, it only requires inverting the posterior covariance for parameters of interest. Fifth and perhaps most importantly, when a random sample from the posterior distribution (such as MCMC output) is available, the proposed statistic can be easily obtained as a by-product of posterior simulation. In addition, the numerical standard error of the estimated proposed statistic can be computed based on random samples. A robust version of the test statistic is developed under model misspecification and inherits many nice properties of the new posterior statistic. The finite sample performance of the statistics is examined in Monte Carlo studies. The method is applied to two latent variable models used in microeconometrics and financial econometrics.

Bayesian analysis of quasar light curves with a running optimal average: new time delay measurements of COSMOGRAIL gravitationally lensed quasars

ABSTRACT We present a new method of modelling time-series data based on the running optimal average (ROA). By identifying the effective number of parameters for the ROA model, in terms of the shape and width of its window function and the times and accuracies of the data, we enable a Bayesian analysis, optimizing the ROA width, along with other model parameters, by minimizing the Bayesian information criterion (BIC) and sampling joint posterior parameter distributions using MCMC methods. For analysis of quasar light curves, our implementation of ROA modelling can measure time delays among light curves at different wavelengths or from different images of a lensed quasar and, in future work, be used to inter-calibrate light-curve data from different telescopes and estimate the shape and thus the power-density spectrum of the light curve. Our noise model implements a robust treatment of outliers and error-bar adjustments to account for additional variance or poorly quantified uncertainties. Tests with simulated data validate the parameter uncertainty estimates. We compare ROA delay measurements with results from cross-correlation and from javelin, which models light curves with a prior on the power-density spectrum. We analyse published COSMOGRAIL light curves of multilensed quasar light curves and present the resulting measurements of the inter-image time delays and detection of microlensing effects.

The global prevalence of dry eye disease: A Bayesian view.

To provide estimates for the prevalence of dry eye disease globally and in sub-groups defined by: diagnostic criterion, sex, geographic location and age, using a Bayesian approach. Modelling prevalence as a Beta distribution, estimates were inferred from Bayesian posterior distributions obtained by combing an uninformed prior with likelihood functions generated from all relevant studies reporting dry eye prevalence between 1997 and 2021. Global prevalence of dry eye disease was estimated at 11.59% (standard deviation (SD)=0.04). For symptomatic disease, the estimate was 9.12% (SD=0.04), with women 9.5% (SD=0.05) and men 6.8% (SD=0.06); prevalence was lowest in North America, 4.6% (SD=0.03) and highest in Africa, 47.9% (SD=1.8). For signs, prevalence was 35.2% (SD=0.3), with woman 34.7% (SD=0.7) and men 37.6% (SD=0.7); North America showed the lowest regional prevalence, 3.5%, (SD=0.4) with Eastern Asia the highest, 42.8% (SD=0.4). Using TFOS DEWS II diagnostic criteria resulted in a global prevalence of 29.5% (SD=0.8), with women 28.1% (SD=1.2) and men 24.9% (SD=1.4). Prevalence was lowest during the fifth decade, increasing approximately linearly with age thereafter. Estimates for other categories are given in accompanying tables. A simple, flexible, yet powerful means of combining data from multiple sources to yield prevalence estimates across a range of circumstances is described, that is compatible with published guidelines for conducting meta-analysis. Estimates can be readily updated as new information emerges, or according to need. Understanding the specific characteristics of studies chosen for inclusion is critical to the validity of the outcome. Although dry eye disease is evidently common, affecting about one in 11 people world-wide, data are sparse for the young and all geographical locations except Eastern Asia.

1215Considerations for using multiple imputation in propensity score-weighted analysis

1215Considerations for using multiple imputation in propensity score-weighted analysis

Calibration and Uncertainty Quantification of Convective Parameters in an Idealized GCM

AbstractParameters in climate models are usually calibrated manually, exploiting only small subsets of the available data. This precludes both optimal calibration and quantification of uncertainties. Traditional Bayesian calibration methods that allow uncertainty quantification are too expensive for climate models; they are also not robust in the presence of internal climate variability. For example, Markov chain Monte Carlo (MCMC) methods typically require model runs and are sensitive to internal variability noise, rendering them infeasible for climate models. Here we demonstrate an approach to model calibration and uncertainty quantification that requires only model runs and can accommodate internal climate variability. The approach consists of three stages: (a) a calibration stage uses variants of ensemble Kalman inversion to calibrate a model by minimizing mismatches between model and data statistics; (b) an emulation stage emulates the parameter‐to‐data map with Gaussian processes (GP), using the model runs in the calibration stage for training; (c) a sampling stage approximates the Bayesian posterior distributions by sampling the GP emulator with MCMC. We demonstrate the feasibility and computational efficiency of this calibrate‐emulate‐sample (CES) approach in a perfect‐model setting. Using an idealized general circulation model, we estimate parameters in a simple convection scheme from synthetic data generated with the model. The CES approach generates probability distributions of the parameters that are good approximations of the Bayesian posteriors, at a fraction of the computational cost usually required to obtain them. Sampling from this approximate posterior allows the generation of climate predictions with quantified parametric uncertainties.

Quantifying malaria acquired during travel and its role in malaria elimination on Bioko Island

BackgroundMalaria elimination is the goal for Bioko Island, Equatorial Guinea. Intensive interventions implemented since 2004 have reduced prevalence, but progress has stalled in recent years. A challenge for elimination has been malaria infections in residents acquired during travel to mainland Equatorial Guinea. The present article quantifies how off-island contributes to remaining malaria prevalence on Bioko Island, and investigates the potential role of a pre-erythrocytic vaccine in making further progress towards elimination.MethodsMalaria transmission on Bioko Island was simulated using a model calibrated based on data from the Malaria Indicator Surveys (MIS) from 2015 to 2018, including detailed travel histories and malaria positivity by rapid-diagnostic tests (RDTs), as well as geospatial estimates of malaria prevalence. Mosquito population density was adjusted to fit local transmission, conditional on importation rates under current levels of control and within-island mobility. The simulations were then used to evaluate the impact of two pre-erythrocytic vaccine distribution strategies: mass treat and vaccinate, and prophylactic vaccination for off-island travellers. Lastly, a sensitivity analysis was performed through an ensemble of simulations fit to the Bayesian joint posterior probability distribution of the geospatial prevalence estimates.ResultsThe simulations suggest that in Malabo, an urban city containing 80% of the population, there are some pockets of residual transmission, but a large proportion of infections are acquired off-island by travellers to the mainland. Outside of Malabo, prevalence was mainly attributable to local transmission. The uncertainty in the local transmission vs. importation is lowest within Malabo and highest outside. Using a pre-erythrocytic vaccine to protect travellers would have larger benefits than using the vaccine to protect residents of Bioko Island from local transmission. In simulations, mass treatment and vaccination had short-lived benefits, as malaria prevalence returned to current levels as the vaccine’s efficacy waned. Prophylactic vaccination of travellers resulted in longer-lasting reductions in prevalence. These projections were robust to underlying uncertainty in prevalence estimates.ConclusionsThe modelled outcomes suggest that the volume of malaria cases imported from the mainland is a partial driver of continued endemic malaria on Bioko Island, and that continued elimination efforts on must account for human travel activity.

Gaussian Mixture Filter for Angles-Only Orbit Determination in Modified Equinoctial Elements

A Gaussian mixture orbit determination filter is developed using a state vector that consists of equinoctial-like osculating elements. This new filter seeks to reduce the number of Gaussian mixture elements that are needed in order to accurately model the Bayesian posterior distributions that apply to the angles-only orbit determination problem. The modified version of the equinoctial elements replaces the elements and , whose sum squared is bounded by 1. The two replacement elements are unbounded, and and can be determined from them. This modification allows the orbit determination filter to operate in an unbounded state space. The new state requires the development of a corresponding mixand covariance upper bound due to the particular type of Gaussian mixture filter that is implemented. The upper bound is used in a mixture resampling algorithm in a way which ensures that extended Kalman filter calculations will be sufficiently accurate for each mixand’s computations. The resulting filter is able to reduce the required number of mixands from 5000 to 500 for angles-only orbit determination of a geosynchronous spacecraft.

Interpretable biomanufacturing process risk and sensitivity analyses for quality‐by‐design and stability control

AbstractWhile biomanufacturing plays a significant role in supporting the economy and ensuring public health, it faces critical challenges, including complexity, high variability, lengthy lead time, and very limited process data, especially for personalized new cell and gene biotherapeutics. Driven by these challenges, we propose an interpretable semantic bioprocess probabilistic knowledge graph and develop a game theory based risk and sensitivity analyses for production process to facilitate quality‐by‐design and stability control. Specifically, by exploring the causal relationships and interactions of critical process parameters and product quality attributes, we create a Bayesian network based probabilistic knowledge graph characterizing the complex causal interdependencies of all factors. Then, we introduce a Shapley value based sensitivity analysis, which can correctly quantify the variation contribution from each input factor on the outputs (i.e., productivity, product quality). Since the bioprocess model coefficients are learned from limited process observations, we derive the Bayesian posterior distribution to quantify model uncertainty and further develop the Shapley value based sensitivity analysis to evaluate the impact of estimation uncertainty from each set of model coefficients. Therefore, the proposed bioprocess risk and sensitivity analyses can identify the bottlenecks, guide the reliable process specifications and the most informative data collection, and improve production stability.

Maximum Likelihood Multiple Imputation: Faster Imputations and Consistent Standard Errors Without Posterior Draws

Multiple imputation (MI) is a method for repairing and analyzing data with missing values. MI replaces missing values with a sample of random values drawn from an imputation model. The most popular form of MI, which we call posterior draw multiple imputation (PDMI), draws the parameters of the imputation model from a Bayesian posterior distribution. An alternative, which we call maximum likelihood multiple imputation (MLMI), estimates the parameters of the imputation model using maximum likelihood (or equivalent). Compared to PDMI, MLMI is faster and yields slightly more efficient point estimates. A past barrier to using MLMI was the difficulty of estimating the standard errors of MLMI point estimates. We derive, implement and evaluate three consistent standard error formulas: (1) one combines variances within and between the imputed datasets, (2) one uses the score function and (3) one uses the bootstrap with two imputations of each bootstrapped sample. Formula (1) modifies for MLMI a formula that has long been used under PDMI, while formulas (2) and (3) can be used without modification under either PDMI or MLMI. We have implemented MLMI and the standard error estimators in the mlmi and bootImpute packages for R.

On the Gaussian Approximation to Bayesian Posterior Distributions

The present article derives the minimal number N of observations needed to approximate a Bayesian posterior distribution by a Gaussian. The derivation is based on an invariance requirement for the likelihood <img src=image/13422860_01.gif>. This requirement is defined by a Lie group that leaves the <img src=image/13422860_01.gif> unchanged, when applied both to the observation(s) <img src=image/13422860_05.gif> and to the parameter <img src=image/13422860_02.gif> to be estimated. It leads, in turn, to a class of specific priors. In general, the criterion for the Gaussian approximation is found to depend on (i) the Fisher information related to the likelihood <img src=image/13422860_01.gif>, and (ii) on the lowest non-vanishing order in the Taylor expansion of the Kullback-Leibler distance between <img src=image/13422860_01.gif> and <img src=image/13422860_03.gif>, where <img src=image/13422860_04.gif> is the maximum-likelihood estimator of <img src=image/13422860_02.gif>, given by the observations <img src=image/13422860_05.gif>. Two examples are presented, widespread in various statistical analyses. In the first one, a chi-squared distribution, both the observations <img src=image/13422860_05.gif> and the parameter <img src=image/13422860_02.gif> are defined all over the real axis. In the other one, the binomial distribution, the observation is a binary number, while the parameter is defined on a finite interval of the real axis. Analytic expressions for the required minimal N are given in both cases. The necessary N is an order of magnitude larger for the chi-squared model (continuous <img src=image/13422860_05.gif>) than for the binomial model (binary <img src=image/13422860_05.gif>). The difference is traced back to symmetry properties of the likelihood function <img src=image/13422860_01.gif>. We see considerable practical interest in our results since the normal distribution is the basis of parametric methods of applied statistics widely used in diverse areas of research (education, medicine, physics, astronomy etc.). To have an analytical criterion whether the normal distribution is applicable or not, appears relevant for practitioners in these fields.

Bayesian bootstrapping for symmetric distributions

In this paper, we describe a Bayesian nonparametric approach to make inference for a spherically symmetric distribution. We consider a Dirichlet invariant process prior on the set of all spherically symmetric distributions and we derive the Dirichlet invariant process posterior. Indeed, our approach is an extension of the Dirichlet invariant process to a spherically symmetric distribution. In addition, we obtain the Dirichlet invariant process posterior for the infinite transformation group and we prove that it approaches the mixtures of Dirichlet processes. Moreover, we develop our approach to obtain the Bayesian nonparametric posterior distribution for functionals of the distribution's support when the support is symmetric with respect to the parallel lines of axes. This suggests a Bayesian nonparametric bootstrapping scheme. The estimates can be derived based on posterior averaging. Then, our simulation results demonstrate that our suggested bootstrapping technique improves the accuracy of the estimates.