Posterior Probability Distribution Functions Research Articles

Expanding dendrochronology to palms: A Bayesian approach to the visual estimate of a palm tree age in urban and natural spaces

The age of trees and palms is fundamental with respect to their probability of survival, the quality and quantity of their production and their value as unique specimens. Determining these ages is necessary in different contexts (natural, forest, agriculture, urban trees and landscaping). Dendrochronology makes it possible to determine the age of trees, but for palms (Arecaceae) it is still lacking. Here we present and use a method based on the study of whole palm tree images and linear regression of stem/crown ratio and age in years, created with individuals of known age, and posterior probability distribution functions using Bayesian and Monte Carlo methods. This methodology is applicable to the estimate of adult palm individuals of different Arecaceae genera that reach the maximum dimensions of crown once became adult, provided an ensemble of individuals with known age is available for comparison. This approach is here applied to the estimation of the age of Canary Islands palm trees. The proposed methodology shows that the age in years of a Canary Islands palm tree is 28.33 × stipe (S)/crown (C) ratio + 7.03 ± s. The application of the methodology allowed the discovery of a dispersal event around 1840–1845, unknown until now, and revealed two palms from Tenoya (Gran Canaria, Spain) as the oldest known living Canary Islands palms, with an estimated age of over three hundred years.

Bayesian inference of in-medium baryon-baryon scattering cross sections from HADES proton flow data

Within a Bayesian statistical framework using a Gaussian Process emulator for an isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model simulator of heavy-ion reactions at intermediate energies, we infer from the HADES proton flow data the posterior probability distribution functions of in-medium baryon-baryon scattering cross section modification factor X with respect to free-space and the corresponding incompressibility K of nuclear matter as well as their correlation function. The mean value of X is found to be X=1.32−0.40+0.28 at 68% confidence level assuming the nuclear incompressibility K will not exceed 400 MeV, providing circumstantial evidence for enhanced baryon-braryon scattering cross sections in hot and dense nuclear matter.

Mesh-Based Consensus Distributed Particle Filtering for Sensor Networks

Following the Bayesian inference framework, this paper investigates the problem of distributed particle filtering over a sensor network to achieve consensus. The objective of the posterior-consensus strategy is to fuse the posterior probability distribution functions (PDFs) at different sensor nodes, so that an agreement of belief can be established in terms of the Kullback-Leibler average (KLA). To facilitate the consensus process and reduce the communication load, the local PDFs are approximated with weighted meshes and transmitted between neighboring nodes. The mesh representations are constructed by resorting to a grid partition of the state space, such that the PDF can be approximated by a linear combination of indicator functions. To derive a particle representation of the fused PDFs, a novel importance density function is designed to draw particles with respect to the information from all neighboring nodes. The weights of the particles are calculated via the recursive solution of the KLA. The effectiveness of the proposed filtering approach is demonstrated through two target tracking examples.

Bayesian Monte Carlo Evaluation of Imperfect (n, 233U) Data and Model

Conventional nuclear data evaluation methods using generalized linear least squares make the following assumptions: prior and posterior probability distribution functions (PDFs) of all model parameters and data are normal (Gaussian); the linear approximation is sufficiently accurate to minimize the cost function (even for nonlinear models); the model (e.g., of neutron cross section) and experimental data (including covariance data) are without defect and prior PDFs of parameters and measured data are known perfectly. Neglect of covariance between model parameters and measured data in conventional evaluations contributes to imperfections. These assumptions are inherent to the generalized linear least squares minimization method commonly used for resolved resonance region neutron cross section evaluations but are often not justified due to the presence of non-normal PDFs, nonlinear models (e.g., R-matrix formalism), and inherent imperfections in data and models (e.g. imperfect covariance data). Here, these assumptions are removed in a mathematical framework of Bayes’ theorem, which is implemented using the Metropolis-Hastings Monte Carlo method. Most importantly, new parameters are introduced to parameterize discrepancies between the theoretical model and measured data to quantify judgement about discrepancies or imperfections in a reproducible manner. An evaluation of 233U in the eV region using the ENDF-B/VIII.0 library and transmission data (Guber et al.) is presented, and posterior parameters are compared to those obtained by conventional evaluation methods. This example illustrates the effects of removing the most harmful assumption: that of model-data perfection.

The eROSITA Final Equatorial-Depth Survey (eFEDS)

Context.The eROSITA Final Equatorial Depth Survey (eFEDS), observed with eROSITA ahead of its planned 4-yr all-sky survey, is the largest contiguous-field X-ray survey at present. It yielded a large sample of X-ray sources with very rich multiband photometric and spectroscopic coverage.Aims.We present here the eFEDS active galactic nuclei (AGN) catalog and the eROSITA X-ray spectral properties of the eFEDS sources.Methods.Using a Bayesian method, we performed a systematic X-ray spectral analysis for all the eFEDS sources. We adopted multiple spectral models, including single-component power-law or hot-plasma models and double-component models of a power law plus soft excess. We investigated the capacity of eROSITA X-ray spectra for constraining AGN spectral shapes through a detailed analysis of the posterior parameter probability distribution functions. Hierarchical Bayesian modeling was used to recover the spectral parameter distribution of the sample. The source fluxes and luminosities were measured from the posterior of the spectral fitting.Results.The eFEDS AGN catalog (22 079 sources) comprises ~80% of the eFEDS point sources. Despite a large number of faint sources, our spectral fitting provides reasonable measurements of spectral shapes and intrinsic luminosities for a majority of the sources. Because of sample selection bias, this AGN catalog is dominated by X-ray unobscured sources, with an obscured (logNH> 21.5) fraction of 8%; the power-law emission of the hot corona is also relatively soft, with a typical slope of 2.0. For type-I AGN, the X-ray emission is well correlated with the UV emission with the usual anticorrelation between the X-ray to UV spectral slopeαOXand the UV luminosity. The X-ray spectral properties measured with various models are presented for all the eFEDS sources.

The Sensitivity of GPz Estimates of Photo-z Posterior PDFs to Realistically Complex Training Set Imperfections

The accurate estimation of photometric redshifts is crucial to many upcoming galaxy surveys, for example, the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST). Almost all Rubin extragalactic and cosmological science requires accurate and precise calculation of photometric redshifts; many diverse approaches to this problem are currently in the process of being developed, validated, and tested. In this work, we use the photometric redshift code GPz to examine two realistically complex training set imperfections scenarios for machine learning based photometric redshift calculation: (i) where the spectroscopic training set has a very different distribution in color–magnitude space to the test set, and (ii) where the effect of emission line confusion causes a fraction of the training spectroscopic sample to not have the true redshift. By evaluating the sensitivity of GPz to a range of increasingly severe imperfections, with a range of metrics (both of photo-z point estimates as well as posterior probability distribution functions, PDFs), we quantify the degree to which predictions get worse with higher degrees of degradation. In particular, we find that there is a substantial drop-off in photo-z quality when line-confusion goes above ∼1%, and sample incompleteness below a redshift of 1.5, for an experimental setup using data from the Buzzard Flock synthetic sky catalogs.

Probing the nuclear equation of state using the neutron star observables within the Bayesian inference approach

The constraint on nuclear matter's equation of state has been a shared goal in nuclear physics and astrophysics. The most difficult one to constrain the equation of state is the constraint on the symmetry energy in nuclear matter. In this study, we adopt a theoretical framework that combines an isospin-dependent parametric equation of state for neutron star cores with a Bayesian inference approach to calculate the posterior probability distribution functions of the parameters of the equation of state in the nuclear matter based on mass and radius data reported by the LIGO/Virgo and NICER Collaborations, and thus constrain both the equation of state in symmetric nuclear matter and the equation of state in non-symmetric nuclear matter. It is found that a stiffer equation of state is supported by the data from the NICER Collaboration, and the improvement in the equation of state is not observed. The 68% credible intervals for the parameters are $L~=~45^{+15}_{-9}$ MeV for the slope, $K_{\mathrm{sym}}$ = $-100_{-110}^{+50}$ MeV for the curvature parameters of the symmetry energy and $J_0~=~-180_{-40}^{+60}$ MeV for the skewness parameter of the equation of state of symmetric nuclear matter. The symmetry energy values at two and three times normal density are, respectively, $E_{\mathrm{sym}}(2\rho_0)~=~48.9_{-16}^{+7.8}$ MeV and $E_{\mathrm{sym}}(3\rho_0)~=~81.8_{-57.2}^{+21.1}$ MeV.

Bayesian uncertainty quantification for nuclear matter incompressibility

Within a Bayesian statistical framework using the standard Skyrme-Hartree-Fock model, the maximum {\it a posteriori} (MAP) values and uncertainties of nuclear matter incompressibility and isovector interaction parameters are inferred from the experimental data of giant resonances and neutron-skin thicknesses of typical heavy nuclei. With the uncertainties of the isovector interaction parameters constrained by the data of the isovector giant dipole resonance and the neutron-skin thickness, we have obtained $K_0 = 223_{-8}^{+7}$ MeV at 68% confidence level using the data of the isoscalar giant monopole resonance in $^{208}$Pb measured at the Research Center for Nuclear Physics (RCNP), Japan, and at the Texas A&M University (TAMU), USA. Although the corresponding $^{120}$Sn data gives a MAP value for $K_0$ about 5 MeV smaller than the $^{208}$Pb data, there are significant overlaps in their posterior probability distribution functions.

Bayesian inference of the dense-matter equation of state encapsulating a first-order hadron-quark phase transition from observables of canonical neutron stars

[Purpose:] We infer the posterior probability distribution functions (PDFs) and correlations of nine parameters characterizing the EOS of dense neutron-rich matter encapsulating a first-order hadron-quark phase transition from the radius data of canonical NSs reported by LIGO/VIRGO, NICER and Chandra Collaborations. We also infer the quark matter (QM) mass fraction and its radius in a 1.4 M$_{\odot}$ NS and predict their values in more massive NSs. [Method:] Meta-modelings are used to generate both hadronic and QM EOSs in the Markov-Chain Monte Carlo sampling process within the Bayesian statistical framework. An explicitly isospin-dependent parametric EOS for the $npe\mu$ matter in NSs at $\beta$ equilibrium is connected through the Maxwell construction to the QM EOS described by the constant speed of sound (CSS) model of Alford, Han and Prakash. [Results:] (1) The most probable values of the hadron-quark transition density $\rho_t/\rho_0$ and the relative energy density jump there $\De\ep/\ep_t$ are $\rho_t/\rho_0=1.6^{+1.2}_{-0.4}$ and $\De\ep/\ep_t=0.4^{+0.20}_{-0.15}$ at 68\% confidence level, respectively. The corresponding probability distribution of QM fraction in a 1.4 M$_{\odot}$ NS peaks around 0.9 in a 10 km sphere. Strongly correlated to the PDFs of $\rho_t$ and $\De\ep/\ep_t$, the PDF of the QM speed of sound squared $\cQMsq/c^2$ peaks at $0.95^{+0.05}_{-0.35}$, and the total probability of being less than 1/3 is very small. (2) The correlations between PDFs of hadronic and QM EOS parameters are very weak. [Conclusions:] The available astrophysical data considered together with all known EOS constraints from theories and terrestrial nuclear experiments prefer the formation of a large volume of QM even in canonical NSs.

Bayesian inference of nuclear symmetry energy from measured and imagined neutron skin thickness in Sn116,118,120,122,124,130,132, Pb208 , and Ca48

The neutron skin thickness $\Delta r_{np}$ in heavy nuclei has been known as one of the most sensitive terrestrial probes of the nuclear symmetry energy around $\frac{2}{3}$ of the saturation density $\rho_0$ of nuclear matter. Existing neutron skin data mostly from hadronic observables suffer from large uncertainties and their extraction from experiments are often strongly model dependent. While waiting eagerly for the promised model-independent and high-precision neutron skin data for $^{208}$Pb and $^{48}$Ca from the parity-violating electron scattering experiments (PREX-II and CREX at JLab as well as MREX at MESA), within the Bayesian statistical framework using the Skyrme-Hartree-Fock model we infer the posterior probability distribution functions (PDFs) of the slope parameter $L$ of the nuclear symmetry energy at $\rho_0$ from imagined $\Delta r_{np}(^{208}$Pb$)=0.15$, 0.20, and 0.30 fm as well as $\Delta r_{np}(^{48}$Ca$)=0.12$, 0.15, and 0.25 fm, with different $1\sigma$ error bars. The results are compared with the PDFs of $L$ inferred using the same approach from the available $\Delta r_{np}$ data for Sn isotopes from hadronic probes. They are also compared with results from a recent Bayesian analysis of the radius and tidal deformability data of canonical neutron stars from GW170817 and NICER. The neutron skin data for Sn isotopes gives $L=45.5^{+26.5}_{-21.6}$ MeV surrounding its mean value or $L=53.4^{+18.6}_{-29.5}$ MeV surrounding its maximum {\it a posteriori} value, respectively, with the latter smaller than but consistent with the $L=66_{-20}^{+12}$ MeV from the neutron star data within their 68\% confidence intervals. In order to provide additionally useful information on $L$ extracted from the $\Delta r_{np}$ of Sn isotopes, the experimental error bar of $\Delta r_{np}(^{208}$Pb) should be at least smaller than 0.06 fm aimed by some current experiments.

Bayesian Inference of the Symmetry Energy of Superdense Neutron-rich Matter from Future Radius Measurements of Massive Neutron Stars

Abstract Using as references the posterior probability distribution functions of the equation of state (EOS) parameters inferred from the radii of canonical neutron stars (NSs) reported by the LIGO/VIRGO and NICER Collaborations based on their observations of GW170817 and PSR J0030+0451, we investigate how future radius measurements of more massive NSs will improve our current knowledge about the EOS of superdense neutron-rich nuclear matter, especially its symmetry energy term. Within the Bayesian statistical approach using an explicitly isospin-dependent parametric EOS for the core of NSs, we infer the EOS parameters of superdense neutron-rich nuclear matter from three sets of imagined mass–radius correlation data representing typical predictions by various nuclear many-body theories, that is, the radius stays the same, decreases, or increases with increasing NS mass within ±15% between 1.4 and 2.0 M ⊙. The corresponding NS average density increases quickly or slowly or slightly decreases as the NS mass increases from 1.4 to 2.0 M ⊙. While the EOSs of symmetric nuclear matter (SNM) inferred from the three data sets are approximately the same, the corresponding symmetry energies above about twice the saturation density of nuclear matter are very different, indicating that the radii of massive NSs carry important information about the high-density behavior of nuclear symmetry energy with little influence from the remaining uncertainties of the SNM EOS at suprasaturation densities.

An Adaptive Bayesian Parameter Estimation of a Synchronous Generator Under Gross Errors

Polynomial-chaos-expansion-based surrogate models have recently been advocated in the literature for power system dynamic parameter estimation. Regarding the estimation of the uncertain generator parameters, a Bayesian inference framework has been proposed based on a polynomial-based reduced-order representation of the synchronous machines using assumed parameter values. Then, the non-Gaussian posterior probability distribution functions (pdfs) of these parameters are recovered through the stochastic sampling approach efficiently. However, facing very large parameter errors, the reliability of the surrogate model decreases, yielding biased estimation results. To overcome this problem, this article develops a hierarchical Bayesian inference framework that processes the measurements provided by phasor measurement units, while making use of multifidelity surrogates together with the importance sampling method. The latter allows us to estimate in an efficient manner the posterior pdfs of the uncertain parameters through the normalized weights of the prior samples. To improve the accuracy of the posterior pdfs, an adaptive procedure is further adopted in the importance sampling for the gradual evolution of its proposal functions. The new proposals assist in fine-tuning the sample space and thereby help to construct surrogates with higher fidelity. Through an iterative process, this approach is able to estimate accurately and efficiently non-Gaussian posterior pdfs of the uncertain generator parameters subject to gross errors.

Bayesian Markov-Chain Monte Carlo Inversion of Low-Temperature Thermochronology Around Two 8 − 10 m Wide Columbia River Flood Basalt Dikes

Flood basalt volcanism involves large volumes of magma emplaced into the crust and surface environment on geologically short timescales. The mechanics of flood basalt emplacement, including dynamics of the crustal magma transport system and the tempo of individual eruptions, are not well constrained. Here we study two exhumed dikes from the Columbia River Flood Basalt province in northeast Oregon, USA, using apatite and zircon (U-Th)/He thermochronology to constrain dike emplacement histories. Sample transects perpendicular to the dike margins document transient heating of granitic host rocks. We model heating as due to dike emplacement, considering a thermal model with distinct melt-fraction temperature relationships for basaltic magma and granitic wallrock, and a parameterization of unsteady flow within the dike. We model partial resetting of thermochronometers by considering He diffusion in spherical grains as a response to dike heating. A Bayesian Markov-Chain Monte Carlo framework is used to jointly invert for six parameters related to dike emplacement and grain-scale He diffusion. We find that the two dikes, despite similar dimensions on an outcrop scale, exhibit different spatial patterns of thermochronometer partial resetting away from the dike. These patterns result in distinct predicted emplacement histories. We extend previous modeling of a presumed feeder dike at Maxwell Lake in the Wallowa Mountains of northeastern Oregon, finding posterior probability distribution functions (PDFs) that predict steady heating from sustained magma flow over $1-6$ years and elevated farfield host rock temperatures. This suggests regional-scale heating in the vicinity of Maxwell Lake, which might arise from nearby intrusions. The other dike, within the Cornucopia subswarm, is predicted to have a $1-4$ year thermally active lifespan with an unsteady heating rate suggestive of magma low flow rate compared to Maxwell Lake, in a cool near-surface thermal environment. In both cases, misfit of near-dike partial resetting of thermochronometers by models suggests either heat transfer via fluid advection in host rocks or pulsed magma flow in the dikes. Our results highlight the diversity of dike emplacement histories within the Columbia River Flood Basalt province and the power of Bayesian inversion methods for quantifying parameter trade-offs and uncertainty in thermal models.

Evaluating the contribution of the climate change and human activities to runoff change under uncertainty

Climate change and human activities are two important factors affecting surface runoff. In water resource planning and management, it is usually important to separate the contribution of the mentioned factors to the runoff change. In this paper, a new methodology is proposed to obtain the relative impact of human activities and climate change on streamflow under uncertainty. In this methodology, the breakpoint of the annual time series of the observed runoff is estimated and the time series is divided into two study periods; before and after the breakpoint, namely the “natural” and “impacted” periods. The Jazim monthly water balance model is used for monthly runoff simulation. To incorporate the effect of snow in simulating runoff, two Jazim-based monthly water balance models with Rao and Al Wagdany (1995) and McCabe and Markstrom (2007) snow modules are used. Then, the monthly water balance models are calibrated and verified for the natural and impacted periods. The main parameters of the water balance models are optimized using the Genetic Algorithms. The uncertainty of the parameters of the models is analyzed using differential evolution adaptive Metropolis (DREAM) algorithm, which is based on the Markov Chain Monte Carlo (MCMC) algorithm. This algorithm provides the posterior probability distribution functions (PDFs) of parameters of the runoff simulation models. Next, the PDFs of the contribution rate (CR) of the climate change and human activities to runoff are determined using the fixing-changing method by taking into account the PDFs of the models’ parameters. To show the applicability and efficiency of the proposed methodology, it is applied to the Zayandehrud sub-basin in Iran. Using the Standard Normal Homogeneity Test (SNHT), Pettitt and Buishand range tests, one breakpoint at 2006 is detected in runoff time series. Through the monthly rainfall-runoff simulations and the fixing-changing method, it is estimated that decrease in the runoff after the breakpoint can be attributed to climate change by 57.8% and 45%, using the Jazim-Rao & Al Wagdany model and the Jazim-McCabe & Markstrom model, respectively. The results of the uncertainty analysis of the parameters of the water balance models show that the parameters of LSMX (maximum moisture capacity of the lower soil layer) and K3 (deep percolation coefficient) are the most significant parameters in the uncertainty of runoff simulated by both the models. According to the PDFs of contribution rate (CR) of climate change and human activities which are obtained using each rainfall-runoff model, it can be inferred that the Jazim-McCabe & Markstrom model significantly decreased the uncertainty of estimations.

The COS CGM Compendium. II. Metallicities of the Partial and Lyman Limit Systems at z ≲ 1

Abstract We present the results from our COS circumgalactic medium (CGM) compendium (CCC), a survey of the CGM at z ≲ 1 using H i-selected absorbers with 15 < < 19. We focus here on 82 partial Lyman limit systems (pLLSs, 16.2 ≤ < 17.2) and 29 LLSs (17.2 ≤ < 19). Using Bayesian techniques and Markov Chain Monte Carlo sampling of a grid of photoionization models, we derive the posterior probability distribution functions (PDFs) for the metallicity of each absorber in CCC. We show that the combined pLLS metallicity PDF at z ≲ 1 has two main peaks at ≃ −1.7 and −0.4, with a strong dip at ≃ −1. The metallicity PDF of the LLSs might be more complicated than a unimodal or bimodal distribution. The pLLSs and LLSs probe a similar range of metallicities −3 ≲ ≲ +0.4, but the fraction of very metal-poor absorbers with ≲ −1.4 is much larger for the pLLSs than the LLSs. In contrast, absorbers with ≥ 19 have mostly −1 ≲ ≲ 0 at z ≲ 1. The metal-enriched gas probed by pLLSs and LLSs confirms that galaxies that have been enriching their CGM over billions of years. Surprisingly, despite this enrichment, there is also abundant metal-poor CGM gas (41%–59% of the pLLSs have ≲ −1.4), representing a reservoir of near-pristine gas around z ≲ 1 galaxies. We compare our empirical results to recent cosmological zoom simulations, finding some discrepancies, including an overabundance of metal-enriched CGM gas in simulations.

The Evolution of Molecular Gas Fraction Traced by the CO Tully–Fisher Relation

Abstract Carbon monoxide (CO) observations show a luminosity−line width correlation that evolves with redshift. We present a method to use CO measurements alone to infer the molecular gas fraction (f mol) and constrain the CO−H2 conversion factor (α CO). We compile from the literature spatially integrated low-J CO observations of six galaxy populations, including a total of 449 galaxies between 0.01 ≤ z ≤ 3.26. The CO data of each population provide an estimate of the -normalized mean molecular gas fraction (f mol/α CO). The redshift evolution of the luminosity−line width correlation thus indicates an evolution of f mol/α CO. We use a Bayesian-based Monte Carlo Markov Chain sampler to derive the posterior probability distribution functions of f mol/α CO for these galaxy populations, accounting for random inclination angles and measurement errors in the likelihood function. We find that the molecular gas fraction evolves rapidly with redshift, with β ≃ 2, for both normal star-forming and starburst galaxies. Furthermore, the evolution trend agrees well with that inferred from the Kennicutt–Schmidt relation and the star-forming main sequence. Finally, at z < 0.1 normal star-forming galaxies (SFGs) require a ∼5× larger α CO than starburst galaxies to match their molecular gas fractions, but at z > 1 both star-forming types exhibit sub-Galactic α CO values and normal SFGs appear more gas rich than starbursts. Future applications of this method include calibrating Tully–Fisher relations without inclination correction and inferring the evolution of the atomic gas fraction with H i observations.

Universality of the halo mass function in screened gravity theories

We investigate the efficacy of the halo mass function (HMF) as a probe of f(ℛ), symmetron and Dvali-Gabadadze-Porrati (DGP) gravity. In this regard, we develop an excursion-set method to generalise a range of popular HMF fitting functions from General Relativity (GR) to screened Modified Gravity (MG), considering the HMF dependence on the critical density parameter δc. In particular, we propose a variety of new methods to account for the environmental dependence of chameleon screening and compare their accuracy to existing ones by submitting them to N-body simulation acid tests. Using the nested sampling routine MultiNest, we then examine two propositions: can the MG N-body results be accurately described by a ΛCDM GR HMF, and do MG HMFs display universality. More specifically, the values of the free parameters defining a given HMF can always be determined by fitting N-body simulation data, but the question arises as to the dependency of these values on cosmological parameters and halo finding algorithm in the GR case, and indeed on the gravitational force itself when considering MG theories. We find that haloes extracted from any one of the MG N-body simulations can plausibly be fit by a ΛCDM GR HMF, albeit with unusual best fit parameter values. In other words, despite the MG N-body data failing to support a clear deviation in the functional form of the HMF, it indicates that one needs to carefully establish ΛCDM-expected credible regions for the values of the best fit parameters before attributing any difference in these values to MG. Alternatively, one can ask whether the MG N-body data is compatible with the existence of a truly universal (as in completely independent of the gravity model) HMF. Although we find that some HMFs display more universality than others—in the sense that they exhibit better overlap of their best fit parameters posterior probability distribution functions (PDFs) for different screening models –, the general trend, regardless of the HMF considered, is for DGP HMF parameter PDFs to overlap with those of ΛCDM, symmetron HMF parameter PDFs to overlap with those of |fℛ0|=10−6, and |fℛ0|=10−5 HMF parameter PDFs to remain somewhat distinct from all the others. These results strongly suggest that a truly universal HMF will remain elusive.

Physical parameter estimation with MCMC from observations of Vela X-1

We present a parameter estimate for continua, and He-like triplets of the high resolution X-ray spectra with a Bayesian inference and a Markov Chain Monte Carlo (MCMC) tool. The method is applied for Vela X-1 with three different orbital phases ($\unicode[STIX]{x1D719}$), Eclipse, $\unicode[STIX]{x1D719}=0.25$, and $\unicode[STIX]{x1D719}=0.5$, which are adopted from the Chandra High-Energy Transmission Grating Spectrometer (HETGS). A parameterized two-component power-law model [Sako et al., Astrophys. J. 525, 921 (1999)] and a multi-Gaussian model are applied to model these continua and He-like triplets, respectively. A uniform distribution over each parameter is used as the prior belief. Posterior probability distribution functions of parameters and the covariances among them are explored by using the MCMC method. The main advantages are (i) all model-based parameters are set to be free instead of artificially fixing some of the parameters during the data-model fitting; (ii) the contributions from satellite lines are considered; (iii) backgrounds are treated as a correction to the observation errors; and (iv) the confidence interval of each parameter is given. The fitted results show that the column density of scatter component ($N_{\text{H}}^{\text{scat}}$) varies from phase to phase, which imply a non-spherical structure of the stellar wind in Vela X-1. Moreover, the wind velocities derived from main lines of each set of He-like triplets show better self-consistency than those in previous publications, which could provide a reliable approach for the diagnostics of photoionized plasma in astrophysical objects and the laboratory.

Bayesian truncation errors in chiral effective field theory: Nucleon-nucleon observables

Chiral effective field theory (EFT) predictions are necessarily truncated at some order in the EFT expansion, which induces an error that must be quantified for robust statistical comparisons to experiment. In previous work, a Bayesian model for truncation errors of perturbative expansions was adapted to EFTs. The model yields posterior probability distribution functions (pdfs) for these errors based on expectations of naturalness encoded in Bayesian priors and the observed order-by-order convergence pattern of the EFT. A first application was made to chiral EFT for neutron-proton scattering using the semi-local potentials of Epelbaum, Krebs, and Mei{\ss}ner (EKM). Here we extend this application to consider a larger set of regulator parameters, energies, and observables as a general example of a statistical approach to truncation errors. The Bayesian approach allows for statistical validations of the assumptions and enables the calculation of posterior pdfs for the EFT breakdown scale. The statistical model is validated for EKM potentials whose convergence behavior is not distorted by regulator artifacts. For these cases, the posterior for the breakdown scale is consistent with EKM assumptions.

THE PAndAS VIEW OF THE ANDROMEDA SATELLITE SYSTEM. II. DETAILED PROPERTIES OF 23 M31 DWARF SPHEROIDAL GALAXIES

ABSTRACT We present a comprehensive analysis of the structural properties and luminosities of the 23 dwarf spheroidal galaxies that fall within the footprint of the Pan-Andromeda Archaeological Survey (PAndAS). These dwarf galaxies represent the large majority of Andromeda’s known satellite dwarf galaxies and cover a wide range in luminosity ( or ) and surface brightness ( mag arcsec−2). We confirm most previous measurements, but we find And XIX to be significantly larger than before ( , ) and cannot derive parameters for And XXVII as it is likely not a bound stellar system. We also significantly revise downward the luminosities of And XV and And XVI, which are now or . Finally, we provide the first detailed analysis of Cas II/And XXX, a fairly faint system ( ) of typical size ( ), located in close proximity to the two bright elliptical dwarf galaxies NGC 147 and NGC 185. Combined with the set of homogeneous distances published in an earlier contribution, our analysis dutifully tracks all relevant sources of uncertainty in the determination of the properties of the dwarf galaxies from the PAndAS photometric catalog. We further publish the posterior probability distribution functions of all the parameters we fit for in the form of MCMC chains available online; these inputs should be used in any analysis that aims to remain truthful to the data and properly account for covariance between parameters.