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All sourcesNASA NTRSDOE OSTI
DOE OSTI2013

REVISITING THE SCALING RELATIONS OF BLACK HOLE MASSES AND HOST GALAXY PROPERTIES

New kinematic data and modeling efforts in the past few years have substantially expanded and revised dynamical measurements of black hole masses (M {sub .}) at the centers of nearby galaxies. Here we compile an updated sample of 72 black holes and their host galaxies, and present revised scaling relations between M {sub .} and stellar velocity dispersion ({sigma}), V-band luminosity (L), and bulge stellar mass (M {sub bulge}), for different galaxy subsamples. Our best-fitting power-law relations for the full galaxy sample are log{sub 10}(M {sub .}) = 8.32 + 5.64log{sub 10}({sigma}/200 km s{sup -1}), log{sub 10}(M {sub .}) = 9.23 + 1.11log{sub 10}(L/10{sup 11} L {sub Sun }), and log{sub 10}(M {sub .}) = 8.46 + 1.05log{sub 10}(M {sub bulge}/10{sup 11} M {sub Sun }). A log-quadratic fit to the M {sub .}-{sigma} relation with an additional term of {beta}{sub 2} [log{sub 10}({sigma}/200 km s{sup -1})]{sup 2} gives {beta}{sub 2} = 1.68 {+-} 1.82 and does not decrease the intrinsic scatter in M {sub .}. Including 92 additional upper limits on M {sub .} does not change the slope of the M {sub .}-{sigma} relation. When the early- and late-type galaxies are fit separately, we obtain similar slopes of 5.20 and 5.06 for the M {sub .}-{sigma} relation but significantly different intercepts-M {sub .} in early-type galaxies are about two times higher than in late types at a given sigma. Within early-type galaxies, our fits to M {sub .}({sigma}) give M {sub .} that is about two times higher in galaxies with central core profiles than those with central power-law profiles. Our M {sub .}-L and M {sub .}-M {sub bulge} relations for early-type galaxies are similar to those from earlier compilations, and core and power-law galaxies yield similar L- and M {sub bulge}-based predictions for M {sub .}. When the conventional quadrature method is used to determine the intrinsic scatter in M {sub .}, our data set shows weak evidence for increased scatter at M {sub bulge} < 10{sup 11} M {sub Sun} or L{sub V} < 10{sup 10.3} L {sub Sun }, while the scatter stays constant for 10{sup 11} < M {sub bulge} < 10{sup 12.3} M {sub Sun} and 10{sup 10.3} < L{sub V} < 10{sup 11.5} L {sub Sun }. A Bayesian analysis indicates that a larger sample of M {sub .} measurements would be needed to detect any statistically significant trend in the scatter with galaxy properties.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

From chiral effective field theory to perturbative QCD: A Bayesian model mixing approach to symmetric nuclear matter

Constraining the equation of state (EOS) of strongly interacting, dense matter is the focus of intense experimental, observational, and theoretical effort. Chiral effective field theory (𝜒⁢EFT ) can describe the EOS between the typical densities of nuclei and those in the outer cores of neutron stars, while perturbative QCD (pQCD) can be applied to properties of deconfined quark matter, both with quantified theoretical uncertainties. However, describing the full range of densities in between with a single EOS that has well-quantified uncertainties is a challenging problem. Bayesian multimodel inference from 𝜒⁢EFT and pQCD can help bridge the gap between the two theories. In this work, we introduce a correlated Bayesian model mixing framework that uses a Gaussian process (GP) to assimilate different information into a single QCD EOS for symmetric nuclear matter. The present implementation uses a stationary GP to infer this mixed EOS solely from the EOSs of 𝜒⁢EFT and pQCD while accounting for the truncation errors of each theory. The GP is trained on the pressure as a function of number density in the low- and high-density regions where 𝜒⁢EFT and pQCD are, respectively, valid. We impose priors on the GP kernel hyperparameters to suppress unphysical correlations between these regimes. This, together with the assumption of stationarity, results in smooth 𝜒⁢EFT-to-pQCD curves for both the pressure and the speed of sound. We show that using uncorrelated mixing requires uncontrolled extrapolation of at least one of 𝜒⁢EFT or pQCD into regions where the perturbative series breaks down and leads to an acausal EOS. Here, we also discuss extensions of this framework to nonstationary and less differentiable GP kernels, its future application to neutron-star matter, and the incorporation of additional constraints from nuclear theory, experiment, and multimessenger astronomy.

Bayesian methods↗
DOE OSTI2023

BeyondPlanck X. Bandpass and beam leakage corrections

We discuss the treatment of bandpass and beam leakage corrections in the Bayesian BeyondPlanck CMB analysis pipeline as applied to the Planck LFI measurements. As a preparatory step, we first apply three corrections to the nominal LFI bandpass profiles including removal of a known systematic effect in the ground measuring equipment at 61 GHz; smoothing of standing wave ripples; and edge regularization. The main net impact of these modifications is an overall shift in the 70 GHz bandpass of +0.6 GHz; we argue that any analysis of LFI data products, either from Planck or BeyondPlanck, should use these new bandpasses. In addition, we fit a single free bandpass parameter for each radiometer of the form Δ i = Δ 0 + $δ$ i , where Δ 0 represents an absolute frequency shift per frequency band and $δ$i is a relative shift per detector. The absolute correction is only fitted at 30 GHz with a full $\chi$ 2 -based likelihood, resulting in a correction of Δ 30 = 0.24 ± 0.03 GHz. The relative corrections are fitted using a spurious map approach, fundamentally similar to the method pioneered by the WMAP team, but without introducing many additional degrees of freedom. All bandpass parameters are sampled using a standard Metropolis sampler within the main BeyondPlanck Gibbs chain, and bandpass uncertainties are thus propagated to all other data products in the analysis. In total, we find that our bandpass model significantly reduces leakage effects. For beam leakage corrections, we adopt the official Planck LFI beam estimates without additional degrees of freedom, and only marginalize over the underlying sky model. We note that this is the first-time leakage from beam mismatch has been included for Planck LFI maps.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

Planck 2015 results: XVIII. Background geometry and topology of the Universe

We report that maps of cosmic microwave background (CMB) temperature and polarization from the 2015 release of Planck data provide the highestquality full-sky view of the surface of last scattering available to date. This enables us to detect possible departures from a globally isotropic cosmology. We present the first searches using CMB polarization for correlations induced by a possible non-trivial topology with a fundamental domain that intersects, or nearly intersects, the last-scattering surface (at comoving distance χ rec ), both via a direct scan for matched circular patterns at the intersections and by an optimal likelihood calculation for specific topologies. We specialize to flat spaces with cubic toroidal (T3) and slab (T1) topologies, finding that explicit searches for the latter are sensitive to other topologies with antipodal symmetry. These searches yield no detection of a compact topology with a scale below the diameter of the last-scattering surface. The limits on the radius R i of the largest sphere inscribed in the fundamental domain (at log-likelihood ratio ΔlnL > -5 relative to a simply-connected flat Planck best-fit model) are: R i > 0.97 χ rec for the T3 cubic torus; and R i > 0.56 χ rec for the T1 slab. The limit for the T3 cubic torus from the matched-circles search is numerically equivalent, R i > 0.97 χ rec at 99% confidence level from polarization data alone. We also perform a Bayesian search for an anisotropic global Bianchi VII h geometry. In the non-physical setting, where the Bianchi cosmology is decoupled from the standard cosmology, Planck temperature data favour the inclusion of a Bianchi component with a Bayes factor of at least 2.3 units of log-evidence. However, the cosmological parameters that generate this pattern are in strong disagreement with those found from CMB anisotropy data alone. Fitting the induced polarization pattern for this model to the Planck data requires an amplitude of -0.10 ± 0.04 compared to the value of + 1 if the model were to be correct. In the physically motivated setting, where the Bianchi parameters are coupled and fitted simultaneously with the standard cosmological parameters, we find no evidence for a Bianchi VII h cosmology and constrain the vorticity of such models to (ω/H) 0 < 7.6 × 10 -10 (95% CL).

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Exploring the contamination of the DES-Y1 cluster sample with SPT-SZ selected clusters

ABSTRACT We perform a cross validation of the cluster catalogue selected by the red-sequence Matched-filter Probabilistic Percolation algorithm (redMaPPer) in Dark Energy Survey year 1 (DES-Y1) data by matching it with the Sunyaev–Zel’dovich effect (SZE) selected cluster catalogue from the South Pole Telescope SPT-SZ survey. Of the 1005 redMaPPer selected clusters with measured richness $\hat{\lambda }\gt 40$ in the joint footprint, 207 are confirmed by SPT-SZ. Using the mass information from the SZE signal, we calibrate the richness–mass relation using a Bayesian cluster population model. We find a mass trend λ ∝ MB consistent with a linear relation (B ∼ 1), no significant redshift evolution and an intrinsic scatter in richness of σλ = 0.22 ± 0.06. By considering two error models, we explore the impact of projection effects on the richness–mass modelling, confirming that such effects are not detectable at the current level of systematic uncertainties. At low richness SPT-SZ confirms fewer redMaPPer clusters than expected. We interpret this richness dependent deficit in confirmed systems as due to the increased presence at low richness of low-mass objects not correctly accounted for by our richness-mass scatter model, which we call contaminants. At a richness $\hat{\lambda }=40$, this population makes up ${\gt}12{{\ \rm per\ cent}}$ (97.5 percentile) of the total population. Extrapolating this to a measured richness $\hat{\lambda }=20$ yields ${\gt}22{{\ \rm per\ cent}}$ (97.5 percentile). With these contamination fractions, the predicted redMaPPer number counts in different plausible cosmologies are compatible with the measured abundance. The presence of such a population is also a plausible explanation for the different mass trends (B ∼ 0.75) obtained from mass calibration using purely optically selected clusters. The mean mass from stacked weak lensing (WL) measurements suggests that these low-mass contaminants are galaxy groups with masses ∼3–5 × 1013 M⊙ which are beyond the sensitivity of current SZE and X-ray surveys but a natural target for SPT-3G and eROSITA.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2003

Intelligent machines in the twenty-first century: foundations of inference and inquiry

The last century saw the application of Boolean algebra to the construction of computing machines, which work by applying logical transformations to information contained in their memory. The development of information theory and the generalization of Boolean algebra to Bayesian inference have enabled these computing machines, in the last quarter of the twentieth century, to be endowed with the ability to learn by making inferences from data. This revolution is just beginning as new computational techniques continue to make difficult problems more accessible. Recent advances in our understanding of the foundations of probability theory have revealed implications for areas other than logic. Of relevance to intelligent machines, we recently identified the algebra of questions as the free distributive algebra, which will now allow us to work with questions in a way analogous to that which Boolean algebra enables us to work with logical statements. In this paper, we examine the foundations of inference and inquiry. We begin with a history of inferential reasoning, highlighting key concepts that have led to the automation of inference in modern machine-learning systems. We then discuss the foundations of inference in more detail using a modern viewpoint that relies on the mathematics of partially ordered sets and the scaffolding of lattice theory. This new viewpoint allows us to develop the logic of inquiry and introduce a measure describing the relevance of a proposed question to an unresolved issue. Last, we will demonstrate the automation of inference, and discuss how this new logic of inquiry will enable intelligent machines to ask questions. Automation of both inference and inquiry promises to allow robots to perform science in the far reaches of our solar system and in other star systems by enabling them not only to make inferences from data, but also to decide which question to ask, which experiment to perform, or which measurement to take given what they have learned and what they are designed to understand.

Review↗
DOE OSTI2023

BEYONDPLANCK XI. Bayesian CMB analysis with sample-based end-to-end error propagation

We present posterior sample-based cosmic microwave background (CMB) constraints from Planck LFI and WMAP observations as derived through global end-to-end Bayesian processing within the BeyondPlanck framework. We first use these samples to study correlations between CMB, foreground, and instrumental parameters, and we identify a particularly strong degeneracy between CMB temperature fluctuations and free-free emission on intermediate angular scales (400 ≲ ℓ ≲ 600), which is mitigated through model reduction, masking, and resampling. We compare our posterior-based CMB results with previous Planck products, and find generally good agreement, although with notably higher noise due to our exclusion of HFI data. We find a best-fit CMB dipole amplitude of 3362.7 ± 1.4 µK, in excellent agreement with previous Planck results. The quoted dipole uncertainty is derived directly from the sampled posterior distribution, and does not involve any ad hoc contributions for Planck instrumental systematic effects. Similarly, we find a temperature quadrupole amplitude of ${σ}_{2}^{TT}$ = 229 ± 97 µK 2 , which is in good agreement with previous results in terms of the amplitude, but the uncertainty is an order of magnitude larger than the naive diagonal Fisher uncertainty. Relatedly, we find lower evidence for a possible alignment between the quadrupole and octopole than previously reported due to a much larger scatter in the individual quadrupole coefficients, caused both by marginalizing over a more complete set of systematic effects, but also by our more conservative analysis mask required to mitigate the free-free degeneracy. For higher multipoles, we find that the angular temperature power spectrum is generally in good agreement with both Planck and WMAP. At the same time, we note that this is the first time the sample-based asymptotically exact Blackwell-Rao estimator has been successfully established for multipoles up to ℓ ≤ 600, and it now accounts for the majority of the cosmologically important information. Overall, this analysis demonstrates the unique capabilities of the Bayesian approach with respect to end-to-end systematic uncertainty propagation, and we believe it can and should play an important role in the analysis of future CMB experiments. Cosmological parameter constraints are presented in a companion paper.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

DAmodel: hierarchical Bayesian modelling of DA white dwarfs for spectrophotometric calibration

We use hierarchical Bayesian modelling to calibrate a network of 32 all-sky faint DA white dwarf (DA WD) spectrophotometric standards (⁠16.5 < V , 19.5⁠) alongside three CALSPEC standards, from 912 Å to 32 μm. The framework is the first of its kind to jointly infer photometric zero points and WD parameters (surface gravity log g⁠, effective temperature T eff ⁠, extinction A V ⁠, dust relation parameter R V ) by simultaneously modelling both photometric and spectroscopic data. We model panchromatic Hubble Space Telescope Wide Field Camera 3 (HST/WFC3) UVIS and IR photometry, HST/STIS UV spectroscopy, and ground-based optical spectroscopy to sub-per cent precision. Photometric residuals for the sample are the lowest yet yielding < 0.004 mag RMS on average from the UV to the NIR, achieved by jointly inferring time-dependent changes in system sensitivity and WFC3/IR count-rate nonlinearity. Our GPU-accelerated implementation enables efficient sampling via Hamiltonian Monte Carlo, critical for exploring the high-dimensional posterior space. The hierarchical nature of the model enables population analysis of intrinsic WD and dust parameters. Inferred spectral energy distributions from this model will be essential for calibrating the James Webb Space Telescope as well as next-generation surveys, including Vera Rubin Observatory’s Legacy Survey of Space and Time and the Nancy Grace Roman Space Telescope.

methods: statistical↗
DOE OSTI2016

Development of a Severe Accident Analysis Engine Using Approximate Reasoning

In hypothetical severe accident scenarios, nuclear power plant safety is challenged by core degradation and release of fission products. To aid the plant operator and technical support center staff, research has been performed to develop understanding of severe accident phenomenology, assessing their risk and devise measures to mitigate accident consequences (Sehgal, 2011). Nuclear power plant accidents involve highly complex physical and chemical interactions, a large number of safety related systems, and emergency operating procedures (EOP) and severe accident management guidelines (SAMG). Large number of factors with uncertainty and stress can overwhelm the operators. The overall objective of the present research is to develop a methodology and tools that an operator can use to predict severe accident progression and make effective decisions to arrest and manage the accident under large uncertainties. Towards this goal, this study examines a novel approach that brings to bear a combined use of techniques in artificial intelligence (Russel and Norvig, 2010), including Bayesian Network (BN), Answer Set Programming (ASP) and Fuzzy Inference (FI) to achieve faster-than-real-time, yet sufficiently accurate analysis of severe accident progression. Bayesian network (BN) is a type of probabilistic graphical models that can be used to represent knowledge and conditional dependencies about an uncertain realm. It is useful for capturing the causal relationships in a complex system. It also allows prediction via Bayesian inference (Kelly and Smith, 2012). It is composed of nodes and edges, which express random variables and probabilistic relationships between the nodes, respectively. This relationship, given by conditional probability is calculated using stochastic methods. The BN is used in several systems for analysis and management of accident risk; e.g. (Knochenhauer and Frid, 2010). In those systems, the nodes or so-called 'effective factors' are presumably known within the network by system descriptions in SAMGs. There are two steps to configure the BN, namely (i) identifying compositions within the network and (ii) defining conditional dependency between nodes. The first step is performed following the spirit of ASP (Baral, 2003) using Prolog (Wielemaker, 2005). Knowledge representation is used to build databases required for processing by ASP. Knowledge representation is defined as the expression of knowledge in a way that is understandable to both human and computers. Originally knowledge indicates data, information and facts. To focus, knowledge should be formed to achieve a goal. In a severe accident, key physics and safety related components status are defined according to a failure mode. The conditional dependencies between nodes are generated by approximate reasoning using Fuzzy inference (Zadeh, 1995). Fuzzy logic is a multiple-valued logic to treat ambiguous features such as linguistic description. It indicates possibility, which is distinguished from probability. Fuzzy set consists of linguistic variables according to the fixed rule. In other words, even uncertain states could be expressed as not binary logic, but as many-value logic by rule-based technology. Fuzzy control system is comprised of Fuzzy inference engine that is performed based on the 'If-Then Rule'. The 'If-Then Rule' is of the following form. If Y1 is B11 and Y2 is B12 then Z is C1. If Y1 is B21 and Y2 is B22 then Z is C2. Y1 and Y2 are inputs, which indicate conditional variables. Z indicates decision-making and both B and C are linguistic variables. With this rule-based technique, the approximate reasoning can be realized by Fuzzy inference. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2026

Cosmological neutrino mass: a frequentist overview in light of DESI

We derive constraints on the neutrino mass using a variety of recent cosmological datasets, including DESI BAO, the full-shape analysis of the DESI matter power spectrum and the one-dimensional power spectrum of the Lyman-α forest (P1D) from eBOSS quasars as well as the cosmic microwave background (CMB). The constraints are obtained in the frequentist formalism by constructing profile likelihoods and applying the Feldman-Cousins prescription to compute confidence intervals. This method avoids potential prior and volume effects that may arise in a comparable Bayesian analysis. Parabolic fits to the profiles allow one to distinguish changes in the upper limits from variations in the constraining power σ of the different data combinations. We find that all profiles in the ΛCDM model are cut off by the ∑m ν ≥ 0 bound, meaning that the corresponding parabolas reach their minimum in the unphysical sector. The most stringent 95% C.L. upper limit is obtained by the combination of DESI DR2 BAO, Planck PR4 and CMB lensing at 53 meV, below the minimum of 59 meV set by the normal ordering. The corresponding constraining power σ is 43 meV, which highlights the importance of the cut-off by negative values in the determination of the upper limit. Extending ΛCDM to non-zero curvature and w 0 w a CDM relaxes the constraints past 59 meV again, but only w 0 w a CDM exhibits profiles with a minimum at a positive value. Additionally, we extend the formalism to constrain the lightest neutrino mass. For DESI DR2 BAO, Planck PR4 and CMB lensing, we find confidence limits at 20 and 19 meV for normal and inverted ordering, respectively. Using a combination of DESI DR1 full-shape, BBN and eBOSS Lyman-α P1D, we successfully constrain the neutrino mass independently of the CMB. This combination yields m l ≤ 97 and 98 meV in the normal and inverted orderings, and total neutrino mass ∑m ν ≤ 285 meV (95% C.L.). The addition of DESI full-shape or Lyman-α P1D to CMB and DESI BAO results in small but noticeable improvement of the constraining power of the data. Lyman-α free-streaming measurements especially improve the constraint. Since they are based on eBOSS data, this sets a promising precedent for upcoming DESI data.

Frequentist statistics↗
DOE OSTI2025

High-significance detection of correlation between the unresolved gamma-ray background and the large-scale cosmic structure

Our understanding of the γ-ray sky has improved dramatically in the past decade, however, the unresolved γ-ray background (UGRB) still has a potential wealth of information about the faintest γ-ray sources pervading the Universe. Statistical cross-correlations with tracers of cosmic structure can indirectly identify the populations that most characterize the γ-ray background. In this study, we analyze the angular correlation between the γ-ray background and the matter distribution in the Universe as traced by gravitational lensing, leveraging more than a decade of observations from the Fermi-Large Area Telescope (LAT) and 3 years of data from the Dark Energy Survey (DES). We detect a correlation at signal-to-noise ratio of 8.9. Most of the statistical significance comes from large scales, demonstrating, for the first time, that a substantial portion of the UGRB aligns with the mass clustering of the Universe as traced by weak lensing. Blazars provide a plausible explanation for this signal, especially if those contributing to the correlation reside in halos of large mass (∼ 10 14 M ⊙ ) and account for approximately 30–40% of the UGRB above 10 GeV. Additionally, we observe a preference for a curved γ-ray energy spectrum, with a log-parabolic shape being favored over a power-law. We also discuss the possibility of modifications to the blazar model and the inclusion of additional γ-ray sources, such as star-forming galaxies, misalinged active galactic nuclei, or particle dark matter.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Counterpart identification and classification for eRASS1 and characterisation of the active galactic nuclei content

Context. Accurately accounting for the Active Galactic Nucleus (AGN) phase in galaxy evolution requires a large, clean AGN sample. This is now possible with SRG/eROSITA, which completed its first all-sky X-ray survey (eRASS1) on June 12, 2020. The public Data Release 1 (DR1, Jan 31, 2024) includes 930,203 sources from the western Galactic hemisphere. Aims. The data enable the selection of a large AGN sample and the discovery of rare sources. However, scientific return depends on accurate characterisation of the X-ray emitters, requiring high-quality multi-wavelength data. This paper presents the identification and classification of optical and infrared counterparts to eRASS1 sources. Methods. Counterparts to eRASS1 X-ray point sources were identified using Gaia DR3, CatWISE2020, and Legacy Survey DR10 (LS10) with the Bayesian NWAY algorithm and trained priors. Sources were classified as Galactic or extragalactic via a machine-learning model combining optical/IR and X-ray properties, trained on a reference sample. For extragalactic LS10 sources, photometric redshifts were computed using CIRCLEZ. Results. Within the LS10 footprint, all 656,614 eROSITA/DR1 sources have at least one possible optical counterpart; ∼570 000 are extragalactic and likely AGN. Half are new detections compared to AllWISE, Gaia, and Quaia AGN catalogues. Gaia and CatWISE2020 counterparts are less reliable, due to the survey’s shallowness and the limited amount of features available to assess the probability of being an X-ray emitter. In the Galactic plane, where the overdensity of stellar sources also increases the chance of associations, using conservative reliability cuts, we identified approximately 18 000 Gaia and 55 000 CatWISE2020 extragalactic sources. Conclusions. We have released three high-quality counterpart catalogues – plus the training and validation sets – as a benchmark for the field. These datasets have many applications, but in particular, they empower researchers to build AGN samples tailored for completeness and purity, accelerating the hunt for the Universe’s most energetic engines.

X-rays: general↗
DOE OSTI2026

Frequentist cosmological constraints from full-shape clustering measurements in DESI DR1

We present a frequentist analysis of clustering measurements from Data Release 1 of the Dark Energy Spectroscopic Instrument (DESI) using the standard profile likelihood method. While Bayesian inferences for effective field theory models of galaxy clustering can be highly sensitive to prior choices for extended cosmological models, frequentist inferences are not susceptible to such effects. We compare frequentist and Bayesian constraints for the parameter set {σ 8 , H 0 , Ω m , w 0 , w a } using the full-shape power spectrum multipoles, post-reconstruction baryon acoustic oscillation (BAO) measurements, and external datasets from the CMB and type Ia supernovae measurements. The frequentist confidence intervals are significantly shifted relative to the Bayesian credible intervals for the w 0 w a CDM model, unless supernovae data are included. When DESI full-shape and BAO data are fit jointly, we obtain the following 1σ frequentist confidence intervals for ΛCDM (w 0 w a CDM): σ 8 = 0.863 +0.048 -0.040 , H 0 = 68.96 +0.81 -0.80 km s -1 Mpc -1 , Ω m = 0.3034 ± 0.0110 (σ 8 = 0.782 +0.060 -0.036 , H 0 = 63.7 +4.2 -2.0 km s -1 Mpc -1 , Ω m = 0.378 +0.024 -0.047 , w 0 = -0.16 +0.10 -0.50 , w a = -3.0 +1.7 ), corresponding to 0.8σ, 0.3σ, 0.7σ (2.1σ, 4.1σ, 6.5σ, 6.3σ, 6.6σ) shifts between the maximum likelihood estimate and the Bayesian posterior mean for ΛCDM (w 0 w a CDM) respectively.

Bayesian reasoning↗