Measurement errors and scaling relations in astrophysics: a review
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paper_title: Bayes in the sky: Bayesian inference and model selection in cosmology

paper_content:

The application of Bayesian methods in cosmology and astrophysics has flourished over the past decade, spurred by data sets of increasing size and complexity. In many respects, Bayesian methods have proven to be vastly superior to more traditional statistical tools, offering the advantage of higher efficiency and of a consistent conceptual basis for dealing with the problem of induction in the presence of uncertainty. This trend is likely to continue in the future, when the way we collect, manipulate and analyse observations and compare them with theoretical models will assume an even more central role in cosmology. This review is an introduction to Bayesian methods in cosmology and astrophysics and recent results in the field. I first present Bayesian probability theory and its conceptual underpinnings, Bayes' Theorem and the role of priors. I discuss the problem of parameter inference and its general solution, along with numerical techniques such as Monte Carlo Markov Chain methods. I then review the th...
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paper_title: The scaling relation between richness and mass of galaxy clusters: a Bayesian approach

paper_content:

We use a sample of 53 galaxy clusters at 0.03<z<0.1 with available masses derived from the caustic technique and with velocity dispersions computed using 208 galaxies on average per cluster, in order to investigate the scaling between richness, mass and velocity dispersion. A tight scaling between richness and mass is found, with an intrinsic scatter of only 0.19 dex in mass and with a slope one, i.e. clusters which have twice as many galaxies are twice as massive. When richness is measured without any knowledge of the cluster mass or linked parameters (such as r200), it can predict mass with an uncertainty of 0.29+/-0.01 dex. As a mass proxy, richness competes favourably with both direct measurements of mass given by the caustic method, which has typically 0.14 dex errors (vs 0.29) and X-ray luminosity, which offers a similar 0.30 dex uncertainty. The similar performances of X-ray luminosity and richness in predicting cluster masses has been confirmed using cluster masses derived from velocity dispersion fixed by numerical simulations. These results suggest that cluster masses can be reliably estimated from simple galaxy counts, at least at the redshift and masses explored in this work. This has important applications in the estimation of cosmological parameters from optical cluster surveys, because in current surveys clusters detected in the optical range outnumber, by at least one order of magnitude, those detected in X-ray. Our analysis is robust from astrophysical and statistical perspectives. The data and code used for the stochastic computation is distributed with the paper. [Abridged]
---
paper_title: Linear Regression for Astronomical Data with Measurement Errors and Intrinsic Scatter

paper_content:

Two new methods are proposed for linear regression analysis for data with measurement errors. Both methods are designed to accommodate intrinsic scatter in addition to measurement errors. The first (BCES) is a direct extension of the ordinary least squares (OLS) estimator to allow for measurement errors. It is quite general, allowing a) for measurement errors on both variables, b) the measurement errors for the two variables to be dependent, c) the magnitudes of the measurement errors to depend on the measurements, and d) other `symmetric' lines such as the bisector and the orthogonal regression can be constructed. The second method is a weighted least squares (WLS) estimator, which applies only in the case where the `independent' variable is measured without error and the magnitudes of the measurement errors on the 'dependent' variable are independent from the measurements. Several applications are made to extragalactic astronomy: The BCES method, when applied to data describing the color-luminosity relations for field galaxies, yields significantly different slopes than OLS and other estimators used in the literature. Simulations with artificial data sets are used to evaluate the small sample performance of the estimators. Unsurprisingly, the least-biased results are obtained when color is treated as the dependent variable. The Tully-Fisher relation is another example where the BCES method should be used because errors in luminosity and velocity are correlated due to inclination corrections. We also find, via simulations, that the WLS method is by far the best method for the Tolman surface-brightness test, producing the smallest variance in slope by an order of magnitude. Moreover, with WLS it is not necessary to ``reduce'' galaxies to a fiducial surface-brightness, since this model incorporates intrinsic scatter.
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paper_title: The scaling relation between richness and mass of galaxy clusters: a Bayesian approach

paper_content:

We use a sample of 53 galaxy clusters at 0.03<z<0.1 with available masses derived from the caustic technique and with velocity dispersions computed using 208 galaxies on average per cluster, in order to investigate the scaling between richness, mass and velocity dispersion. A tight scaling between richness and mass is found, with an intrinsic scatter of only 0.19 dex in mass and with a slope one, i.e. clusters which have twice as many galaxies are twice as massive. When richness is measured without any knowledge of the cluster mass or linked parameters (such as r200), it can predict mass with an uncertainty of 0.29+/-0.01 dex. As a mass proxy, richness competes favourably with both direct measurements of mass given by the caustic method, which has typically 0.14 dex errors (vs 0.29) and X-ray luminosity, which offers a similar 0.30 dex uncertainty. The similar performances of X-ray luminosity and richness in predicting cluster masses has been confirmed using cluster masses derived from velocity dispersion fixed by numerical simulations. These results suggest that cluster masses can be reliably estimated from simple galaxy counts, at least at the redshift and masses explored in this work. This has important applications in the estimation of cosmological parameters from optical cluster surveys, because in current surveys clusters detected in the optical range outnumber, by at least one order of magnitude, those detected in X-ray. Our analysis is robust from astrophysical and statistical perspectives. The data and code used for the stochastic computation is distributed with the paper. [Abridged]
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paper_title: Highly accurate H-2 Lyman and Werner band laboratory measurements and an improved constraint on a cosmological variation of the proton-to-electron mass ratio

paper_content:

The search for a cosmological variability of fundamental physical constants has become an active field of research now that accurate spectroscopic data can be obtained from quasars that have emitted their radiation more than ten billion years ago. A comparison between laboratory data, obtained at the highest accuracy in the modern epoch, and analysis of corresponding redshifted spectra from distant objects allows for stringent constraints on variability of the physical constants under
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paper_title: Measurement Error: Models, Methods, and Applications

paper_content:

Introduction What is measurement error? Some examples The main ingredients Some terminology A look ahead Misclassification in Estimating a Proportion Motivating examples A model for the true values Misclassification models and naive analyses Correcting for misclassification Finite populations Multiple measures with no direct validation The multinomial case Mathematical developments Misclassification in Two-Way Tables Introduction Models for true values Misclassification models and naive estimators Behavior of naive analyses Correcting using external validation data Correcting using internal validation data General two-way tables Mathematical developments Simple Linear Regression Introduction The additive Berkson model and consequences The additive measurement error model The behavior of naive analyses Correcting for additive measurement error Examples Residual analysis Prediction Mathematical developments Multiple Linear Regression Introduction Model for true values Models and bias in naive estimators Correcting for measurement error Weighted and other estimators Examples Instrumental variables Mathematical developments Measurement Error in Regression: A General Overview Introduction Models for true values Analyses without measurement error Measurement error models Extra data Assessing bias in naive estimators Assessing bias using induced models Assessing bias via estimating equations Moment based and direct bias corrections Regression calibration and quasi-likelihood methods Simulation extrapolation (SIMEX) Correcting using likelihood methods Modified estimating equation approaches Correcting for misclassification Overview on use of validation data Bootstrapping Mathematical developments Binary Regression Introduction Additive measurement error Using validation data Misclassification of predictors Linear Models with Nonadditive Error Introduction Quadratic regression First-order models with interaction General nonlinear functions of the predictors Linear measurement error with validation data Misclassification of a categorical predictor Miscellaneous Nonlinear Regression Poisson regression: Cigarettes and cancer rates General nonlinear models Error in the Response Introduction Additive error in a single sample Linear measurement error in the one-way setting Measurement error in the response in linear models Mixed/Longitudinal Models Introduction, overview, and some examples Berkson error in designed repeated measures Additive error in the linear mixed model Time Series Introduction Random walk/population viability models Linear autoregressive models Background Material Notation for vectors, covariance matrices, etc. Double expectations Approximate Wald inferences The delta-method: approximate moments of nonlinear functions Fieller's method for ratios References Author Index Subject Index
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paper_title: Highly accurate H-2 Lyman and Werner band laboratory measurements and an improved constraint on a cosmological variation of the proton-to-electron mass ratio

paper_content:

The search for a cosmological variability of fundamental physical constants has become an active field of research now that accurate spectroscopic data can be obtained from quasars that have emitted their radiation more than ten billion years ago. A comparison between laboratory data, obtained at the highest accuracy in the modern epoch, and analysis of corresponding redshifted spectra from distant objects allows for stringent constraints on variability of the physical constants under
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paper_title: Do X-ray dark or underluminous galaxy clusters exist?

paper_content:

We study the X-ray properties of a color-selected sample of clusters at 0.1 < z < 0.3, to quantify the real aboundance of the population of X-ray dark or underluminous clusters and at the same time the spurious detection contamination level of color-selected cluster catalogs. Starting from a local sample of color-selected clusters, we restrict our attention to those with sufficiently deep X-ray observations to probe their X-ray luminosity down to very faint values and without introducing any X-ray bias. This allowed us to have an X-ray- unbiased sample of 33 clusters to measure the LX-richness relation. Swift 1.4 Ms X-ray observations show that at least 89% of the color-detected clusters are real objects with a potential well deep enough to heat and retain an intracluster medium. The percentage rises to 94% when one includes the single spectroscopically confirmed color-selected cluster whose X-ray emission is not secured. Looking at our results from the opposite perspective, the percentage of X-ray dark clusters among color-selected clusters is very low: at most about 11 per cent (at 90% confidence). Supplementing our data with those from literature, we conclude that X-rayand color- cluster surveys sample the same population and consequently that in this regard we can safely use clusters selected with any of the two methods for cosmological purposes. This is an essential and promising piece of information for upcoming surveys in both the optical/IR (DES, EUCLID) and X-ray (eRosita). Richness correlates with X-ray luminosity with a large scatter, 0.51 ± 0.08 (0.44 ± 0.07) dex in lgLX at a given richness, when Lx is measured in a 500 (1070) kpc aperture. We release data and software to estimate the X-ray flux, or its upper limit, of a source with over-Poisson background fluctuations (found in this work to be ∼20% on cluster angular scales) and to fit X-ray luminosity vs richness if there is an intrinsic scatter. These Bayesian applications rigorously account for boundaries (e.g., the X-ray luminosity and the richness cannot be negative).
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paper_title: Cosmological Parameters from Observations of Galaxy Clusters

paper_content:

Studies of galaxy clusters have proved crucial in helping to establish the standard model of cosmology, with a Universe dominated by dark matter and dark energy. A theoretical basis that describes clusters as massive, multicomponent, quasi-equilibrium systems is growing in its capability to interpret multiwavelength observations of expanding scope and sensitivity. We review current cosmological results, including contributions to fundamental physics, obtained from observations of galaxy clusters. These results are consistent with and complementary to those from other methods. We highlight several areas of opportunity for the next few years, and emphasize the need for accurate modeling of survey selection and sources of systematic error. Capitalizing on these opportunities will require a multiwavelength approach and the application of rigorous statistical frameworks, utilizing the combined strengths of observers, simulators, and theorists.
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paper_title: New evidence for a linear colour–magnitude relation and a single Schechter function for red galaxies in a nearby cluster of galaxies down to M*+ 8

paper_content:

The colour and luminosity distributions of red galaxies in the cluster Abell 1185 (z = 0.0325) were studied down to M ∗ + 8 in the B, V and R bands. The colour‐magnitude (CM) relation is linear without evidence for significant bending down to absolute magnitudes that are seldom probed in the literature (MR =− 12.5 mag). The CM relation is thin (±0.04 mag) and its thickness is quite independent of the magnitude. The luminosity function (LF) of red galaxies in Abell 1185 is adequately described by a Schechter function, with a characteristic magnitude and a faint end slope that also well describe the LF of red galaxies in other clusters. There is no passband dependence of the LF shape other than an obvious M ∗ shift due to the colour of the considered population. Finally, we conclude that, based on colours and luminosity, red galaxies form a homogeneous population over four decades in stellar mass, providing a second piece of evidence against faint red galaxies being a recent cluster population.
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paper_title: XBOOTES: AN X-RAY SURVEY OF THE NDWFS BOOTES FIELD. II. THE X-RAY SOURCE CATALOG

paper_content:

We present results from a Chandra survey of the 9 deg 2 Bootes field of the NOAO Deep Wide-Field Survey (NDWFS). This XBootes survey consists of 126 separate contiguous ACIS-I observations each of approximately 5000 s in duration. These unique Chandra observations allow us to search for large-scale structure and to calculate X-raysource statistics overawide,contiguousfieldofviewwitharcsecondangularresolutionanduniformcoverage. Opticalspectroscopicfollow-upobservationsandtherichNDWFSdatasetwillallowustoidentifyandclassifythese X-ray‐selected sources. Using wavelet decomposition, we detect 4642 point sources with n � 2 counts. In order to keep our detections � 99% reliable, we limit our list to sources with n � 4 counts. For a 5000 s observation and assuming a canonical unabsorbed active galactic nucleus (AGN) type X-ray spectrum, a 4 count on-axis source correspondsto afluxof4:7 ;10 � 15 ergscm � 2 s � 1 inthe soft (0.5‐2keV) band, 1:5 ; 10 � 14 ergscm � 2 s � 1 in thehard (2‐7 keV) band, and 7:8 ; 10 � 15 ergs cm � 2 s � 1 in the full (0.5‐7 keV) band. The full 0.5‐7 keV band n � 4 count list has 3293 point sources. In addition to the point sources, 43 extended sources have been detected, consistent with the depth of these observations and the number counts of clusters. We present here the X-ray catalog for the XBootes survey, including source positions, X-ray fluxes, hardness ratios, and their uncertainties. We calculate and present the differential number of sources per flux density interval, N(S), for the point sources. In the soft (0.5‐2 keV) band, N(S) is well fitted by a broken power law with slope of 2:60 þ0:11 � 0:12 at bright fluxes and 1:74 þ0:28 � 0:22 for faint fluxes. The hard source N(S) is well described by a single power law with an index of � 2:93 þ0:09 � 0:09.
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paper_title: The scaling relation between richness and mass of galaxy clusters: a Bayesian approach

paper_content:

We use a sample of 53 galaxy clusters at 0.03<z<0.1 with available masses derived from the caustic technique and with velocity dispersions computed using 208 galaxies on average per cluster, in order to investigate the scaling between richness, mass and velocity dispersion. A tight scaling between richness and mass is found, with an intrinsic scatter of only 0.19 dex in mass and with a slope one, i.e. clusters which have twice as many galaxies are twice as massive. When richness is measured without any knowledge of the cluster mass or linked parameters (such as r200), it can predict mass with an uncertainty of 0.29+/-0.01 dex. As a mass proxy, richness competes favourably with both direct measurements of mass given by the caustic method, which has typically 0.14 dex errors (vs 0.29) and X-ray luminosity, which offers a similar 0.30 dex uncertainty. The similar performances of X-ray luminosity and richness in predicting cluster masses has been confirmed using cluster masses derived from velocity dispersion fixed by numerical simulations. These results suggest that cluster masses can be reliably estimated from simple galaxy counts, at least at the redshift and masses explored in this work. This has important applications in the estimation of cosmological parameters from optical cluster surveys, because in current surveys clusters detected in the optical range outnumber, by at least one order of magnitude, those detected in X-ray. Our analysis is robust from astrophysical and statistical perspectives. The data and code used for the stochastic computation is distributed with the paper. [Abridged]
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paper_title: The Build–up of the Red Sequence in the galaxy cluster MS1054-0321 at z = 0.831

paper_content:

Using one of the deepest datasets available, we determine that the red sequence of the massive cluster MS1054-0321 at z=0.831 is well populated at all studied magnitudes, showing no deficit of faint (down to M^*+3.5) red galaxies: the faint end of the colour-magnitude relation is neither empty nor underpopulated. The effect is quantified by the computation of the luminosity function (LF) of red galaxies. We found a flat slope, showing that the abundance of red galaxies is similar at faint and at intermediate magnitudes. Comparison with present-day and z~0.4 LFs suggests that the slope of the LF is not changed, within the errors, between z=0.831 and z=0. Therefore, the analysis of the LF shows no evidence for a decreasing (with magnitude or redshift) number of faint red galaxies. The presence of faint red galaxies in high redshift clusters disfavours scenarios where the evolution of red galaxies is mass-dependent, because the mass dependency should differentially depauperate the red sequence, while the MS1054-0321 colour-magnitude relation is populated as in nearby clusters and as in z~0.4 clusters. The presence of abundant faint red galaxies in the high redshift cluster MS1054-0321 restricts the room for allocating descendants of Butcher-Oemler galaxies, because they should change the faint end slope of the LF of red galaxies, while instead the same faint end slopes are observed in MS1054-0321, at z~0 and at z~0.4. In the rich MS1054-0321 cluster, the colour-magnitude relation seems to be fully in place at z=0.831 and therefore red galaxies of all magnitudes were wholly assembled at higher redshift.
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paper_title: New evidence for a linear colour–magnitude relation and a single Schechter function for red galaxies in a nearby cluster of galaxies down to M*+ 8

paper_content:

The colour and luminosity distributions of red galaxies in the cluster Abell 1185 (z = 0.0325) were studied down to M ∗ + 8 in the B, V and R bands. The colour‐magnitude (CM) relation is linear without evidence for significant bending down to absolute magnitudes that are seldom probed in the literature (MR =− 12.5 mag). The CM relation is thin (±0.04 mag) and its thickness is quite independent of the magnitude. The luminosity function (LF) of red galaxies in Abell 1185 is adequately described by a Schechter function, with a characteristic magnitude and a faint end slope that also well describe the LF of red galaxies in other clusters. There is no passband dependence of the LF shape other than an obvious M ∗ shift due to the colour of the considered population. Finally, we conclude that, based on colours and luminosity, red galaxies form a homogeneous population over four decades in stellar mass, providing a second piece of evidence against faint red galaxies being a recent cluster population.
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paper_title: Estimating Mixtures of Regressions

paper_content:

This article shows how Bayesian inference for switching regression models and their generalizations can be achieved by the specification of loss functions which overcome the label switching problem common to all mixture models. We also derive an extension to models where the number of components in the mixture is unknown, based on the birthand-death technique developed in recent literature. The methods are illustrated on various real datasets.
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paper_title: AN OVERVIEW OF LINEAR STRUCTURAL MOD- ELS IN ERRORS IN VARIABLES REGRESSION

paper_content:

• This paper aims to overview the numerous approaches that have been developed to estimate the parameters of the linear structural model. The linear structural model is an example of an errors in variables model, or measurement error model that has wide practical use. This paper brings together key concepts from a scattered literature to give an accessible account of existing work on this particular errors in variables model.
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paper_title: Measurement Error: Models, Methods, and Applications

paper_content:

Introduction What is measurement error? Some examples The main ingredients Some terminology A look ahead Misclassification in Estimating a Proportion Motivating examples A model for the true values Misclassification models and naive analyses Correcting for misclassification Finite populations Multiple measures with no direct validation The multinomial case Mathematical developments Misclassification in Two-Way Tables Introduction Models for true values Misclassification models and naive estimators Behavior of naive analyses Correcting using external validation data Correcting using internal validation data General two-way tables Mathematical developments Simple Linear Regression Introduction The additive Berkson model and consequences The additive measurement error model The behavior of naive analyses Correcting for additive measurement error Examples Residual analysis Prediction Mathematical developments Multiple Linear Regression Introduction Model for true values Models and bias in naive estimators Correcting for measurement error Weighted and other estimators Examples Instrumental variables Mathematical developments Measurement Error in Regression: A General Overview Introduction Models for true values Analyses without measurement error Measurement error models Extra data Assessing bias in naive estimators Assessing bias using induced models Assessing bias via estimating equations Moment based and direct bias corrections Regression calibration and quasi-likelihood methods Simulation extrapolation (SIMEX) Correcting using likelihood methods Modified estimating equation approaches Correcting for misclassification Overview on use of validation data Bootstrapping Mathematical developments Binary Regression Introduction Additive measurement error Using validation data Misclassification of predictors Linear Models with Nonadditive Error Introduction Quadratic regression First-order models with interaction General nonlinear functions of the predictors Linear measurement error with validation data Misclassification of a categorical predictor Miscellaneous Nonlinear Regression Poisson regression: Cigarettes and cancer rates General nonlinear models Error in the Response Introduction Additive error in a single sample Linear measurement error in the one-way setting Measurement error in the response in linear models Mixed/Longitudinal Models Introduction, overview, and some examples Berkson error in designed repeated measures Additive error in the linear mixed model Time Series Introduction Random walk/population viability models Linear autoregressive models Background Material Notation for vectors, covariance matrices, etc. Double expectations Approximate Wald inferences The delta-method: approximate moments of nonlinear functions Fieller's method for ratios References Author Index Subject Index
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paper_title: Numerical recipes: Cambridge University Press

paper_content:

A polyanhydride copolymer is made having styrene grafted onto the linear backbone chain of a styrene-maleic anhydride copolymer or a 1-alkene-maleic anhydride copolymer. This graft copolymer is also prepared in the presence of an epoxy compound such as monoepoxide or an epoxy resin for subsequent curing.
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paper_title: Parameter estimation in astronomy through application of the likelihood ratio

paper_content:

Many problems in the experimental estimation of parameters for models can be solved through use of the likelihood ratio test. Applications of the likelihood ratio, with particular attention to photon counting experiments, are discussed. The procedures presented solve a greater range of problems than those currently in use, yet are no more difficult to apply. The procedures are proved analytically, and examples from current problems in astronomy are discussed.
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paper_title: The Use and Misuse of Orthogonal Regression in Linear Errors-in-Variables Models

paper_content:

Abstract Orthogonal regression is one of the standard linear regression methods to correct for the effects of measurement error in predictors. We argue that orthogonal regression is often misused in errors-in-variables linear regression because of a failure to account for equation errors. The typical result is to overcorrect for measurement error, that is, overestimate the slope, because equation error is ignored. The use of orthogonal regression must include a careful assessment of equation error, and not merely the usual (often informal) estimation of the ratio of measurement error variances. There are rarer instances, for example, an example from geology discussed here, where the use of orthogonal regression without proper attention to modeling may lead to either overcorrection or undercorrection, depending on the relative sizes of the variances involved. Thus our main point, which does not seem to be widely appreciated, is that orthogonal regression, just like any measurement error analysis, requires ...
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paper_title: Numerical recipes: Cambridge University Press

paper_content:

A polyanhydride copolymer is made having styrene grafted onto the linear backbone chain of a styrene-maleic anhydride copolymer or a 1-alkene-maleic anhydride copolymer. This graft copolymer is also prepared in the presence of an epoxy compound such as monoepoxide or an epoxy resin for subsequent curing.
---
paper_title: Measurement Error: Models, Methods, and Applications

paper_content:

Introduction What is measurement error? Some examples The main ingredients Some terminology A look ahead Misclassification in Estimating a Proportion Motivating examples A model for the true values Misclassification models and naive analyses Correcting for misclassification Finite populations Multiple measures with no direct validation The multinomial case Mathematical developments Misclassification in Two-Way Tables Introduction Models for true values Misclassification models and naive estimators Behavior of naive analyses Correcting using external validation data Correcting using internal validation data General two-way tables Mathematical developments Simple Linear Regression Introduction The additive Berkson model and consequences The additive measurement error model The behavior of naive analyses Correcting for additive measurement error Examples Residual analysis Prediction Mathematical developments Multiple Linear Regression Introduction Model for true values Models and bias in naive estimators Correcting for measurement error Weighted and other estimators Examples Instrumental variables Mathematical developments Measurement Error in Regression: A General Overview Introduction Models for true values Analyses without measurement error Measurement error models Extra data Assessing bias in naive estimators Assessing bias using induced models Assessing bias via estimating equations Moment based and direct bias corrections Regression calibration and quasi-likelihood methods Simulation extrapolation (SIMEX) Correcting using likelihood methods Modified estimating equation approaches Correcting for misclassification Overview on use of validation data Bootstrapping Mathematical developments Binary Regression Introduction Additive measurement error Using validation data Misclassification of predictors Linear Models with Nonadditive Error Introduction Quadratic regression First-order models with interaction General nonlinear functions of the predictors Linear measurement error with validation data Misclassification of a categorical predictor Miscellaneous Nonlinear Regression Poisson regression: Cigarettes and cancer rates General nonlinear models Error in the Response Introduction Additive error in a single sample Linear measurement error in the one-way setting Measurement error in the response in linear models Mixed/Longitudinal Models Introduction, overview, and some examples Berkson error in designed repeated measures Additive error in the linear mixed model Time Series Introduction Random walk/population viability models Linear autoregressive models Background Material Notation for vectors, covariance matrices, etc. Double expectations Approximate Wald inferences The delta-method: approximate moments of nonlinear functions Fieller's method for ratios References Author Index Subject Index
---
paper_title: Linear Regression for Astronomical Data with Measurement Errors and Intrinsic Scatter

paper_content:

Two new methods are proposed for linear regression analysis for data with measurement errors. Both methods are designed to accommodate intrinsic scatter in addition to measurement errors. The first (BCES) is a direct extension of the ordinary least squares (OLS) estimator to allow for measurement errors. It is quite general, allowing a) for measurement errors on both variables, b) the measurement errors for the two variables to be dependent, c) the magnitudes of the measurement errors to depend on the measurements, and d) other `symmetric' lines such as the bisector and the orthogonal regression can be constructed. The second method is a weighted least squares (WLS) estimator, which applies only in the case where the `independent' variable is measured without error and the magnitudes of the measurement errors on the 'dependent' variable are independent from the measurements. Several applications are made to extragalactic astronomy: The BCES method, when applied to data describing the color-luminosity relations for field galaxies, yields significantly different slopes than OLS and other estimators used in the literature. Simulations with artificial data sets are used to evaluate the small sample performance of the estimators. Unsurprisingly, the least-biased results are obtained when color is treated as the dependent variable. The Tully-Fisher relation is another example where the BCES method should be used because errors in luminosity and velocity are correlated due to inclination corrections. We also find, via simulations, that the WLS method is by far the best method for the Tolman surface-brightness test, producing the smallest variance in slope by an order of magnitude. Moreover, with WLS it is not necessary to ``reduce'' galaxies to a fiducial surface-brightness, since this model incorporates intrinsic scatter.
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paper_title: The BUGS project: Evolution, critique and future directions

paper_content:

BUGS is a software package for Bayesian inference using Gibbs sampling. The software has been instrumental in raising awareness of Bayesian modelling among both academic and commercial communities internationally, and has enjoyed considerable success over its 20-year life span. Despite this, the software has a number of shortcomings and a principal aim of this paper is to provide a balanced critical appraisal, in particular highlighting how various ideas have led to unprecedented flexibility while at the same time producing negative side effects. We also present a historical overview of the BUGS project and some future perspectives. Copyright © 2009 John Wiley & Sons, Ltd.
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paper_title: Prior distributions for variance parameters in hierarchical models

paper_content:

Various noninformative prior distributions have been suggested for scale parameters in hierarchical models. We construct a new folded-noncentral- t family of conditionally conjugate priors for hierarchical standard deviation parameters, and then consider noninformative and weakly informative priors in this family. We use an example to illustrate serious problems with the inverse-gamma family of ``noninformative'' prior distributions. We suggest instead to use a uniform prior on the hierarchical standard deviation, using the half-t family when the number of groups is small and in other settings where a weakly informative prior is desired.
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paper_title: BAYESIAN REASONING IN DATA ANALYSIS: A CRITICAL INTRODUCTION

paper_content:

Critical review and outline of the Bayesian alternative: uncertainty in physics and the usual methods of handling it a probabilistic theory of measurement uncertainty. A Bayesian primer: subjective probability and Bayes' theorem probability distributions (a concise reminder) Bayesian inference of continuous quantities Gaussian likelihood counting experiments bypassing Bayes' theorem for routine applications Bayesian unfolding. Further comments, examples and applications: miscellanea on general issues in probability and inference combination of experimental results - a closer look asymmetric uncertainties and nonlinear propagation which priors for frontier physics? Concluding matter: conclusions and bibliography.
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paper_title: BAYESIAN ANALYSIS OF ERRORS-IN-VARIABLES REGRESSION MODELS

paper_content:

SUMMARY Use of errors-in-variables models is appropriate in many practical experimental problems. However, inference based on such models is by no means straightforward. In previous analyses, simplifying assumptions have been made in order to ease this intractability, but assumptions of this nature are unfortunate and restrictive. In this paper, we analyse errors-in-variables models in full generality under a Bayesian formulation. In order to compute the necessary posterior distributions, we utilize various computational techniques. Two specific non-linear errors-in-variables regression examples are considered; the first is a re-analysed Berkson-type model, and the second is a classical errors-in-variables model. Our analyses are compared and contrasted with those presented elsewhere in the literature.
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paper_title: Fits , and especially linear fits , with errors on both axes , extra variance of the data points and other complications

paper_content:

The aim of this paper, triggered by some discussions in the astrophysics community raised by astro-ph/0508529, is to introduce the issue of `fits' from a probabilistic perspective (also known as Bayesian), with special attention to the construction of model that describes the `network of dependences' (a Bayesian network) that connects experimental observations to model parameters and upon which the probabilistic inference relies. The particular case of linear fit with errors on both axes and extra variance of the data points around the straight line (i.e. not accounted by the experimental errors) is shown in detail. Some questions related to the use of linear fit formulas to log-linearized exponential and power laws are also sketched, as well as the issue of systematic errors.
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paper_title: Linear Regression for Astronomical Data with Measurement Errors and Intrinsic Scatter

paper_content:

Two new methods are proposed for linear regression analysis for data with measurement errors. Both methods are designed to accommodate intrinsic scatter in addition to measurement errors. The first (BCES) is a direct extension of the ordinary least squares (OLS) estimator to allow for measurement errors. It is quite general, allowing a) for measurement errors on both variables, b) the measurement errors for the two variables to be dependent, c) the magnitudes of the measurement errors to depend on the measurements, and d) other `symmetric' lines such as the bisector and the orthogonal regression can be constructed. The second method is a weighted least squares (WLS) estimator, which applies only in the case where the `independent' variable is measured without error and the magnitudes of the measurement errors on the 'dependent' variable are independent from the measurements. Several applications are made to extragalactic astronomy: The BCES method, when applied to data describing the color-luminosity relations for field galaxies, yields significantly different slopes than OLS and other estimators used in the literature. Simulations with artificial data sets are used to evaluate the small sample performance of the estimators. Unsurprisingly, the least-biased results are obtained when color is treated as the dependent variable. The Tully-Fisher relation is another example where the BCES method should be used because errors in luminosity and velocity are correlated due to inclination corrections. We also find, via simulations, that the WLS method is by far the best method for the Tolman surface-brightness test, producing the smallest variance in slope by an order of magnitude. Moreover, with WLS it is not necessary to ``reduce'' galaxies to a fiducial surface-brightness, since this model incorporates intrinsic scatter.
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paper_title: Scaling relations of the colour-detected cluster RzCS 052 at z=1.016 and of some other high redshift clusters

paper_content:

We report on the discovery of the z = 1.016 cluster RzCS 052 using a modified red-sequence method, follow up spectroscopy and X-ray imaging. This cluster has a velocity dispersion of 710 ± 150 km s -1 , a virial mass of 4.0 x 10 14 M ⊙ (based on 21 spectroscopically confirmed members) and an X-ray luminosity of (0.68 ± 0.47) x 10 44 erg s -1 in the [1-4] keV band. This optically selected cluster appears to be of richness class 3 and to follow the known L X -σ v relation for high-redshift X-ray selected clusters. Using these data, we find that the halo occupation number for this cluster is only marginally consistent with what was expected assuming a self-similar evolution of cluster scaling relations, suggesting perhaps a break of them at z ∼ 1. We also rule out a strong galaxy merging activity between z = 1 and today. Finally, we present a Bayesian approach to measuring cluster velocity dispersions and X-ray luminosities in the presence of a background: we critically reanalyse recent claims for X-ray underluminous clusters using these techniques and find that the clusters can be accommodated within the existing L X -σ v relation.
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paper_title: BAYESIAN ANALYSIS OF ERRORS-IN-VARIABLES REGRESSION MODELS

paper_content:

SUMMARY Use of errors-in-variables models is appropriate in many practical experimental problems. However, inference based on such models is by no means straightforward. In previous analyses, simplifying assumptions have been made in order to ease this intractability, but assumptions of this nature are unfortunate and restrictive. In this paper, we analyse errors-in-variables models in full generality under a Bayesian formulation. In order to compute the necessary posterior distributions, we utilize various computational techniques. Two specific non-linear errors-in-variables regression examples are considered; the first is a re-analysed Berkson-type model, and the second is a classical errors-in-variables model. Our analyses are compared and contrasted with those presented elsewhere in the literature.
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paper_title: New evidence for a linear colour–magnitude relation and a single Schechter function for red galaxies in a nearby cluster of galaxies down to M*+ 8

paper_content:

The colour and luminosity distributions of red galaxies in the cluster Abell 1185 (z = 0.0325) were studied down to M ∗ + 8 in the B, V and R bands. The colour‐magnitude (CM) relation is linear without evidence for significant bending down to absolute magnitudes that are seldom probed in the literature (MR =− 12.5 mag). The CM relation is thin (±0.04 mag) and its thickness is quite independent of the magnitude. The luminosity function (LF) of red galaxies in Abell 1185 is adequately described by a Schechter function, with a characteristic magnitude and a faint end slope that also well describe the LF of red galaxies in other clusters. There is no passband dependence of the LF shape other than an obvious M ∗ shift due to the colour of the considered population. Finally, we conclude that, based on colours and luminosity, red galaxies form a homogeneous population over four decades in stellar mass, providing a second piece of evidence against faint red galaxies being a recent cluster population.
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paper_title: Linear Regression for Astronomical Data with Measurement Errors and Intrinsic Scatter

paper_content:

Two new methods are proposed for linear regression analysis for data with measurement errors. Both methods are designed to accommodate intrinsic scatter in addition to measurement errors. The first (BCES) is a direct extension of the ordinary least squares (OLS) estimator to allow for measurement errors. It is quite general, allowing a) for measurement errors on both variables, b) the measurement errors for the two variables to be dependent, c) the magnitudes of the measurement errors to depend on the measurements, and d) other `symmetric' lines such as the bisector and the orthogonal regression can be constructed. The second method is a weighted least squares (WLS) estimator, which applies only in the case where the `independent' variable is measured without error and the magnitudes of the measurement errors on the 'dependent' variable are independent from the measurements. Several applications are made to extragalactic astronomy: The BCES method, when applied to data describing the color-luminosity relations for field galaxies, yields significantly different slopes than OLS and other estimators used in the literature. Simulations with artificial data sets are used to evaluate the small sample performance of the estimators. Unsurprisingly, the least-biased results are obtained when color is treated as the dependent variable. The Tully-Fisher relation is another example where the BCES method should be used because errors in luminosity and velocity are correlated due to inclination corrections. We also find, via simulations, that the WLS method is by far the best method for the Tolman surface-brightness test, producing the smallest variance in slope by an order of magnitude. Moreover, with WLS it is not necessary to ``reduce'' galaxies to a fiducial surface-brightness, since this model incorporates intrinsic scatter.
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paper_title: Fits , and especially linear fits , with errors on both axes , extra variance of the data points and other complications

paper_content:

The aim of this paper, triggered by some discussions in the astrophysics community raised by astro-ph/0508529, is to introduce the issue of `fits' from a probabilistic perspective (also known as Bayesian), with special attention to the construction of model that describes the `network of dependences' (a Bayesian network) that connects experimental observations to model parameters and upon which the probabilistic inference relies. The particular case of linear fit with errors on both axes and extra variance of the data points around the straight line (i.e. not accounted by the experimental errors) is shown in detail. Some questions related to the use of linear fit formulas to log-linearized exponential and power laws are also sketched, as well as the issue of systematic errors.
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paper_title: The Build–up of the Red Sequence in the galaxy cluster MS1054-0321 at z = 0.831

paper_content:

Using one of the deepest datasets available, we determine that the red sequence of the massive cluster MS1054-0321 at z=0.831 is well populated at all studied magnitudes, showing no deficit of faint (down to M^*+3.5) red galaxies: the faint end of the colour-magnitude relation is neither empty nor underpopulated. The effect is quantified by the computation of the luminosity function (LF) of red galaxies. We found a flat slope, showing that the abundance of red galaxies is similar at faint and at intermediate magnitudes. Comparison with present-day and z~0.4 LFs suggests that the slope of the LF is not changed, within the errors, between z=0.831 and z=0. Therefore, the analysis of the LF shows no evidence for a decreasing (with magnitude or redshift) number of faint red galaxies. The presence of faint red galaxies in high redshift clusters disfavours scenarios where the evolution of red galaxies is mass-dependent, because the mass dependency should differentially depauperate the red sequence, while the MS1054-0321 colour-magnitude relation is populated as in nearby clusters and as in z~0.4 clusters. The presence of abundant faint red galaxies in the high redshift cluster MS1054-0321 restricts the room for allocating descendants of Butcher-Oemler galaxies, because they should change the faint end slope of the LF of red galaxies, while instead the same faint end slopes are observed in MS1054-0321, at z~0 and at z~0.4. In the rich MS1054-0321 cluster, the colour-magnitude relation seems to be fully in place at z=0.831 and therefore red galaxies of all magnitudes were wholly assembled at higher redshift.
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paper_title: New evidence for a linear colour–magnitude relation and a single Schechter function for red galaxies in a nearby cluster of galaxies down to M*+ 8

paper_content:

The colour and luminosity distributions of red galaxies in the cluster Abell 1185 (z = 0.0325) were studied down to M ∗ + 8 in the B, V and R bands. The colour‐magnitude (CM) relation is linear without evidence for significant bending down to absolute magnitudes that are seldom probed in the literature (MR =− 12.5 mag). The CM relation is thin (±0.04 mag) and its thickness is quite independent of the magnitude. The luminosity function (LF) of red galaxies in Abell 1185 is adequately described by a Schechter function, with a characteristic magnitude and a faint end slope that also well describe the LF of red galaxies in other clusters. There is no passband dependence of the LF shape other than an obvious M ∗ shift due to the colour of the considered population. Finally, we conclude that, based on colours and luminosity, red galaxies form a homogeneous population over four decades in stellar mass, providing a second piece of evidence against faint red galaxies being a recent cluster population.
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paper_title: Red sequence determination of the redshift of the cluster of galaxies JKCS041: z~2.2

paper_content:

This paper aims at robustly determining the redshift of the cluster of galaxies JKCS041 and at putting constraints on the formation epoch of the color-magnitude sequence in two very high redshift clusters. New deep z'-J data show a clear narrow red sequence that is co-centered with, and similarly concentrated on, the extended X-ray emission of the cluster of galaxies JKCS041. The JKCS041 red sequence is 0.32+/-0.06 mag redder in z'-J than the red sequence of the zspec=1.62 IRC0218A cluster, putting JKCS041 at z>>1.62 and ruling out z<~1.49 the latter claimed by a recent paper. The color difference of the two red sequences gives a red-sequence-based redshift of z=2.20+/-0.11 for JKCS041, where the uncertainty accounts for uncertainties in stellar synthesis population models, in photometric calibration, and in the red sequence color of both JKCS041 and IRC0218A clusters. We do not observe any sign of truncation of the red sequence for both clusters down to J=23 mag (1.0e+11 solar masses), which suggests that it is already in place in clusters rich and massive enough to heat and retain hot gas at these high redshifts.
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paper_title: Richness-mass relation self-calibration for galaxy clusters

paper_content:

This work attains a threefold objective: first, we derived the richness-mass scaling in the local Universe from data of 53 clusters with individual measurements of mass. We found a 0.46 ± 0.12 slope and a 0.25 ± 0.03 dex scatter measuring richness with a previously developed method. Second, we showed on a real sample of 250 0.06 < z < 0.9 clusters, most of which are at z < 0.3, with spectroscopic redshift that the colour of the red sequence allows us to measure the clusters’ redshift to better than Δz = 0.02. Third, we computed the predicted prior of the richness-mass scaling to forecast the capabilities of future wide-field-area surveys of galaxy clusters to constrain cosmological parameters. To this aim, we generated a simulated universe obeying the richness-mass scaling that we found. We observed it with a PanStarrs 1+Euclid-like survey, allowing for intrinsic scatter between mass and richness, for errors on mass, on richness, and for photometric redshift errors. We fitted the observations with an evolving five-parameter richness-mass scaling with parameters to be determined. Input parameters were recovered, but only if the cluster mass function and the weak-lensing redshift-dependent selection function were accounted for in the fitting of the mass-richness scaling. This emphasizes the limitations of often adopted simplifying assumptions, such as having a mass-complete redshift-independent sample. We derived the uncertainty and the covariance matrix of the (evolving) richness-mass scaling, which are the input ingredients of cosmological forecasts using cluster counts. We find that the richness-mass scaling parameters can be determined 10 5 times better than estimated in previous works that did not use weak-lensing mass estimates, although we emphasize that this high factor was derived with scaling relations with different parameterizations. The better knowledge of the scaling parameters likely has a strong impact on the relative importance of the different probes used to constrain cosmological parameters. The fitting code used for computing the predicted prior, including the treatment of the mass function and of the weak-lensing selection function, is provided in Appendix A. It can be re-used, for example, to derive the predicted prior of other observable-mass scalings, such as the LX-mass relation.
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paper_title: The absolute magnitudes of Type IA supernovae

paper_content:

Absolute magnitudes in the B, V, and I bands are derived for nine well-observed Type Ia supernovae using host galaxy distances estimated via the surface brightness fluctuations or Tully-Fisher methods. These data indicate that there is a significant intrinsic dispersion in the absolute magnitudes at maximum light of Type Ia supernovae, amounting to ±0.8 mag in B, ±0.6 mag in V, and ±0.5 mag in I. Moreover, the absolute magnitudes appear to be tightly correlated with the initial rate of decline of the B light curve, with the slope of the correlation being steepest in B and becoming progressively flatter in the V and I bands
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paper_title: Do X-ray dark or underluminous galaxy clusters exist?

paper_content:

We study the X-ray properties of a color-selected sample of clusters at 0.1 < z < 0.3, to quantify the real aboundance of the population of X-ray dark or underluminous clusters and at the same time the spurious detection contamination level of color-selected cluster catalogs. Starting from a local sample of color-selected clusters, we restrict our attention to those with sufficiently deep X-ray observations to probe their X-ray luminosity down to very faint values and without introducing any X-ray bias. This allowed us to have an X-ray- unbiased sample of 33 clusters to measure the LX-richness relation. Swift 1.4 Ms X-ray observations show that at least 89% of the color-detected clusters are real objects with a potential well deep enough to heat and retain an intracluster medium. The percentage rises to 94% when one includes the single spectroscopically confirmed color-selected cluster whose X-ray emission is not secured. Looking at our results from the opposite perspective, the percentage of X-ray dark clusters among color-selected clusters is very low: at most about 11 per cent (at 90% confidence). Supplementing our data with those from literature, we conclude that X-rayand color- cluster surveys sample the same population and consequently that in this regard we can safely use clusters selected with any of the two methods for cosmological purposes. This is an essential and promising piece of information for upcoming surveys in both the optical/IR (DES, EUCLID) and X-ray (eRosita). Richness correlates with X-ray luminosity with a large scatter, 0.51 ± 0.08 (0.44 ± 0.07) dex in lgLX at a given richness, when Lx is measured in a 500 (1070) kpc aperture. We release data and software to estimate the X-ray flux, or its upper limit, of a source with over-Poisson background fluctuations (found in this work to be ∼20% on cluster angular scales) and to fit X-ray luminosity vs richness if there is an intrinsic scatter. These Bayesian applications rigorously account for boundaries (e.g., the X-ray luminosity and the richness cannot be negative).
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paper_title: The scaling relation between richness and mass of galaxy clusters: a Bayesian approach

paper_content:

We use a sample of 53 galaxy clusters at 0.03<z<0.1 with available masses derived from the caustic technique and with velocity dispersions computed using 208 galaxies on average per cluster, in order to investigate the scaling between richness, mass and velocity dispersion. A tight scaling between richness and mass is found, with an intrinsic scatter of only 0.19 dex in mass and with a slope one, i.e. clusters which have twice as many galaxies are twice as massive. When richness is measured without any knowledge of the cluster mass or linked parameters (such as r200), it can predict mass with an uncertainty of 0.29+/-0.01 dex. As a mass proxy, richness competes favourably with both direct measurements of mass given by the caustic method, which has typically 0.14 dex errors (vs 0.29) and X-ray luminosity, which offers a similar 0.30 dex uncertainty. The similar performances of X-ray luminosity and richness in predicting cluster masses has been confirmed using cluster masses derived from velocity dispersion fixed by numerical simulations. These results suggest that cluster masses can be reliably estimated from simple galaxy counts, at least at the redshift and masses explored in this work. This has important applications in the estimation of cosmological parameters from optical cluster surveys, because in current surveys clusters detected in the optical range outnumber, by at least one order of magnitude, those detected in X-ray. Our analysis is robust from astrophysical and statistical perspectives. The data and code used for the stochastic computation is distributed with the paper. [Abridged]
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paper_title: The enrichment history of the intracluster medium: a Bayesian approach

paper_content:

This work measures the evolution of the iron content in galaxy clusters by a rigorous analysis of the data of 130 clusters at 0.1 < z < 1.3. This task is made difficult by a) the low signal-to-noise ratio of abundance measurements and the upper limits; b) possible selection effects; c) boundaries in the parameter space; d) non-Gaussian errors; e) the intrinsic variety of the objects studied; and f) abundance systematics. We introduce a Bayesian model to address all these issues at the same time, thus allowing cross-talk (covariance). On simulated data, the Bayesian fit recovers the input enrichment history, unlike in standard analysis. After accounting for a possible dependence on X-ray temperature, for metal abundance systematics, and for the intrinsic variety of studied objects, we found that the present-day metal content is not reached either at high or at low redshifts, but gradually over time: iron abundance increases by a factor 1.5 in the 7 Gyr sampled by the data. Therefore, feedback in metal abundance does not end at high redshift. Evolution is established with a moderate amount of evidence, 19 to 1 odds against faster or slower metal enrichment histories. We quantify, for the first time, the intrinsic spread in metal abundance, 18 ± 3%, after correcting for the effect of evolution, X-ray temperature, and metal abundance systematics. Finally, we also present an analytic approximation of the X-ray temperature and metal abundance likelihood functions, which are useful for other regression fitting involving these parameters. The data for the 130 clusters and code used for the stochastic computation are provided with the paper.
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Title: Measurement errors and scaling relations in astrophysics: a review
Section 1: Introduction
Description 1: Provide an overview of scaling relations in astrophysics, including their importance and common examples.

Section 2: Why astronomers regress
Description 2: Discuss the reasons why astronomers perform regression analyses, such as parameter estimation, prediction, and model selection.

Section 3: Heteroscedastic error structure
Description 3: Explain the complications introduced by heteroscedastic error structures in astronomical data and how to address them.

Section 4: Intrinsic scatter
Description 4: Elaborate on the concept of intrinsic scatter and differentiate it from measurement noise in regression models.

Section 5: Non-ignorable data collection and selection effects
Description 5: Discuss the challenges posed by non-random data collection and selection effects on the regression analyses.

Section 6: Data structure and non-uniform populations
Description 6: Describe how the structure of the data and the presence of non-uniform populations can lead to biases in regression parameter estimation.

Section 7: Non Gaussian data, outliers and other complications
Description 7: Address issues arising from non-Gaussian data, outliers, and other complications in regression models.

Section 8: Commonly used regression techniques in astronomy
Description 8: Provide an overview of various regression methods used in astronomy, including their strengths and weaknesses.

Section 9: Ordinary Least Squares regression
Description 9: Explain the Ordinary Least Squares (OLS) regression method and its application in astronomy.

Section 10: Weighted least squares fits
Description 10: Describe the weighted least squares regression method and scenarios where it is more appropriate than OLS.

Section 11: Maximum Likelihood Estimation
Description 11: Elaborate on the Maximum Likelihood Estimation (MLE) method and its advantages over OLS and weighted least squares.

Section 12: Robust estimation
Description 12: Introduce robust estimation methods to handle outliers and violations of parametric assumptions.

Section 13: Bivariate Correlated Errors and Intrinsic Scatter (BCES)
Description 13: Review the BCES method developed to handle correlated errors and intrinsic scatter in both axes of regression models.

Section 14: Astronomy Survival Analysis (ASURV)
Description 14: Discuss the ASURV method for handling censored data in regression analysis.

Section 15: Errors-in-variable Bayesian regression
Description 15: Review the Bayesian regression method for errors-in-variables models and its advantages in handling complicated data structures.

Section 16: Performance comparisons
Description 16: Compare the performance of different regression techniques in estimating regression parameters and prediction accuracy.

Section 17: Including more features of astronomical data in Bayesian regression
Description 17: Discuss how Bayesian regression can be adapted to include common features of astronomical data such as heteroscedastic errors, intrinsic scatter, and non-ignorable selection effects.

Section 18: Conclusions
Description 18: Summarize the key points discussed in the review and highlight the importance of Bayesian approaches in addressing complex regression problems in astrophysics.