Data augmentation and Gibbs sampling are two closely related, sampling-based approaches to the calculation of posterior moments. The fact that each produces a sample whose constituents are neither independent nor identically distributed complicates the assessment of convergence and numerical accuracy of the approximations to the expected value of functions of interest under the posterior. In this paper methods from spectral analysis are used to evaluate numerical accuracy formally and construct diagnostics for convergence. These methods are illustrated in the normal linear model with informative priors, and in the Tobit-censored regression model.

Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments

We measure star formation rates (SFRs) of ≈50,000 optically selected galaxies in the local universe (z ≈ 0.1)—from gas-rich dwarfs to massive ellipticals. We obtain dust-corrected SFRs by fitting the GALEX (ultraviolet) and SDSS photometry to a library of dust-attenuated population synthesis models. For star-forming galaxies, our UV-based SFRs compare remarkably well with those from SDSS-measured emission lines (Hα). Deviations from perfect agreement are shown to be due to differences in the dust attenuation estimates. In contrast to Hα measurements, UV provides reliable SFRs for galaxies with weak Hα, and where Hα is contaminated with AGN emission (1/2 of the sample). Using full-SED SFRs, we calibrate a simple prescription that uses GALEX far- and near-UV magnitudes to produce dust-corrected SFRs for normal star-forming galaxies. The specific SFR is considered as a function of stellar mass for (1) star-forming galaxies with no AGNs, (2) those hosting an AGN, and (3) galaxies without Hα emission. We find that the three have distinct star formation histories, with AGNs lying intermediate between the star-forming and the quiescent galaxies. Star-forming galaxies without an AGN lie on a relatively narrow linear sequence. Remarkably, galaxies hosting a strong AGN appear to represent the massive continuation of this sequence. On the other hand, weak AGNs, while also massive, have lower SFRs, sometimes extending to the realm of quiescent galaxies. We propose an evolutionary sequence for massive galaxies that smoothly connects normal star-forming galaxies to quiescent galaxies via strong and weak AGNs. We confirm that some galaxies with no Hα show signs of star formation in the UV. We derive a cosmic star formation density at z = 0.1 with significantly smaller total error than previous measurements.

/pdf/uv-star-formation-rates-in-the-local-universe-4du2bu5i6w.pdf

UV Star Formation Rates in the Local Universe

We introduce the weighted likelihood bootstrap (WLB) as a way to simulate approximately from a posterior distribution This method is often easy to implement, requiring only an algorithm for calculating the maximum likelihood estimator, such as iteratively reweighted least squares In the generic weighting scheme, the WLB is first order correct under quite general conditions Inaccuracies can be removed by using the WLB as a source of samples in the sampling-importance resampling (SIR) algorithm, which also allows incorporation of particular prior information The SIR-adjusted WLB can be a competitive alternative to other integration methods in certain models Asymptotic expansions elucidate the second-order properties of the WLB, which is a generalization of Rubin's Bayesian bootstrap The calculation of approximate Bayes factors for model comparison is also considered We note that, given a sample simulated from the posterior distribution, the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values; a modification of this estimator that avoids instability is also noted These methods provide simple ways of calculating approximate Bayes factors and posterior model probabilities for a very wide class of models

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1994.tb01956.x

Approximate Bayesian-inference With the Weighted Likelihood Bootstrap

Simulation and the monte carlo method

Nonlinear mixed-effects models have received a great deal of attention in the statistical literature in recent years because of the flexibility they offer in handling the unbalanced repeated-measures data that arise in different areas of investigation, such as pharmacokinetics and economics. Several different methods for estimating the parameters in nonlinear mixed-effects model have been proposed. We concentrate here on two of them—maximum likelihood and restricted maximum likelihood. A rather complex numerical issue for (restricted) maximum likelihood estimation in nonlinear mixed-effects models is the evaluation of the log-likelihood function of the data, because it involves the evaluation of a multiple integral that, in most cases, does not have a closed-form expression. We consider here four different approximations to the log-likelihood, comparing their computational and statistical properties. We conclude that the linear mixed-effects (LME) approximation suggested by Lindstrom and Bates, t...

/pdf/approximations-to-the-log-likelihood-function-in-the-30ub6xsa3z.pdf

Approximations to the Log-Likelihood Function in the Nonlinear Mixed-Effects Model

1. Introductory statistical concepts 2. The discrete version of Bayes' theorem 3. Models with a single unknown parameter 4. The expected utility hypothesis and its alternatives 5. Models with several unknown parameters 6. Prior structures, posterior smoothing, and Bayes-Stein estimation Guide to worked examples Guide to self-study exercises.

Bayesian methods : an analysis for statisticians and interdisciplinary researchers

A flexible class of prior distributions is proposed, for the covariance matrix of a multivariate normal distribution, yielding much more general hierarchical and empirical Bayes smoothing and inference, when compared with a conjugate analysis involving an inverted Wishart distribution. A likelihood approximation is obtained for the matrix logarithm of the covariance matrix, via Bellman's iterative solution to a Volterra integral equation. Exact and approximate Bayesian, empirical and hierarchical Bayesian estimation and finite sample inference techniques are developed. Some risk and asymptotic frequency properties are investigated. A subset of the Project Talent American High School data is analyzed. Applications and extensions to multivariate analysis, including a generalized linear model for covariance matrices, are indicated.

https://projecteuclid.org/download/pdf_1/euclid.aos/1176348885

Bayesian Inference for a Covariance Matrix

Bayesian Methods in Finance

A method is proposed for approximating the marginal posterior density of a continuous function of several unknown parameters, thus permitting inferences about any parameter of interest for nonlinear models when the sample size is finite. Possibly tedious numerical integrations are replaced by conditional maximizations, which are shown to be quite accurate in a number of special cases. There are similarities with the profile likelihood ideas originated by Kalbfleisch and Sprott (1970), and the method is contrasted with a Laplacian approximation recommended by Kass, Tierney, and Kadane (1988, in press), referred to here as the “KTK procedure.” The methods are used to approximate the marginal posterior densities of the log-linear interaction effects and an overall measure of association in a two-way contingency table. Snee's (1974) hair/eye color data are reanalyzed, and adjustments are proposed to Goodman's (1964) analysis for the full-rank interaction model. Another application concerns marginaliz...

Bayesian Marginal Inference

An exact Bayesian analysis can be performed for normal theory variance components models, using importance sampling. But remarkably accurate approximations are available using the Laplacian T-approximation introduced by Leonard, Hsu, and Ritter. Instead of maximizing the joint posterior density, conditional upon the parameter of interest, a device described by O'Hagan is used first. The Bayesian estimators are compared to the Lindley—Stein shrinkage estimators and the Lindley—Smith joint modal estimators. It is confirmed that joint modes can overcollapse toward prior hypotheses, when compared with more sensible Bayesian procedures. This is referred to as a “collapsing phenomenon.” A numerical example from the one-way random-effects model is considered, and the risks of the different estimators are simulated under a variety of loss functions. It is concluded that although the Lindley—Stein estimator performs well, a full hierarchical Bayesian analysis performs at least equally well, while permitti...

John S. J. Hsu

Papers

Bayesian methods : an analysis for statisticians and interdisciplinary researchers

Bayesian Inference for a Covariance Matrix

Bayesian Methods in Finance

Bayesian Marginal Inference

Bayesian Methods for Variance Component Models