A Robust Bayesian Meta-Analysis for Estimating the Hubble Constant via Time Delay Cosmography

We propose a Bayesian meta-analysis to infer the current expansion rate of the Universe, called the Hubble constant ($H_0$), via time delay cosmography. Inputs of the meta-analysis are estimates of two properties for each pair of gravitationally lensed images; time delay and Fermat potential difference estimates with their standard errors. A meta-analysis can be appealing in practice because obtaining each estimate from even a single lens system involves substantial human efforts, and thus estimates are often separately obtained and published. This work focuses on combining these estimates from independent studies to infer $H_0$ in a robust manner. For this purpose, we adopt Student's $t$ error for the inputs of the meta-analysis. We investigate properties of the resulting $H_0$ estimate via two simulation studies with realistic imaging data. It turns out that the meta-analysis can infer $H_0$ with sub-percent bias and about 1 percent level of coefficient of variation, even when 30 percent of inputs are manipulated to be outliers. We also apply the meta-analysis to three gravitationally lensed systems, and estimate $H_0$ by $75.632 \pm 6.918$ (km/second/Mpc), which covers a wide range of $H_0$ estimates obtained under different physical processes. An R package, h0, is publicly available for fitting the proposed meta-analysis

A Repelling-Attracting Metropolis Algorithm for Multimodality

Although the Metropolis algorithm is simple to implement, it often has difficulties exploring multimodal distributions. We propose the repelling-attracting Metropolis (RAM) algorithm that maintains the simple-to-implement nature of the Metropolis algorithm, but is more likely to jump between modes. The RAM algorithm is a Metropolis-Hastings algorithm with a proposal that consists of a downhill move in density that aims to make local modes repelling, followed by an uphill move in density that aims to make local modes attracting. The downhill move is achieved via a reciprocal Metropolis ratio so that the algorithm prefers downward movement. The uphill move does the opposite using the standard Metropolis ratio which prefers upward movement. This down-up movement in density increases the probability of a proposed move to a different mode. Because the acceptance probability of the proposal involves a ratio of intractable integrals, we introduce an auxiliary variable which creates a term in the acceptance probability that cancels with the intractable ratio. Using several examples, we demonstrate the potential for the RAM algorithm to explore a multimodal distribution more efficiently than a Metropolis algorithm and with less tuning than is commonly required by tempering-based methods

Rgbp: An R Package for Gaussian, Poisson, and Binomial Random Effects Models with Frequency Coverage Evaluations

Rgbp is an R package that provides estimates and verifiable confidence intervals for random effects in two-level conjugate hierarchical models for overdispersed Gaussian, Poisson, and binomial data. Rgbp models aggregate data from k independent groups summarized by observed sufficient statistics for each random effect, such as sample means, possibly with covariates. Rgbp uses approximate Bayesian machinery with unique improper priors for the hyper-parameters, which leads to good repeated sampling coverage properties for random effects. A special feature of Rgbp is an option that generates synthetic data sets to check whether the interval estimates for random effects actually meet the nominal confidence levels. Additionally, Rgbp provides inference statistics for the hyper-parameters, e.g., regression coefficients

TD-CARMA: Painless, accurate, and scalable estimates of gravitational-lens time delays with flexible CARMA processes

Cosmological parameters encoding our current understanding of the expansion history of the Universe can be constrained by the accurate estimation of time delays arising in gravitationally lensed systems. We propose TD-CARMA, a Bayesian method to estimate cosmological time delays by modelling the observed and irregularly sampled light curves as realizations of a Continuous Auto-Regressive Moving Average (CARMA) process. Our model accounts for heteroskedastic measurement errors and microlensing, an additional source of independent extrinsic long-term variability in the source brightness. The CARMA formulation admits a linear state-space representation, that allows for efficient and scalable likelihood computation using the Kalman Filter. We obtain a sample from the joint posterior distribution of the model parameters using a nested sampling approach. This allows for "painless" Bayesian Computation, dealing with the expected multi-modality of the posterior distribution in a straightforward manner and not requiring the specification of starting values or an initial guess for the time delay, unlike existing methods. In addition, the proposed sampling procedure automatically evaluates the Bayesian evidence, allowing us to perform principled Bayesian model selection. TD-CARMA is parsimonious, and typically includes no more than a dozen unknown parameters. We apply TD-CARMA to three doubly lensed quasars HS 2209+1914, SDSS J1001+5027 and SDSS J1206+4332, estimating their time delays as $-21.96 \pm 1.448$ (6.6$\%$ precision), $120.93 \pm 1.015$ (0.8$\%$), and $111.51 \pm 1.452$ (1.3$\%$), respectively. A python package, TD-CARMA, is publicly available to implement the proposed method