The Efficiency of Next-Generation Gibbs-Type Samplers: An Illustration Using a Hierarchical Model in Cosmology
Supernovae occur when a star’s life ends in a violent thermonuclear explosion, briefly outshining an entire galaxy before fading from view over a period of weeks or months. Because so-called Type Ia supernovae occur only in a particular physical scenario,
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Abstract Supernovae occur when a star’s life ends in a violent thermonuclear explosion, briefly outshining an entire galaxy before fading from view over a period of weeks or months. Because so-called Type Ia supernovae occur only in a particular physical scenario, their explosions have similar intrinsic brightnesses which allows us to accurately estimate their distances. This in turn allows us to constrain the parameters of cosmological models that characterize the expansion history of the universe. In this paper, we show how a cosmological model can be embedded into a Gaussian hierarchical model and fit using observations of Type Ia supernovae. The overall model is an ideal testing ground of new computational methods. AncillaritySufficiency Interweaving Strategy (ASIS) and Partially Collapsed Gibbs (PCG) are effective tools to improve the convergence of Gibbs samplers. Besides using either of them alone, we can combine PCG and/or ASIS along with Metropolis-Hastings algorithm to simplify implementation and further improve convergence. We use four samplers to draw from the posterior distribution of the cosmological hierarchical model, and confirm the efficiency of both PCG and ASIS. Furthermore, we find that we can gain more efficiency by combining two or more strategies into one sampler.
1 Introduction The Physics Nobel Prize (2011) was awarded for the discovery that the expansion of the universe is accelerating, a phenomenon attributed to the existence of “dark energy”. Type Ia supernova (SN) observations have been instrumental in this discovery and remain an important tool to quantify the characteristics of dark energy (March et al. 2011). Although details remain unclear, it is thought that a Type Ia SN occurs when a compact, carbon-oxygen white dwarf star accumulates extra
X. Jiao () • D.A. van Dyk Statistics Section, Imperial College, London, UK e-mail: [email protected]; [email protected] R. Trotta • H. Shariff Astrophysics Group, Imperial College, London, UK © Springer International Publishing Switzerland 2016 Z. Jin et al. (eds.), New Developments in Statistical Modeling, Inference and Application, ICSA Book Series in Statistics, DOI 10.1007/978-3-319-42571-9_9
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material until its mass approaches a critical threshold (“Chandrasekhar threshold”: 1.44 Mˇ , where Mˇ is the mass of the sun). Because of their common formation mechanism, all Type Ia SNe have similar absolute luminosity (which is measured in absolute magnitudes, i.e., the negative logarithm of flux). This means that their distance can be estimated from their apparent magnitude (i.e., their brightness as viewed from earth). We can also directly measure their redshift due to the expansion of the universe, a stretching of the wavelength of light emanating from objects moving away from us. The underlying cosmological models of interest predict the relationship between redshift and the difference between apparent and absolute magnitudes, called “distance modulus”. We embed the cosmological model into a Gaussian
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