The AstroStat Slog

In the past month, I’ve noticed relatively frequent paper appearance in arxiv/astro-ph whose title includes Bayesian or Markov Chain Monte Carlo (MCMC). Those papers are:

• [astro-ph:0709.1058v1] Joint Bayesian Component Separation and CMB Power Spectrum Estimation by H.K.Eriksen et. al.
• [astro-ph:0709.1104v1] Monolithic or hierarchical star formation? A new statistical analysis by M. Kampakoglou, R. Trotta, and J. Silk
• [astro-ph:0411573v2] A Bayesian analysis of the primordial power spectrum by M.Bridges, A.N.Lasenby, M.P.Hobson
• [astro-ph:0709.0596v1] Bayesian inversion of Stokes profiles by A. A. Ramos, M.J.M. Gonzales, and J.A. Rubino-Martin
• [astro-ph:0709.0711v1] Bayesian posterior classification of planetary nebulae according to the Peimbert types by C. Quireza, H.J.Rocha-Pinto, and W.J. Maciel
• [astro-ph:0708.2340v1] Bayesian Galaxy Shape Measurement for Weak Lensing Surveys -I. Methodology and a Fast Fitting Algorithm by L. Miller et. al.
• [astro-ph:0708.1871v1] Dark energy and cosmic curvature: Monte-Carlo Markov Chain approach by Y. Gong et. al.

in addition to the previous posting in this slog, [ArXiv] Bayesian Star Formation Study.

Without knowing astrophysics in each paper, no intention exists to criticize/comment/summarize their Bayesian approaches to astrophysical research. I’d rather see these papers and other papers titled with Bayesian that I met at arxiv from a how-to-practice-Bayesian-statistic viewpoint. Naively speaking, astronomical papers discussing Bayesian analysis mainly serve as Bayesian analysis tutorials in the astronomical subfields of authors’ expertise. These papers introduce Bayesian analysis tools (Dark energy and cosmic curvature: Monte-Carlo Markov Chain approach introduced a MCMC tool, COSMOMC by Lewi and Bridle^[1] ). They explain basic but fundamental Bayesian ideas such as Bayes rule, likelihoods, priors, posterior, marginalization, Bayes factor, and Bayesian evidence. They describe well known algorithms such as gibbs sampler, Metropolis-Hasting algorithm, Metropolis algorithms, important sampling, nested sampling, and so forth, and their applications in the astronomical data analysis. They compare Bayesian to classical (frequentists) statistics and discuss the advantages of Bayesian analysis such as getting credible regions from posterior distributions corresponding frequentists’ confidence intervals. Many papers tried to make an asymptotic connection from Bayesian analysis to the о‡^2 method or incorporate the о‡^2 function into the likelihood function. As in Joint Bayesian Component Separation and CMB Power Spectrum Estimation, the о‡^2 method, non-Bayesian method, sometimes has been inserted to get constraints on the parameter of interest. Overall, these papers are nice tutorials and reading one or two helps readers to apply Bayesian methodologies to similar type data sets as illustrated in the papers for acquiring parameter distributions (joint distributions on multiple parameters frequently appeared).

Not many astronomical papers actually introduce new Bayesian analysis methods, statistical theories, and the stochastic properties of new MCMC algorithms; therefore, don’t be panic because of the title including Bayesian. Since Bayesian analysis involves choosing the likelihood function and priors from parametric distribution families, to a large extent, a priori knowledge on data, i.e. experience in physical data generating models, instruments effects, and suitable priors/likelihoods selection is most adequate for Bayesian analysis even without MCMC theories. Bayesian analysis appreciates understandings of astronomical data more than measure theory, where astronomers are able to perform statistical analysis better than statisticians. Generally, in terms of getting a parameter estimate and its error, Bayesian method is easier to be interpreted than the conventional о‡^2 method.

Surely, frequent and practical applications of Bayesian methods will help Bayesian statisticians to develop more practical Bayesian methodologies for general applications.

1. Cosmological parameters from CMB and other data: A Monte Carlo approach, 2002, Phys. Rev. Letter, 66(10), p.3511[↩]