Variational Inference as an alternative to MCMC for parameter estimation and model selection

Most applications of Bayesian Inference for parameter estimation and model selection in astrophysics involve the use of Monte Carlo techniques such as Markov Chain Monte Carlo (MCMC) and nested sampling. However, these techniques are time consuming and their convergence to the posterior could be difficult to determine. In this work, we advocate Variational inference as an alternative to solve the above problems, and demonstrate its usefulness for parameter estimation and model selection in Astrophysics. Variational inference converts the inference problem into an optimization problem by approximating the posterior from a known family of distributions and using Kullback-Leibler divergence to characterize the difference. It takes advantage of fast optimization techniques, which make it ideal to deal with large datasets and makes it trivial to parallelize on a multicore platform. We also derive a new approximate evidence estimation based on variational posterior, and importance sampling technique called posterior weighted importance sampling for the calculation of evidence (PWISE), which is useful to perform Bayesian model selection. As a proof of principle, we apply variational inference to five different problems in astrophysics, where Monte Carlo techniques were previously used. These include assessment of significance of annual modulation in the COSINE-100 dark matter experiment, measuring exoplanet orbital parameters from radial velocity data, tests of periodicities in measurements of Newton's constant $G$, assessing the significance of a turnover in the spectral lag data of GRB 160625B and estimating the mass of a galaxy cluster using weak gravitational lensing. We find that variational inference is much faster than MCMC and nested sampling techniques for most of these problems while providing competitive results. All our analysis codes have been made publicly available.