  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.
