  Importance sampling is used to approximate Bayes' rule in many computational
approaches to Bayesian inverse problems, data assimilation and machine
learning. This paper reviews and further investigates the required sample size
for importance sampling in terms of the $\chi^2$-divergence between target and
proposal. We develop general abstract theory and illustrate through numerous
examples the roles that dimension, noise-level and other model parameters play
in approximating the Bayesian update with importance sampling. Our examples
also facilitate a new direct comparison of standard and optimal proposals for
particle filtering.
