with a given probability (typically 90% or 99%). Initial studies approximated the posterior probabilities of causality under a single causal variant assumption. Under this assumption, posterior probabilities of causality can be estimated from z-scores without the need for LD information45; this approach is both practical and computationally efficient. More recent studies have computed posterior probabilities of causality under a multiple causal variant assumption46. As in the case of imputation using summary statistics, the likelihoods of the observed z-scores can be computed based on the multi-variate normal (MVN) distribution with variance equal to the LD correlation matrix, with LD estimated from population reference panels using regularization techniques. Unlike imputation using summary statistics, which uses the null model of no association (i.e. a mean of 0 in the MVN), in fine-mapping the mean is a function of causal effect sizes, which can be heuristically approximated or integrated out using conjugate priors46,47. These methods often restrict computations to a maximum number of causal variants (e.g. 3 or 6); more recent studies have shown that further speed-ups can be achieved through matrix factorizations48 or stochastic search49. Methods that model multiple causal variants generally improve the accuracy (and calibration) of credible sets at loci with
