This allows calculation of the posterior distribution of θ , P ( θ | b$_{1}$ , b$_{2}$ ), assuming uninformative priors for θ and β . Inference is based on posterior predictive p values where T $^{∗}$( θ ) is the p value associated with T (tan( θ )). Full mathematical details are given in the supplementary material of Wallace et al. [2012], but it is worth revisiting here the justification of a flat prior for θ . If β$_{1}$, β$_{2}$ were univariate with Gaussian priors and mean 0, then tan( θ ) = β$_{1}$ β$_{2}$ would have a Cauchy(0 , k ) prior where k is the ratio of the prior variances of β$_{1}$ and β$_{2}$ . Thus it seems an appropriate form to consider for the prior for θ in the multivariate case. k is unknown, but Wallace et al. [2012] found that varying k had a negligible effect on the posterior predictive p value for the sample sizes common in GWAS and eQTL studies (100s to 1000s of subjects) and we have set k = 1, implying a uniform prior for θ , for all analyses in this paper. Posterior predictive p values have a somewhat different interpretation than and appear conser- vative in comparison to standard p values [Rubin, 1984; Meng, Sep., 1994]. However, they avoid assuming the log-likelihood for η is approximately quadratic near its maximum which is not always the case. In practice, Wallace et al. [2012] found standard and posterior predictive p values to be almost identical in large samples. H prop 0 is not the same as H cond 0 as it does not explicitly condition on the most associated SNPs, but is a general property expected to be true of any pair of traits which share a common causal variant. While a shared causal variant should imply H prop 0 is true at any pair of SNPs, and that H cond 0 is true if k = k ′ is the causal variant, the reverse is not the case as it is possible that two traits have distinct causal variants in complete LD. Thus, failure to reject the null hypothesis indicates only that the data are consistent with a shared causal variant. Note that colocalisation testing may be applied equally to case control data (using logistic regression), expression data (using linear regression) or to compare case control results against expression results for a specific gene. Most commonly, this last approach might be applied in turn to all genes with a known eQTL signal in the neighbourhood of the disease association signal. However, it is assumed that cov( b$_{1}$ , b$_{2}$ ) = 1, meaning that case control studies may only be compared if they do not share a common control group. 6 ∫ T $^{∗}$( θ ) P ( θ | b$_{1}$ , b$_{2}$ ) d θ (3)