tokens: the position angle and distance modulus measurements with no further adjustment. In the Appendix, we discuss the validity of the approximation used in equation ( 8 ) by comparison to models which explicitly employ the posterior samples from Martin etal. ( 2016c ). We find that the posterior distributions on the model parameters are very similar when using the posterior samples compared to the approximation employed throughout the paper. Additionally, in the absence of similar sets of posterior samples for the other M31 dSphs and all of the MW dSphs, we are forced to employ an approximation. We now detail each of the models employed. The first model assumes the dSphs are randomly aligned with respect to the observer, so does not utilize the position angle information. The second model assumes the major axis of the dSphs are preferentially aligned with the galactocentric radial vector. Our final model assumes the major axes of the dSphs preferentially lie in a plane. 3.2 Random alignment We attempt to infer the distribution of T and E given the observed shapes and associated uncertainties under the assumption that there is no preferred orientation for the dSphs. Implicitly, this assumes the distribution of observed position angles is uniform. Our model is described as where is the modified normal distribution to account for the asymmetric uncertainties, and subscript i corresponds to the i th of N dSphs. is a uniform distribution between a and b and is a Cauchy distribution centred on a with scale parameter b . From dark-matter-only simulations, the triaxiality and ellipticity distribution are well approximated by Gaussians (Allgood etal. 2006 ; Knebe etal. 2008a ). The natural choice for the prior on a scale parameter would be a Jeffreys prior of 1/. Instead, we heed the advice of Gelman ( 2006 ) and adopt a half-Cauchy distribution as a weakly informative prior. Very large values of are disfavoured, whilst very small values are not strongly favoured. This matches nicely our expectation that the intrinsic triaxiality and ellipticity distributions will not be -function spikes nor will they lack any structure at all. The choice of the scale as unity is inspired by the finite range of both the triaxiality and the ellipticity. One complication is that some of our variables have a discrete range so the pdf that describe them must be normalized over this range. In the case of the uniformly distributed variables, the normalization is a constant factor but for the normally distributed variables (i.e. T i and E i ) the normalization depends on the parameters and that we are attempting to constrain. Therefore, any distributions f ( x ) for a variable x constrained to lie in the interval ( a, b ) that are quoted in this paper implicitly contain a factor of N 1 where We additionally introduce the pdf given by For clarity, the log-likelihood for this model is given by where p (cos, , T, E ) is a -function related to equation ( 2 ). 3.2.1 Results The
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