### The intrinsic shapes of prestellar cores

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Using observations of prestellar cores to infer their intrinsic properties requires the solution of several poorly constrained inverse problems. Here we address one of these problems, namely to deduce from the observed aspect ratios of prestellar cores their intrinsic three-dimensional shapes. Four models are proposed, all based on the standard assumption that prestellar cores are ellipsoidal, and on the further assumption that a core's shape is not correlated with its absolute size. The first and simplest model, M1, has a single free parameter, and assumes that the relative axes of a prestellar core are drawn randomly from a log-normal distribution with zero mean and standard deviation \tauO. The second model, M2a, has two free parameters, and assumes that the log-normal distribution (with standard deviation \tauO) has a finite mean, \nuO, defined so that \nuO<0 means elongated (or filamentary) cores are favoured, whereas \nuO>0 means flattened (or disc-like) cores are favoured. Details of the third model (M2b, two free parameters) and the fourth model (M4, four free parameters) are given in the text. Markov chain Monte Carlo sampling and Bayesian analysis are used to map out the posterior probability density functions of the model paramaters, and the relative merits of the models are compared using Bayes factors. We show that M1 provides an acceptable fit to the data with \tauO=0.57+/-0.06; and that, although the other models sometimes provide an improved fit, there is no strong justification for the introduction of their additional parameters.