$\alpha$-stable distributions are utilised as models for heavy-tailed noise
in many areas of statistics, finance and signal processing engineering.

However, in general, neither univariate nor multivariate $\alpha$-stable
models admit closed form densities which can be evaluated pointwise. This
complicates the inferential procedure.

As a result, $\alpha$-stable models are practically limited to the univariate
setting under the Bayesian paradigm, and to bivariate models under the
classical framework.

In this article we develop a novel Bayesian approach to modelling univariate
and multivariate $\alpha$-stable distributions based on recent advances in
"likelihood-free" inference.

We present an evaluation of the performance of this procedure in 1, 2 and 3
dimensions, and provide an analysis of real daily currency exchange rate data.
The proposed approach provides a feasible inferential methodology at a moderate
computational cost.