On Latent Variable Models for Bayesian Inference with Stable Distributions and Processes

Extreme values and skewness in time-series are often observed in engineering, financial and biological applications. This thesis is a study motivated by the need of efficient and reliable Bayesian inference methods when the $\alpha$-stable model is selected to represent such data. The class of stable distributions is the limit of the generalized central limit theorem (CLT), having a key role in representing phenomena that can be thought of as the sum of many perturbations, with potentially unbounded variance. Besides the ability to model heavy-tailedness, another consequence of the generalized CLT is a further degree of freedom of stable distributions, namely their potential skewness. However, stable distributions are, at the same time, highly intractable for inference purposes. Several approximate methods are available in the literature, in both the frequentist and Bayesian paradigms, but they suffer from a number of deficiencies, the greatest of which is the lack of quantification of the approximation in place. This thesis proposes Bayesian inference schemes for two different latent variable models, with the aim of providing guarantees of accuracy when the $\alpha$-stable model is used. In the first part of the thesis, a marginal representation of the $\alpha$-stable density is used to develop a novel, asymptotically exact, Bayesian method for parameter inference. This is based on the pseudo-marginal Markov chain Monte Carlo (MCMC) approach, that requires only unbiased estimates of the intractable likelihood, computed through adaptive importance sampling for the marginal representation. The results obtained are comparable to a state of the art conditional Gibbs sampler, but do not introduce any approximation, while allowing for better control of the quality of the inference. The focus of the second and central part of the thesis is the Poisson series representation (PSR) of $\alpha$-stable random variables. An approach that turns the infinite-dimensional PSR into an approximately conditionally Gaussian representation, by means of Gaussian approximation of the residual of the series, has been presented in previous literature, together with inference procedures such as MCMC and Particle Filtering. In this setting, the first contribution of this dissertation is the formulation of a CLT for the PSR residual, which serves to justify the existing approximation. Moreover, numerical and theoretical results on the rate of convergence for finite values of the truncation parameter are presented. The convergence is examined directly in terms of Kolmogorov distance between distribution functions, through the application of probability theoretic results, such as the Ess$\acute e$en’s smoothing lemma. This analysis allows for the selection of appropriate truncations for different $\alpha$-stable parameter configurations and gives theoretical guarantees on the accuracy achieved when using the PSR model. Furthermore, superior behaviour of the proposed approximation is found, compared to the simple series truncation, justifying its use for inference tasks. In the third and final part of this thesis, an extension of the modified Poisson series representation (MPSR) of linear continuous-time models driven by $\alpha$-stable L$\acute e$vy processes to the multivariate case is presented. Stable L$\acute e$vy processes are suitable to model jumps and discontinuities in the state, while possessing the self-similarity property, which makes these processes a very natural class for the driving noise in continuous time models. A scheme for approximate simulation from the multivariate linear models, namely multivariate stable vectors evolving in time, is presented. While stable random vectors are parametrized by a function, the presented approximate approach involves only finite dimensional parameters. This will facilitate inference methods, to be developed in future work, towards which the proposed simulation methods constitute the foundational work