1,150 research outputs found
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Fully Bayesian inference for α-stable distributions using a Poisson series representation
In this paper we develop an approach to Bayesian Monte Carlo inference for skewed α-stable distributions. Based on a series representation of the stable law in terms of infinite summations of random Poisson process arrival times, our framework leads to a simple representation in terms of conditionally Gaussian distributions for certain latent variables. Inference can therefore be carried out straightforwardly using techniques such as auxiliary variables versions of Markov chain Monte Carlo (MCMC) methods. The Poisson series representation (PSR) is further extended to practical application by introducing an approximation of the series residual terms based on exact moment calculations. Simulations illustrate the proposed framework applied to skewed α-stable simulated and real-world data, successfully estimating the distribution parameter values and being consistent with other (non-Bayesian) approaches. The methods are highly suitable for incorporation into hierarchical Bayesian models, and in this case the conditionally Gaussian structure of our model will lead to very efficient computations compared to other approaches.Godsill acknowledges partial funding for the work from the EPSRC BTaRoT project EP/K020153/1, and Tatjana Lemke acknowledges PhD funding from Fraunhofer ITWM, Kaiserslautern.This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.dsp.2015.08.01
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Simulated convergence rates with application to an intractable α-stable inference problem
© 2017 IEEE. We report the results of a series of numerical studies examining the convergence rate for some approximate representations of α-stable distributions, which are a highly intractable class of distributions for inference purposes. Our proposed representation turns the intractable inference for an infinite-dimensional series of parameters into an (approximately) conditionally Gaussian representation, to which standard inference procedures such as Expectation-Maximization (EM), Markov chain Monte Carlo (MCMC) and Particle Filtering can be readily applied. While we have previously proved the asymptotic convergence of this representation, here we study the rate of this convergence for finite values of a truncation parameter, c. This allows the selection of appropriate truncations for different parameter configurations and for the accuracy required for the model. The convergence is examined directly in terms of cumulative distribution functions and densities, through the application of the Berry theorems and Parseval theorems. Our results indicate that the behaviour of our representations is significantly superior to that of representations that simply truncate the series with no Gaussian residual term
Pseudo-Marginal MCMC for Parameter Estimation in α-Stable Distributions
The α-stable distribution is very useful for modelling data with extreme values and skewed behaviour. The distribution is governed by two key parameters, tail thickness and skewness, in addition to scale and location. Inferring these parameters is difficult due to the lack of a closed form expression of the probability density. We develop a Bayesian method, based on the pseudo-marginal MCMC approach, that requires only unbiased estimates of the intractable likelihood. To compute these estimates we build an adaptive importance sampler for a latentvariable-representation of the α-stable density. This representation has previously been used in the literature for conditional MCMC sampling of the parameters, and we compare our method with this approach.This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.ifacol.2015.12.17
Sharp Gaussian Approximation Bounds for Linear Systems with α-stable Noise
We report the results of several theoretical
studies into the convergence rate for certain random series
representations of α-stable random variables, which are
motivated by and find application in modelling heavy-tailed
noise in time series analysis, inference, and stochastic processes.
The use of α-stable noise distributions generally leads
to analytically intractable inference problems. The particular
version of the Poisson series representation invoked here
implies that the resulting distributions are “conditionally
Gaussian,” for which inference is relatively straightforward,
although an infinite series is still involved. Our approach is
to approximate the residual (or “tail”) part of the series from
some point, c > 0, say, to ∞, as a Gaussian random variable.
Empirically, this approximation has been found to be very
accurate for large c. We study the rate of convergence, as
c → ∞, of this Gaussian approximation. This allows the
selection of appropriate truncation parameters, so that a
desired level of accuracy for the approximate model can be
achieved. Explicit, nonasymptotic bounds are obtained for
the Kolmogorov distance between the relevant distribution
functions, through the application of probability-theoretic
tools. The theoretical results obtained are found to be in very
close agreement with numerical results obtained in earlier
work
DPpackage: Bayesian Semi- and Nonparametric Modeling in R
Data analysis sometimes requires the relaxation of parametric assumptions in order to gain modeling flexibility and robustness against mis-specification of the probability model. In the Bayesian context, this is accomplished by placing a prior distribution on a function space, such as the space of all probability distributions or the space of all regression functions. Unfortunately, posterior distributions ranging over function spaces are highly complex and hence sampling methods play a key role. This paper provides an introduction to a simple, yet comprehensive, set of programs for the implementation of some Bayesian nonparametric and semiparametric models in R, DPpackage. Currently, DPpackage includes models for marginal and conditional density estimation, receiver operating characteristic curve analysis, interval-censored data, binary regression data, item response data, longitudinal and clustered data using generalized linear mixed models, and regression data using generalized additive models. The package also contains functions to compute pseudo-Bayes factors for model comparison and for eliciting the precision parameter of the Dirichlet process prior, and a general purpose Metropolis sampling algorithm. To maximize computational efficiency, the actual sampling for each model is carried out using compiled C, C++ or Fortran code.
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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 -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 -stable model is used.
In the first part of the thesis, a marginal representation of the -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 -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 Essen’s smoothing lemma. This analysis
allows for the selection of appropriate truncations for different -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 -stable
Lvy processes to the multivariate case is presented. Stable Lvy 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
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