Augmentation schemes for particle MCMC

Particle MCMC involves using a particle filter within an MCMC algorithm. For inference of a model which involves an unobserved stochastic process, the standard implementation uses the particle filter to propose new values for the stochastic process, and MCMC moves to propose new values for the parameters. We show how particle MCMC can be generalised beyond this. Our key idea is to introduce new latent variables. We then use the MCMC moves to update the latent variables, and the particle filter to propose new values for the parameters and stochastic process given the latent variables. A generic way of defining these latent variables is to model them as pseudo-observations of the parameters or of the stochastic process. By choosing the amount of information these latent variables have about the parameters and the stochastic process we can often improve the mixing of the particle MCMC algorithm by trading off the Monte Carlo error of the particle filter and the mixing of the MCMC moves. We show that using pseudo-observations within particle MCMC can improve its efficiency in certain scenarios: dealing with initialisation problems of the particle filter; speeding up the mixing of particle Gibbs when there is strong dependence between the parameters and the stochastic process; and enabling further MCMC steps to be used within the particle filter.     

Markov chain Monte Carlo algorithms with sequential proposals

We explore a general framework in Markov chain Monte Carlo (MCMC) sampling where sequential proposals are tried as a candidate for the next state of the Markov chain. This sequential-proposal framework can be applied to various existing MCMC methods, including Metropolis–Hastings algorithms using random proposals and methods that use deterministic proposals such as Hamiltonian Monte Carlo (HMC) or the bouncy particle sampler. Sequential-proposal MCMC methods construct the same Markov chains as those constructed by the delayed rejection method under certain circumstances. In the context of HMC, the sequential-proposal approach has been proposed as extra chance generalized hybrid Monte Carlo (XCGHMC). We develop two novel methods in which the trajectories leading to proposals in HMC are automatically tuned to avoid doubling back, as in the No-U-Turn sampler (NUTS). The numerical efficiency of these new methods compare favorably to the NUTS. We additionally show that the sequential-proposal bouncy particle sampler enables the constructed Markov chain to pass through regions of low target density and thus facilitates better mixing of the chain when the target density is multimodal.

     

Bayesian inference for generalized extreme value distributions via Hamiltonian Monte Carlo

In this article, we propose to evaluate and compare Markov chain Monte Carlo (MCMC) methods to estimate the parameters in a generalized extreme value model. We employed the Bayesian approach using traditional Metropolis-Hastings methods, Hamiltonian Monte Carlo (HMC), and Riemann manifold HMC (RMHMC) methods to obtain the approximations to the posterior marginal distributions of interest. Applications to real datasets and simulation studies provide evidence that the extra analytical work involved in Hamiltonian Monte Carlo algorithms is compensated by a more efficient exploration of the parameter space.     

Particle learning for Bayesian semi-parametric stochastic volatility model

This article designs a Sequential Monte Carlo (SMC) algorithm for estimation of Bayesian semi-parametric Stochastic Volatility model for financial data. In particular, it makes use of one of the most recent particle filters called Particle Learning (PL). SMC methods are especially well suited for state-space models and can be seen as a cost-efficient alternative to Markov Chain Monte Carlo (MCMC), since they allow for online type inference. The posterior distributions are updated as new data is observed, which is exceedingly costly using MCMC. Also, PL allows for consistent online model comparison using sequential predictive log Bayes factors. A simulated data is used in order to compare the posterior outputs for the PL and MCMC schemes, which are shown to be almost identical. Finally, a short real data application is included.     

Value at risk estimation under stochastic volatility models using adaptive PMCMC methods

In this paper, we propose a value-at-risk (VaR) estimation technique based on a new stochastic volatility model with leverage effect, nonconstant conditional mean and jump. In order to estimate the model parameters and latent state variables, we integrate the particle filter and adaptive Markov Chain Monte Carlo (MCMC) algorithms to develop a novel adaptive particle MCMC (A-PMCMC) algorithm. Comprehensive simulation experiments based on three stock indices and two foreign exchange time series show effectiveness of the proposed A-PMCMC algorithm and the VaR estimation technique.     

Inference for a Class of Stochastic Volatility Models Using Option and Spot Prices: Application of a Bivariate Kalman Filter

In this paper Bayesian methods are applied to a stochastic volatility model using both the prices of the asset and the prices of options written on the asset. Posterior densities for all model parameters, latent volatilities and the market price of volatility risk are produced via a Markov Chain Monte Carlo (MCMC) sampling algorithm. Candidate draws for the unobserved volatilities are obtained in blocks by applying the Kalman filter and simulation smoother to a linearization of a nonlinear state space representation of the model. Crucially, information from both the spot and option prices affects the draws via the specification of a bivariate measurement equation, with implied Black–Scholes volatilities used to proxy observed option prices in the candidate model. Alternative models nested within the Heston (1993) framework are ranked via posterior odds ratios, as well as via fit, predictive and hedging performance. The method is illustrated using Australian News Corporation spot and option price data.     

Regression models for time-varying extremes

A common approach to modelling extreme data are to consider the distribution of the exceedance value over a high threshold. This approach is based on the distribution of excess, which follows the generalized Pareto distribution (GPD) and has shown to be adequate for this type of situation. As with all data involving analysis in time, excesses above a threshold may also vary and suffer from the influence of covariates. Thus, the GPD distribution can be modelled by entering the presence of these factors. This paper presents a new model for extreme values, where GPD parameters are written on the basis of a dynamic regression model. The estimation of the model parameters is made under the Bayesian paradigm, with sampling points via MCMC. As with environmental data, behaviour data are related to other factors such as time and covariates such as latitude and distance from the sea. Simulation studies have shown the efficiency and identifiability of the model, and applying real rain data from the state of Piaui, Brazil, shows the advantage in predicting and interpreting the model against other similar models proposed in the literature.     

Efficient $$\hbox {SMC}^2$$ schemes for stochastic kinetic models

Fitting stochastic kinetic models represented by Markov jump processes within the Bayesian paradigm is complicated by the intractability of the observed-data likelihood. There has therefore been considerable attention given to the design of pseudo-marginal Markov chain Monte Carlo algorithms for such models. However, these methods are typically computationally intensive, often require careful tuning and must be restarted from scratch upon receipt of new observations. Sequential Monte Carlo (SMC) methods on the other hand aim to efficiently reuse posterior samples at each time point. Despite their appeal, applying SMC schemes in scenarios with both dynamic states and static parameters is made difficult by the problem of particle degeneracy. A principled approach for overcoming this problem is to move each parameter particle through a Metropolis-Hastings kernel that leaves the target invariant. This rejuvenation step is key to a recently proposed \(\hbox {SMC}^2\) algorithm, which can be seen as the pseudo-marginal analogue of an idealised scheme known as iterated batch importance sampling. Computing the parameter weights in \(\hbox {SMC}^2\) requires running a particle filter over dynamic states to unbiasedly estimate the intractable observed-data likelihood up to the current time point. In this paper, we propose to use an auxiliary particle filter inside the \(\hbox {SMC}^2\) scheme. Our method uses two recently proposed constructs for sampling conditioned jump processes, and we find that the resulting inference schemes typically require fewer state particles than when using a simple bootstrap filter. Using two applications, we compare the performance of the proposed approach with various competing methods, including two global MCMC schemes.     

Inference for a Class of Stochastic Volatility Models Using Option and Spot Prices: Application of a Bivariate Kalman Filter

In this paper Bayesian methods are applied to a stochastic volatility model using both the prices of the asset and the prices of options written on the asset. Posterior densities for all model parameters, latent volatilities and the market price of volatility risk are produced via a Markov Chain Monte Carlo (MCMC) sampling algorithm. Candidate draws for the unobserved volatilities are obtained in blocks by applying the Kalman filter and simulation smoother to a linearization of a nonlinear state space representation of the model. Crucially, information from both the spot and option prices affects the draws via the specification of a bivariate measurement equation, with implied Black-Scholes volatilities used to proxy observed option prices in the candidate model. Alternative models nested within the Heston (1993) framework are ranked via posterior odds ratios, as well as via fit, predictive and hedging performance. The method is illustrated using Australian News Corporation spot and option price data.     

A comparison of Bayesian model selection based on MCMC with an application to GARCH-type models

This paper presents a comprehensive review and comparison of five computational methods for Bayesian model selection, based on MCMC simulations from posterior model parameter distributions. We apply these methods to a well-known and important class of models in financial time series analysis, namely GARCH and GARCH-t models for conditional return distributions (assuming normal and t-distributions). We compare their performance with the more common maximum likelihood-based model selection for simulated and real market data. All five MCMC methods proved reliable in the simulation study, although differing in their computational demands. Results on simulated data also show that for large degrees of freedom (where the t-distribution becomes more similar to a normal one), Bayesian model selection results in better decisions in favor of the true model than maximum likelihood. Results on market data show the instability of the harmonic mean estimator and reliability of the advanced model selection methods.     

Inference for Adaptive Time Series Models: Stochastic Volatility and Conditionally Gaussian State Space Form

In this paper we model the Gaussian errors in the standard Gaussian linear state space model as stochastic volatility processes. We show that conventional MCMC algorithms for this class of models are ineffective, but that the problem can be alleviated by reparameterizing the model. Instead of sampling the unobserved variance series directly, we sample in the space of the disturbances, which proves to lower correlation in the sampler and thus increases the quality of the Markov chain.

Using our reparameterized MCMC sampler, it is possible to estimate an unobserved factor model for exchange rates between a group of n countries. The underlying n + 1 country-specific currency strength factors and the n + 1 currency volatility factors can be extracted using the new methodology. With the factors, a more detailed image of the events around the 1992 EMS crisis is obtained.

We assess the fit of competitive models on the panels of exchange rates with an effective particle filter and find that indeed the factor model is strongly preferred by the data.     

Bayesian inference of asymmetric stochastic conditional duration models

This paper extends stochastic conditional duration (SCD) models for financial transaction data to allow for correlation between error processes and innovations of observed duration process and latent log duration process. Suitable algorithms of Markov Chain Monte Carlo (MCMC) are developed to fit the resulting SCD models under various distributional assumptions about the innovation of the measurement equation. Unlike the estimation methods commonly used to estimate the SCD models in the literature, we work with the original specification of the model, without subjecting the observation equation to a logarithmic transformation. Results of simulation studies suggest that our proposed models and corresponding estimation methodology perform quite well. We also apply an auxiliary particle filter technique to construct one-step-ahead in-sample and out-of-sample duration forecasts of the fitted models. Applications to the IBM transaction data allow comparison of our models and methods to those existing in the literature.     

A Bayesian approach to zero-inflated data in extremes

Abstract

The generalized extreme value (GEV) distribution is known as the limiting result for the modeling of maxima blocks of size n, which is used in the modeling of extreme events. However, it is possible for the data to present an excessive number of zeros when dealing with extreme data, making it difficult to analyze and estimate these events by using the usual GEV distribution. The Zero-Inflated Distribution (ZID) is widely known in literature for modeling data with inflated zeros, where the inflator parameter w is inserted. The present work aims to create a new approach to analyze zero-inflated extreme values, that will be applied in data of monthly maximum precipitation, that can occur during months where there was no precipitation, being these computed as zero. An inference was made on the Bayesian paradigm, and the parameter estimation was made by numerical approximations of the posterior distribution using Markov Chain Monte Carlo (MCMC) methods. Time series of some cities in the northeastern region of Brazil were analyzed, some of them with predominance of non-rainy months. The results of these applications showed the need to use this approach to obtain more accurate and with better adjustment measures results when compared to the standard distribution of extreme value analysis.     

Mixing of MCMC algorithms

We analyse MCMC chains focusing on how to find simulation parameters that give good mixing for discrete time, Harris ergodic Markov chains on a general state space X having invariant distribution π. The analysis uses an upper bound for the variance of the probability estimate. For each simulation parameter set, the bound is estimated from an MCMC chain using recurrence intervals. Recurrence intervals are a generalization of recurrence periods for discrete Markov chains. It is easy to compare the mixing properties for different simulation parameters. The paper gives general advice on how to improve the mixing of the MCMC chains and a new methodology for how to find an optimal acceptance rate for the Metropolis-Hastings algorithm. Several examples, both toy examples and large complex ones, illustrate how to apply the methodology in practice. We find that the optimal acceptance rate is smaller than the general recommendation in the literature in some of these examples.     

On Bayesian model and variable selection using MCMC

Several MCMC methods have been proposed for estimating probabilities of models and associated 'model-averaged' posterior distributions in the presence of model uncertainty. We discuss, compare, develop and illustrate several of these methods, focussing on connections between them.     

Extreme quantile estimation based on financial time series

Estimation of market risk is an important problem in finance. Two well-known risk measures, viz., value at risk and median shortfall, turn out to be extreme quantiles of the marginal distribution of asset return. Time series on asset returns are known to exhibit certain stylized facts, such as heavy tails, skewness, volatility clustering, etc. Therefore, estimation of extreme quantiles in the presence of such features in the data seems to be of natural interest. It is difficult to capture most of these stylized facts using one specific time series model. This motivates nonparametric and extreme value theory-based estimation of extreme quantiles that do not require exact specification of the asset return model. We review these quantile estimators and compare their known properties. Their finite sample performance are compared using Monte Carlo simulation. We propose a new estimator that exhibits encouraging finite sample performance while estimating extreme quantile in the right tail region.     

Variational Bayes for estimating the parameters of a hidden Potts model

Hidden Markov random field models provide an appealing representation of images and other spatial problems. The drawback is that inference is not straightforward for these models as the normalisation constant for the likelihood is generally intractable except for very small observation sets. Variational methods are an emerging tool for Bayesian inference and they have already been successfully applied in other contexts. Focusing on the particular case of a hidden Potts model with Gaussian noise, we show how variational Bayesian methods can be applied to hidden Markov random field inference. To tackle the obstacle of the intractable normalising constant for the likelihood, we explore alternative estimation approaches for incorporation into the variational Bayes algorithm. We consider a pseudo-likelihood approach as well as the more recent reduced dependence approximation of the normalisation constant. To illustrate the effectiveness of these approaches we present empirical results from the analysis of simulated datasets. We also analyse a real dataset and compare results with those of previous analyses as well as those obtained from the recently developed auxiliary variable MCMC method and the recursive MCMC method. Our results show that the variational Bayesian analyses can be carried out much faster than the MCMC analyses and produce good estimates of model parameters. We also found that the reduced dependence approximation of the normalisation constant outperformed the pseudo-likelihood approximation in our analysis of real and synthetic datasets.     

The Ornstein-Uhlenbeck Dirichlet process and other time-varying processes for Bayesian nonparametric inference

This paper introduces a new class of time-varying, measure-valued stochastic processes for Bayesian nonparametric inference. The class of priors is constructed by normalising a stochastic process derived from non-Gaussian Ornstein-Uhlenbeck processes and generalises the class of normalised random measures with independent increments from static problems. Some properties of the normalised measure are investigated. A particle filter and MCMC schemes are described for inference. The methods are applied to an example in the modelling of financial data.     

INLA or MCMC? A tutorial and comparative evaluation for spatial prediction in log-Gaussian Cox processes

We investigate two options for performing Bayesian inference on spatial log-Gaussian Cox processes assuming a spatially continuous latent field: Markov chain Monte Carlo (MCMC) and the integrated nested Laplace approximation (INLA). We first describe the device of approximating a spatially continuous Gaussian field by a Gaussian Markov random field on a discrete lattice, and present a simulation study showing that, with careful choice of parameter values, small neighbourhood sizes can give excellent approximations. We then introduce the spatial log-Gaussian Cox process and describe MCMC and INLA methods for spatial prediction within this model class. We report the results of a simulation study in which we compare the Metropolis-adjusted Langevin Algorithm (MALA) and the technique of approximating the continuous latent field by a discrete one, followed by approximate Bayesian inference via INLA over a selection of 18 simulated scenarios. The results question the notion that the latter technique is both significantly faster and more robust than MCMC in this setting; 100,000 iterations of the MALA algorithm running in 20 min on a desktop PC delivered greater predictive accuracy than the default INLA strategy, which ran in 4 min and gave comparative performance to the full Laplace approximation which ran in 39 min.     

Importance sampling type estimators based on approximate marginal Markov chain Monte Carlo

We consider importance sampling (IS) type weighted estimators based on Markov chain Monte Carlo (MCMC) targeting an approximate marginal of the target distribution. In the context of Bayesian latent variable models, the MCMC typically operates on the hyperparameters, and the subsequent weighting may be based on IS or sequential Monte Carlo (SMC), but allows for multilevel techniques as well. The IS approach provides a natural alternative to delayed acceptance (DA) pseudo-marginal/particle MCMC, and has many advantages over DA, including a straightforward parallelization and additional flexibility in MCMC implementation. We detail minimal conditions which ensure strong consistency of the suggested estimators, and provide central limit theorems with expressions for asymptotic variances. We demonstrate how our method can make use of SMC in the state space models context, using Laplace approximations and time-discretized diffusions. Our experimental results are promising and show that the IS-type approach can provide substantial gains relative to an analogous DA scheme, and is often competitive even without parallelization.