Bayesian inference for pairwise interacting point processes

Pairwise interacting point processes are commonly used to model spatial point patterns. To perform inference, the established frequentist methods can produce good point estimates when the interaction in the data is moderate, but some methods may produce severely biased estimates when the interaction in strong. Furthermore, because the sampling distributions of the estimates are unclear, interval estimates are typically obtained by parametric bootstrap methods. In the current setting however, the behavior of such estimates is not well understood. In this article we propose Bayesian methods for obtaining inferences in pairwise interacting point processes. The requisite application of Markov chain Monte Carlo (MCMC) techniques is complicated by an intractable function of the parameters in the likelihood. The acceptance probability in a Metropolis-Hastings algorithm involves the ratio of two likelihoods evaluated at differing parameter values. The intractable functions do not cancel, and hence an intractable ratio r must be estimated within each iteration of a Metropolis-Hastings sampler. We propose the use of importance sampling techniques within MCMC to address this problem. While r may be estimated by other methods, these, in general, are not readily applied in a Bayesian setting. We demonstrate the validity of our importance sampling approach with a small simulation study. Finally, we analyze the Swedish pine sapling dataset (Strand 1972) and contrast the results with those in the literature.     

Bayesian inference for Heston-STAR models

The Heston-STAR model is a new class of stochastic volatility models defined by generalizing the Heston model to allow the volatility of the volatility process as well as the correlation between asset log-returns and variance shocks to change across different regimes via smooth transition autoregressive (STAR) functions. The form of the STAR functions is very flexible, much more so than the functions introduced in Jones (J Econom 116:181–224, 2003), and provides the framework for a wide range of stochastic volatility models. A Bayesian inference approach using data augmentation techniques is used for the parameters of our model. We also explore goodness of fit of our Heston-STAR model. Our analysis of the S&P 500 and VIX index demonstrates that the Heston-STAR model is more capable of dealing with large market fluctuations (such as in 2008) compared to the standard Heston model.