Consistent nonparametric Bayesian inference for discretely observed scalar diffusions

We study Bayes procedures for the problem of nonparametric drift estimation for one-dimensional, ergodic diffusion models from discrete-time, low-frequency data. We give conditions for posterior consistency and verify these conditions for concrete priors, including priors based on wavelet expansions.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ385 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals

Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case. We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods

Introduction to Automatic Backward Filtering Forward Guiding

In this document I aim to give an informal treatment of automatic Backward Filtering Forward Guiding, a general algorithm for conditional sampling from a Markov process on a directed acyclic graph. I'll show that the underlying ideas can be understood with a basic background in probability and statistics. The more technical treatment is the paper Automatic backward filtering forward guiding for Markov processes and graphical models (Van der Meulen and Schauer, 2021). I specifically assume some background knowledge on likelihood based inference and Bayesian statistics. The final sections are more demanding and assume familiarity with continuous-time stochastic processes constructed from their infinitesimal generator. Clearly, all work discussed here is the result of research carried out over the past decade together with various collaborators, most importantly Moritz Schauer (Chalmers University of Technology and University of Gothenburg, Sweden). Section 8 is based on joint work with Marcin Mider (Trium Analysis Online GmbH, Germany) and Frank Sch\"afer (University of Basel, Switzerland) as well

A non-parametric Bayesian approach to decompounding from high frequency data

Given a sample from a discretely observed compound Poisson process, we consider non-parametric estimation of the density $f_0$ of its jump sizes, as well as of its intensity $\lambda_0.$ We take a Bayesian approach to the problem and specify the prior on $f_0$ as the Dirichlet location mixture of normal densities. An independent prior for $\lambda_0$ is assumed to be compactly supported and to possess a positive density with respect to the Lebesgue measure. We show that under suitable assumptions the posterior contracts around the pair $(\lambda_0,f_0)$ at essentially (up to a logarithmic factor) the $\sqrt{n\Delta}$-rate, where $n$ is the number of observations and $\Delta$ is the mesh size at which the process is sampled. The emphasis is on high frequency data, $\Delta\to 0$, but the obtained results are also valid for fixed $\Delta$. In either case we assume that $n\Delta\rightarrow\infty$. Our main result implies existence of Bayesian point estimates converging (in the frequentist sense, in probability) to $(\lambda_0,f_0)$ at the same rate. We also discuss a practical implementation of our approach. The computational problem is dealt with by inclusion of auxiliary variables and we develop a Markov Chain Monte Carlo algorithm that samples from the joint distribution of the unknown parameters in the mixture density and the introduced auxiliary variables. Numerical examples illustrate the feasibility of this approach

Simulation of elliptic and hypo-elliptic conditional diffusions

Suppose $X$ is a multidimensional diffusion process. Assume that at time zero the state of $X$ is fully observed, but at time $T>0$ only linear combinations of its components are observed. That is, one only observes the vector $L X_T$ for a given matrix $L$. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in Schauer et al. (2017), can be used in a unified way for both uniformly and hypo-elliptic diffusions, also when $L$ is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice integrated diffusion with multiple wells and the partially observed FitzHugh-Nagumo model

Nonparametric Bayesian inference for multidimensional compound Poisson processes

Given a sample from a discretely observed multidimensional compound Poisson process, we study the problem of nonparametric estimation of its jump size density $r_0$ and intensity $\lambda_0$. We take a nonparametric Bayesian approach to the problem and determine posterior contraction rates in this context, which, under some assumptions, we argue to be optimal posterior contraction rates. In particular, our results imply the existence of Bayesian point estimates that converge to the true parameter pair $(r_0,\lambda_0)$ at these rates. To the best of our knowledge, construction of nonparametric density estimators for inference in the class of discretely observed multidimensional L\'{e}vy processes, and the study of their rates of convergence is a new contribution to the literature.Comment: Published at http://dx.doi.org/10.15559/15-VMSTA20 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/

Networks & big data

Once a year, the NWO cluster Stochastics – Theoretical and Applied Research (STAR) organises a STAR Outreach Day, a one-day event around a theme that is of a broad interest to the stochastics community in the Netherlands. The last Outreach Day took place at Eurandom on 12 December 2014. The theme of the day was ‘Networks & Big Data’. The topic is very timely. The Vision document 2025 of the PlatformWiskunde Nederland (PWN) mentions big data as one of the six “major societal and scientific trends that influence the mathematical sciences”. In 2014 a ten-year Gravitation programme ‘NETWORKS’ inmathematics was awarded by NWO. The STAR Outreach Day has presented an exciting opportunity to promote these topics and their numerous applications in life and science. Organisers Nelly Litvak and Frank van der Meulen look back on this event

Fast and scalable non-parametric Bayesian inference for Poisson point processes

We study the problem of non-parametric Bayesian estimation of the intensity function of a Poisson point process. The observations are $n$ independent realisations of a Poisson point process on the interval $[0,T]$. We propose two related approaches. In both approaches we model the intensity function as piecewise constant on $N$ bins forming a partition of the interval $[0,T]$. In the first approach the coefficients of the intensity function are assigned independent gamma priors, leading to a closed form posterior distribution. On the theoretical side, we prove that as $n\rightarrow\infty,$ the posterior asymptotically concentrates around the "true", data-generating intensity function at an optimal rate for $h$-H\"older regular intensity functions ($0 < h\leq 1$). In the second approach we employ a gamma Markov chain prior on the coefficients of the intensity function. The posterior distribution is no longer available in closed form, but inference can be performed using a straightforward version of the Gibbs sampler. Both approaches scale well with sample size, but the second is much less sensitive to the choice of $N$. Practical performance of our methods is first demonstrated via synthetic data examples. We compare our second method with other existing approaches on the UK coal mining disasters data. Furthermore, we apply it to the US mass shootings data and Donald Trump's Twitter data.Comment: 45 pages, 22 figure