# Abstracts

## Talk Abstracts

• Sergios Agapiou (Cyprus) - "Unbiased Estimation and Exact Simulation for Bayesian Inverse Problems"
• Abstract: We will consider the problem of estimating expectations with respect to measures which are intractable in the sense that sampling them requires infinite cost. In particular we are interested in computing expectations with respect to the posterior distribution in the context of Bayesian PDE inverse problems. The standard approach in this context is to construct a Markov chain with limiting distribution the posterior measure and to use samples from this Markov chain to estimate posterior expectations, for example using the ergodic average. In this approach there are two forms of approximation, due to a) the discretization of the possibly infinite dimensional parameter and/or the discretization of the forward PDE in order to solve it numerically; b) the use of finite-time samples from the Markov chain which have not necessarily converged to the limiting posterior distribution. We will first discuss techniques for estimating posterior expectations unbiasedly, building on the recent work by Peter Glynn and Chang-han Rhee. Then, we will discuss techniques for drawing samples from the posterior without any error due to the discretization of the forward PDE solver.
• The first part on unbiased estimation is work done in collaboration with Gareth Roberts (Warwick) and Sebastian Vollmer (Oxford) (arXiv:1411.7713), while the second part on exact estimation is ongoing work in collaboration with Gareth Roberts and Andrew Stuart (Warwick).
• Jose Blanchet (Columbia) - "Efficient Monte Carlo Methods for Spatial Extremes"
• Abstract: Many applications, including modeling extreme weather events, naturally call
for extrapolation techniques of extremes with spatial dependence. To preserve the standard univariate extreme value theory, it is natural to consider random fields, $\{M( t):t\in T\}$ ($T$ may represent a geographical region), satisfying, as their definition, a max-stable property. That is, if $\{M_{i}(\cdot)\}_{i=1}^{n}$ are iid copies of $M(\cdot)$ then there is a sequence of functions $\{a_{n}(t) ,b_{n}(t)\}$ satisfying the equality in distribution $M(\cdot) =\max_{i=1}^{n}(M_{i}(\cdot )-a_{n}(\cdot)) /b_{n}(\cdot)$. It turns out that if $M(\cdot)$ is a max-stable random field then (after applying a simple transformation) $M(\cdot)$ must admit a representation of the form $M( t) =\max_{n=1}^{\infty}\{-\log ( A_{n}) +X_{n}(t) \}%$, where $A_{n}$ is the $n$-th arrival of a Poisson process and$X_{n}(\cdot)$ is an iid sequence of random fields (typically Gaussian) independent of the $A_{n}$s. The goal of this talk is to discuss efficient Monte Carlo techniques for max-stable random fields.
• In particular, we study the following questions:
• a) Is it possible to simulate $\{M t_{1}) ,...M(t_{d})\}$ exactly (i.e. without bias)?
• b) Can exact simulation be performed optimally as $d$ grows to infinity?
• c) Can conditional simulation also be done efficiently and without bias?
• We share surprising good news in response to these questions using ideas borrowed from recent exact simulation algorithms, rare-event simulation, and multilevel Monte Carlo. (This is joint work with Z. Liu, T. Dieker, and T. Mikosch.)
• Hongsheng Dai (Essex) - "Monte Carlo method for conflation of probability distributions"
• Abstract: This presentation will cover a new rejection sampling algorithms, which could potentially be applied Bayesian group decision problems and meta-analysis. The idea depends on a decomposition of the target distribution to a product of two simple distributions. If independent realisations are generated from each of the two simple distributions, then they can be combined into a single realisation in a certain way, which is exactly from the target distribution. The presentation will also discuss the possibility of using random series to improve the efficiency of the algorithm.
• Jing Dong (Northwestern) - "$\epsilon$-strong simulation for multidimensional stochastic differential equations via rough path analysis"
• Abstract: Consider a multidimensional diffusion process, $X$. Let $\epsilon>0$ be a deterministic user defined tolerance error parameter. We develop a systematic way to construct a probability space, supporting both X and a fully simulatable piecewise constant process $X_\epsilon$, such that $X_\epsilon$ is within epsilon distance from $X$ under the uniform metric on compact time intervals with probability one, Our construction requires a detailed study of continuity estimates of the Ito map using Lyons' theory of rough paths. We approximate the underlying Brownian paths, jointly with the Levy areas, with a deterministic error bound in the underlying rough path metric.
• Flávio Gonçalves (UFMG) - "Infinite-dimensional pseudo-marginal MCMC for exact inference in SDE driven models"
• Abstract: Exact methodologies for analytically intractable infinite-dimensional problems have been an area of intensive investigation in the last few years. In particular, models including continuous time stochastic processes which are the solution of some given stochastic differential equation (SDE). The most promising solutions rely on novel simulation techniques, in special, retrospective Monte Carlo ones. In this talk, I describe a novel general methodology do deal with inference problems for SDE driven models in an exact setup. It consists of an infinite-dimensional pseudo-marginal MCMC with Barker's steps. Retrospective sampling and Bernoulli factories play an important role in this context. I will then discuss its implementation for two families of models: jump-diffusions and diffusion-driven Cox processes.
• Peter Glynn (Stanford) - "Randomized MLMC for Markov Chains"
• Abstract: Multi-level Monte Carlo (MLMC) algorithms have been extensively applied in recent years to obtain schemes that often converge at faster rates than corresponding traditional Monte Carlo methods. In this talk, we shall discuss a randomized method introduced in joint work with Chang-han Rhee, and then describe a stratified alternative estimator. Our principal focus in the talk will be on applications to equilibrium computations for Markov chains. computing value functions, spectral densities, and sensitivity estimates, and covers joint work with Rhee and Zeyu Zheng.
• Mark Huber (Claremont McKenna) - "A Bernoulli Factory using the Fundamental Theorem of Perfect Simulation"
• Abstract: Given a sequence of iid coin flips with unknown probability $p$ of heads, a Bernoulli factory builds a single coin whose probability of heads is $f(p)$. This problem has a twenty year history, with applications in perfect simulation for regenerative MCMC and diffusions. When $f(p)$ is analytic, Yuval and Peres showed that the problem can be reduced to a Bernoulli factory for $2p$. Here a probabilistic recursive approach is used for functions of the form $f(p)= Cp$, which gives an algorithm that runs in an expected number of flips that is provably within a constant factor of the best possible. This notion of probabilistic recursion forms the basis of many perfect simulation methods, and the Fundamental Theorem of Perfect Simulation presented here gives a simple criterion for the correctness of such algorithms.
• Chang-han Rhee (CWI Amsterdam) - "Unbiased Multilevel Monte Carlo"
• Abstract: Monte Carlo simulation is a powerful computational tool when the quantity of ones interest can be written as an expectation of a random object. Often, however, such a random object is difficult to generate from its exact distribution, and only approximations are available. The errors from such approximations can lead to slower convergence rates and less reliable error estimates. To address such difficulties, [1] proposes a simple yet effective and broadly applicable idea, and studies the implications of the idea in SDE context. In this talk, we will review the general theory developed in [1] and discuss its close connection to the standard (biased) multilevel Monte Carlo methods. We will then discuss the application of the idea to rare event simulation of stochastic recurrence equations.
• [1] C.-H. Rhee and P. W. Glynn Unbiased estimation with square root convergence for SDE models. Operations Research, 63(5):1026–1043, 2015.
• Gareth Roberts (Warwick) - "The Zig-Zag"
• Abstract: This is a new continuous-time non-reversible MCMC method. It has been known for some time that non-reversible MCMC methods sometimes have potentially large advantages over reversible ones. The difficulty has always been to construct them in a way which is practically possible. The zig-zag is one of an emerging collection of ideas motivated by this problem.