ZZDiffusionBridge

ZigZag sampler is used to explore the conditional measure of a diffusion process. This measure lays in a high dimensional space (infinite dimensional) and could differ significantly from a Gaussian measure (which in our case is the reference measure).

Overview

The main function introduced (available here) is

zz_sampler(X::AbstractModel, T::Float64, L::Int64, u::Float64, v::Float64, clock::Float64; θ = fill(1.0, 2<<L - 1))

It takes a diffusion model with its parameters, its initial and final points u,v, the truncation level of the infinite summation L, the final time of the ZigZag sampler clock and as optional input, the velocity vector for each coordinate. The function returns the skeleton of the ZigZag process with the real event times. This is given in the form of Array{Skeleton, 1}.

Structures

Three main types are defined (here):

1. the abstract type ::AbstractModel, inheriting any specific diffusion model
2. the abstract type ::AbstractDependenceStructure inheriting the types of dependences (for our application we have now two subtypes ::FullIndependence and ::PartialIndependence) acting as a flag
3. the abstract type SamplingScheme inheriting the types of sampling scheme (::Regular and ::PartialIndependence ) acting as a flag
4. the type System, container of all the attributes necessary for the sampler

Examples

We have three examples right now (here running the ZigZag sampler for three different SDEs. how to run them: easy, just download the repo and run the script in the folder src/examples

Developing the ZigZag for a new SDE

If you want to develop your own SDE, you need to:

1. define your ::ModelName:<AbstractModel containing the parameters (and optionally some parameters/vectors/matrices needed to the sampler which is better to precompute.)
2. define the function λbar(n, S::System, X::ModelName , u, v) and λratio(n::Int64, S::System, X::ModelName, u::Float64, v::Float64, t::Float64) (λratio only if the sampling scheme is SubSampling)
3. assign the flags for the sampler depending wheather the sampler is Fullindependence or PartialIndependence; Regular or SubSampling. This is done for example via the flag-funcitons

dependence_strucute(::ModelName) = FullIndependence() sampling_scheme(::ModelName) = SubSampling()

Faber-Schauder functions

The file faber.jl and fs_expansion.jl contains all the functions necessary to work with the Faber-Schauder functions and change of basis to finite elements.

Benchmark

An efficient implementation of the fully local Zig-Zag sampler is implemented in the official package ZigZagBoomerang.jl. We used it to test the performances of the Zig-Zag sampler agaist the Boomerang sampler and the MALA. The effective sample size is computed using functions implemented in [https://github.com/jbierkens/ICML-boomerang/].

Literature

• Joris Bierkens, Sebastiano Grazzi, Frank van der Meulen, Moritz Schauer: A piecewise deterministic Monte Carlo method for diffusion bridges. 2020. [https://arxiv.org/abs/2001.05889].
• Joris Bierkens, Sebastiano Grazzi, Kengo Kamatani and Gareth Robers: The Boomerang Sampler. ICML 2020. [https://arxiv.org/abs/2006.13777].
• Moritz Schauer, Sebastiano Grazzi: ZigZagBoomerang.jl. 2020. [https://github.com/mschauer/ZigZagBoomerang.jl]