Scaling Limits of Markov Chains/Processes in Monte
Carlo Methods
Kengo Kamatani (Institute of Statistical Mathematics, JST CREST)

In this presentation, we will explore the recent results of scaling limit of piecewise deterministic
Markov processes for anisotropic targets. Suppose we wish to sample from
Πpdxq “ expp´Hpxqqdx
where H : Rd Ñ R is a continuously differentiable function. For the Bayesian context, thisprobability distribution is the posterior distribution of interest. If we have an i.i.d. sample fromΠ, we can approximate Π-integral of any function f pxq by the law of large numbers. In mostof the cases, direct i.i.d. sampling is impossible or computationally very expensive. For thesecases, the Markov chain Monte Carlo method is useful which originated with the classic paper byMetropolis et al. (1953) almost 70 years ago. The Markov chain Monte Carlo method is designed
to construct an ergodic Markov kernel P which is Π-invariant. If a Markov chain X1 , X2 , . . . isgeneraetd from the Markov kernel P then the law of large numbers is satisfied. The Markovchain Monte Carlo is now a gold standard for Bayesian inference.
Recently, its continuous process version, the Markov process Monte Carlo method is ofsubstantial interest for Monte Carlo analysis. Known Markov process Monte Carlo methods relyon an auxiliary variable trick which uses an auxiliary variable v with a probability density νon Ξ and considers the joint probability distribution µ :“ Πpdxq b νpdvq as an extended target
distribution on Z “ Rd ˆ Ξ. The original target distribution is a marginal distribution of theextended target distribution. Since Brownian motion does not have an absolutely continuouspath, we can not simulate processes driven by Brownian motion exactly. For our Monte Carloanalysis, exact sampling is necessary. Therefore, the Markov processes of interest should nothave a Brownian part. Known processes consist of a deterministic part and a pure jump part.
These processes are known as the piecewise deterministic Markov processes.
Here we follow Azaïs et al. (2014) for the expression of the piecewise deterministic Markovprocesses. The processes are constructed by characteristics pϕ, λk , Qk : l “ 1, . . . , Kq. The flow
ϕ : Z ˆ R Ñ Z is continuous, ϕp¨, tq is a homeomorphism for each t P R and ϕpϕp¨, sq, tq “ϕp¨, s ` tq. For each k “ 1, . . . , K, the jump rate λk : Z Ñ R` determines the jump time of pureşt
jump processes, and Qk is a Markov kernel on Z. Let Λk pz, tq “ 0 λk pϕpz, sqqds.
The Markov process is defined by the following way. Suppose zp0q “ pxp0q, tp0qq P Z. Let
T1 , . . . , TK be independent processes with PpTk ě tq “ expp´Λk pz, tqq. Let T˚ “ mink“1,...,K Tk .
If Tk “ T˚ , then Z is generated from Qk pϕpz, T˚ q, ¨q and set"ϕpzp0q, tq for t ă T˚Xptq “Zfor t “ T˚ .
After T˚ , the process evolves in the same way with starting value Z. There are several choicesof characteristics. Two popular piecewise deterministic Markov processes use the same flow ϕdefined by x1 ptq “ vptq and v 1 ptq “ 0. The Zig-Zag sampler proposed by Bierkens et al. (2019)1

uses d Markov kernels Q1 , . . . , Qd with d jump rates λ1 , . . . , λd . For each i “ 1, . . . , d, the Markovkernel is a deterministic kernel Qi defined by a map px, vq ÞÑ px, Fi pvqq where Fi is an operator
that flips the i-th coordinate of x. The jump rate is defined by λi ppx, vqq “ maxt0, Bi Hpxqvi u.
The bouncy particle sampler proposed by Peters and de With (2012), Bouchard-Côtéet al. (2018) uses two Markov kernels Qbounce and Qref with corresponding jump rates λbounceand λref . The kernel Qbounce is a deterministic kernel defined by a map px, vq ÞÑ px, κpx, vqq:κpx, vq “ v ´ 2x∇Hpxq, vy∇Hpxq}∇Hpxq}2and λbounce px, vq “ maxt0, x∇Hpxq, vyu. The jump rate λref is a positive constant, and Qref is a µinvariant Markov kernel. For our analysis, for simplicity, we assume Qref ppx, vq, dpy, wqq “ νpdwq.
We have several critical findings. For the Zig-Zag algorithm, its performance is intricately
linked to the orientation of the target’s anisotropy; specific alignments with the algorithm’soperational axes lead to enhanced efficiency, while others can hinder its effectiveness. The BPSalgorithm, on the other hand, exhibits a deterministic dynamical behaviour in its limiting form
with a better rate of convergence.
This is joint work with Joris Bierkens (TU Delft) and Gareth O. Roberts (Warwick). See ourpaper on arxiv https://arxiv.org/abs/2305.00694 for the detail.

References
Christophe Andrieu, Alain Durmus, Nikolas Nüsken, and Julien Roussel. Hypocoercivity ofpiecewise deterministic markov process-monte carlo. The Annals of Applied Probability, 31, 102021. ISSN 1050-5164. doi: 10.1214/20-AAP1653.
Romain Azaïs, Jean-Baptiste Bardet, Alexandre Génadot, Nathalie Krell, and Pierre-André Zitt.
Piecewise deterministic Markov process—recent results. In Journées MAS 2012, volume 44 ofESAIM Proc., pages 276–290. EDP Sci., Les Ulis, 2014. doi: 10.1051/proc/201444017.
Joris Bierkens, Paul Fearnhead, and Gareth Roberts. The zig-zag process and super-efficient
sampling for Bayesian analysis of big data. Ann. Statist., 47(3):1288–1320, 2019. ISSN 00905364. doi: 10.1214/18-AOS1715.
Joris Bierkens, Kengo Kamatani, and Gareth O. Roberts. High-dimensional scaling limits ofpiecewise deterministic sampling algorithms. The Annals of Applied Probability, 32, 10 2022.
ISSN 1050-5164. doi: 10.1214/21-AAP1762.
Alexandre Bouchard-Côté, Sebastian J. Vollmer, and Arnaud Doucet. The bouncy particle sampler: A nonreversible rejection-free markov chain monte carlo method. Journal of the American
Statistical Association, 113(522):855–867, 2018. doi: 10.1080/01621459.2017.1294075.
George Deligiannidis, Daniel Paulin, Alexandre Bouchard-Côté, and Arnaud Doucet. Randomized hamiltonian monte carlo as scaling limit of the bouncy particle sampler and dimensionfree convergence rates. The Annals of Applied Probability, 31, 12 2021. ISSN 1050-5164. doi:
10.1214/20-AAP1659.
N. Metropolis, W. A. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. Equations ofstate calculations by fast computing machines. Journal of Chemical Physics, 21:1087–1092,
1953.
Elias AJF Peters and G. de With. Rejection-free monte carlo sampling for general potentials.
Physical Review E, 85(2):026703, 2012.