arXiv subjects

Kengo Kamatani

Publications and source records attributed to Kengo Kamatani.

At least 19 recordsLinked to original sources

arXiv2026

Diffusive Scaling Limits of Forward Event-Chain Monte Carlo: Provably Efficient Exploration with Partial Refreshment

Piecewise deterministic Markov process samplers are attractive alternatives to Metropolis--Hastings algorithms. A central design question is how to incorporate partial velocity refreshment to ensure ergodicity without injecting excessive noise. Forward Event-Chain Monte Carlo (FECMC) is a generalization of the Bouncy Particle Sampler (BPS) that addresses this issue through a stochastic reflection mechanism, thereby reducing reliance on global refreshment moves. Despite promising empirical performance, its theoretical efficiency remains largely unexplored. We develop a high-dimensional scaling analysis for standard Gaussian targets and prove that the negative log-density (or potential) process of FECMC converges to an Ornstein--Uhlenbeck diffusion, under the same scaling as BPS. We derive closed-form expressions for the limiting diffusion coefficients of both methods by analyzing their associated radial momentum processes and solving the corresponding Poisson equations. These expressions yield a sharp efficiency comparison: the diffusion coefficient of FECMC is strictly larger than that of optimally tuned BPS, and the optimum for FECMC is attained at zero global refreshment. Specifically, they imply an approximately eightfold increase in effective sample size per event over optimal BPS. Numerical experiments confirm the predicted diffusion coefficients and show that the resulting efficiency gains remain substantial for a range of non-Gaussian targets. Finally, as an application of these results, we propose an asymptotic variance estimator for Piecewise deterministic Markov processes that becomes increasingly efficient in high dimensions by extracting information from the velocity variable.

math.PR↗

arXiv2025

Transient regime of piecewise deterministic Monte Carlo algorithms

Piecewise Deterministic Markov Processes (PDMPs) such as the Bouncy Particle Sampler and the Zig-Zag Sampler, have gained attention as continuous-time counterparts of classical Markov chain Monte Carlo. We study their transient regime under convex potentials, namely how trajectories that start in low-probability regions move toward higher-probability sets. Using fluid-limit arguments with a decomposition of the generator into fast and slow parts, we obtain deterministic ordinary differential equation descriptions of early-stage behaviour. The fast dynamics alone are non-ergodic because once the event rate reaches zero it does not restart. The slow component reactivates the dynamics, so averaging remains valid when taken over short micro-cycles rather than with respect to an invariant law. Using the expected number of jump events as a cost proxy for gradient evaluations, we find that for Gaussian targets the transient cost of PDMP methods is comparable to that of random-walk Metropolis. For convex heavy-tailed families with subquadratic growth, PDMP methods can be more efficient when event simulation is implemented well. Forward Event-Chain and Coordinate Samplers can, under the same assumptions, reach the typical set with an order-one expected number of jumps. For the Zig-Zag Sampler we show that, under a diagonal-dominance condition, the transient choice of direction coincides with the solution of a box-constrained quadratic program; outside that regime we give a formal derivation and a piecewise-smooth update rule that clarifies the roles of the gradient and the Hessian. These results provide theoretical insight and practical guidance for the use of PDMP samplers in large-scale inference.

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arXiv2025

Bayesian Inference for Non-Synchronously Observed Diffusions

We consider the problem of Bayesian inference for bi-variate data observed in time but with observation times which occur non-synchronously. In particular, this occurs in a wide variety of applications in finance, such as high-frequency trading or crude oil futures trading. We adopt a diffusion model for the data and formulate a Bayesian model with priors on unknown parameters along with a latent representation for the the so-called missing data. We then consider computational methodology to fit the model using Markov chain Monte Carlo (MCMC). We have to resort to time-discretization methods as the complete data likelihood is intractable and this can cause considerable issues for MCMC when the data are observed in low frequencies. In a high frequency observation frequencies we present a simple particle MCMC method based on an Euler--Maruyama time discretization, which can be enhanced using multilevel Monte Carlo (MLMC). In the low frequency observation regime we introduce a novel bridging representation of the posterior in continuous time to deal with the issues of MCMC in this case. This representation is discretized and fitted using MCMC and MLMC. We apply our methodology to real and simulated data to establish the efficacy of our methodology.

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arXiv2024

Multilevel Monte Carlo for a class of Partially Observed Processes in Neuroscience

In this paper we consider Bayesian parameter inference associated to a class of partially observed stochastic differential equations (SDE) driven by jump processes. Such type of models can be routinely found in applications, of which we focus upon the case of neuroscience. The data are assumed to be observed regularly in time and driven by the SDE model with unknown parameters. In practice the SDE may not have an analytically tractable solution and this leads naturally to a time-discretization. We adapt the multilevel Markov chain Monte Carlo method of [11], which works with a hierarchy of time discretizations and show empirically and theoretically that this is preferable to using one single time discretization. The improvement is in terms of the computational cost needed to obtain a pre-specified numerical error. Our approach is illustrated on models that are found in neuroscience.

q-bio.NC↗

arXiv2024

Scaling of Piecewise Deterministic Monte Carlo for Anisotropic Targets

Piecewise deterministic Markov processes (PDMPs) are a type of continuous-time Markov process that combine deterministic flows with jumps. Recently, PDMPs have garnered attention within the Monte Carlo community as a potential alternative to traditional Markov chain Monte Carlo (MCMC) methods. The Zig-Zag sampler and the Bouncy Particle Sampler are commonly used examples of the PDMP methodology which have also yielded impressive theoretical properties, but little is known about their robustness to extreme dependence or anisotropy of the target density. It turns out that PDMPs may suffer from poor mixing due to anisotropy and this paper investigates this effect in detail in the stylised but important Gaussian case. To this end, we employ a multi-scale analysis framework in this paper. Our results show that when the Gaussian target distribution has two scales, of order $1$ and $ε$, the computational cost of the Bouncy Particle Sampler is of order $ε^{-1}$, and the computational cost of the Zig-Zag sampler is $ε^{-2}$. In comparison, the cost of the traditional MCMC methods such as RWM is of order $ε^{-2}$, at least when the dimensionality of the small component is more than $1$. Therefore, there is a robustness advantage to using PDMPs in this context.

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arXiv2024

Automated Techniques for Efficient Sampling of Piecewise-Deterministic Markov Processes

Piecewise deterministic Markov processes (PDMPs) are a class of continuous-time Markov processes that were recently used to develop a new class of Markov chain Monte Carlo algorithms. However, the implementation of the processes is challenging due to the continuous-time aspect and the necessity of integrating the rate function. Recently, Corbella, Spencer, and Roberts (2022) proposed a new algorithm to automate the implementation of the Zig-Zag sampler. However, the efficiency of the algorithm highly depends on a hyperparameter ($t_{\text{max}}$) that is fixed all along the run of the algorithm and needs preliminary runs to tune. In this work, we relax this assumption and propose a new variant of their algorithm that let this parameter change over time and automatically adapt to the target distribution. We also replace the Brent optimization algorithm by a grid-based method to compute the upper bound of the rate function. This method is more robust to the regularity of the function and gives a tighter upper bound while being quicker to compute. We also extend the algorithm to other PDMPs and provide a Python implementation of the algorithm based on JAX.

stat.CO↗

arXiv2023

Multilevel Particle Filters for a Class of Partially Observed Piecewise Deterministic Markov Processes

In this paper we consider the filtering of a class of partially observed piecewise deterministic Markov processes (PDMPs). In particular, we assume that an ordinary differential equation (ODE) drives the deterministic element and can only be solved numerically via a time discretization. We develop, based upon the approach in [20], a new particle and multilevel particle filter (MLPF) in order to approximate the filter associated to the discretized ODE. We provide a bound on the mean square error associated to the MLPF which provides guidance on setting the simulation parameter of that algorithm and implies that significant computational gains can be obtained versus using a particle filter. Our theoretical claims are confirmed in several numerical examples.

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arXiv2021

Haar-Weave-Metropolis kernel

Recently, many Markov chain Monte Carlo methods have been developed with deterministic reversible transform proposals inspired by the Hamiltonian Monte Carlo method. The deterministic transform is relatively easy to reconcile with the local information (gradient etc.) of the target distribution. However, as the ergodic theory suggests, these deterministic proposal methods seem to be incompatible with robustness and lead to poor convergence, especially in the case of target distributions with heavy tails. On the other hand, the Markov kernel using the Haar measure is relatively robust since it learns global information about the target distribution introducing global parameters. However, it requires a density preserving condition, and many deterministic proposals break this condition. In this paper, we carefully select deterministic transforms that preserve the structure and create a Markov kernel, the Weave-Metropolis kernel, using the deterministic transforms. By combining with the Haar measure, we also introduce the Haar-Weave-Metropolis kernel. In this way, the Markov kernel can employ the local information of the target distribution using the deterministic proposal, and thanks to the Haar measure, it can employ the global information of the target distribution. Finally, we show through numerical experiments that the performance of the proposed method is superior to other methods in terms of effective sample size and mean square jump distance per second.

stat.ME↗

arXiv2021

MCMC Algorithms for Posteriors on Matrix Spaces

We study Markov chain Monte Carlo (MCMC) algorithms for target distributions defined on matrix spaces. Such an important sampling problem has yet to be analytically explored. We carry out a major step in covering this gap by developing the proper theoretical framework that allows for the identification of ergodicity properties of typical MCMC algorithms, relevant in such a context. Beyond the standard Random-Walk Metropolis (RWM) and preconditioned Crank--Nicolson (pCN), a contribution of this paper in the development of a novel algorithm, termed the `Mixed' pCN (MpCN). RWM and pCN are shown not to be geometrically ergodic for an important class of matrix distributions with heavy tails. In contrast, MpCN is robust across targets with different tail behaviour and has very good empirical performance within the class of heavy-tailed distributions. Geometric ergodicity for MpCN is not fully proven in this work, as some remaining drift conditions are quite challenging to obtain owing to the complexity of the state space. We do, however, make a lot of progress towards a proof, and show in detail the last steps left for future work. We illustrate the computational performance of the various algorithms through numerical applications, including calibration on real data of a challenging model arising in financial statistics.

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arXiv2021

Non-reversible guided Metropolis kernel

We construct a class of non-reversible Metropolis kernels as a multivariate extension of the guided-walk kernel proposed by Gustafson 1998. The main idea of our method is to introduce a projection that maps a state space to a totally ordered group. By using Haar measure, we construct a novel Markov kernel termed Haar-mixture kernel, which is of interest in its own right. This is achieved by inducing a topological structure to the totally ordered group. Our proposed method, the Delta-guided Metropolis--Haar kernel, is constructed by using the Haar-mixture kernel as a proposal kernel. The proposed non-reversible kernel is at least 10 times better than the random-walk Metropolis kernel and Hamiltonian Monte Carlo kernel for the logistic regression and a discretely observed stochastic process in terms of effective sample size per second.

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arXiv2020

The Boomerang Sampler

This paper introduces the Boomerang Sampler as a novel class of continuous-time non-reversible Markov chain Monte Carlo algorithms. The methodology begins by representing the target density as a density, $e^{-U}$, with respect to a prescribed (usually) Gaussian measure and constructs a continuous trajectory consisting of a piecewise elliptical path. The method moves from one elliptical orbit to another according to a rate function which can be written in terms of $U$. We demonstrate that the method is easy to implement and demonstrate empirically that it can out-perform existing benchmark piecewise deterministic Markov processes such as the bouncy particle sampler and the Zig-Zag. In the Bayesian statistics context, these competitor algorithms are of substantial interest in the large data context due to the fact that they can adopt data subsampling techniques which are exact (ie induce no error in the stationary distribution). We demonstrate theoretically and empirically that we can also construct a control-variate subsampling boomerang sampler which is also exact, and which possesses remarkable scaling properties in the large data limit. We furthermore illustrate a factorised version on the simulation of diffusion bridges.

stat.CO↗

arXiv2019

High-dimensional scaling limits of piecewise deterministic sampling algorithms

Piecewise deterministic Markov processes are an important new tool in the design of Markov Chain Monte Carlo algorithms. Two examples of fundamental importance are the Bouncy Particle Sampler (BPS) and the Zig-Zag process (ZZ). In this paper scaling limits for both algorithms are determined. Here the dimensionality of the space tends towards infinity and the target distribution is the multivariate standard normal distribution. For several quantities of interest (angular momentum, first coordinate, and negative log-density) the scaling limits show qualitatively very different and rich behaviour. Based on these scaling limits the performance of the two algorithms in high dimensions can be compared. Although for angular momentum both processes require only a computational effort of $O(d)$ to obtain approximately independent samples, the computational effort for negative log-density and first coordinate differ: for these BPS requires $O(d^2)$ computational effort whereas ZZ requires $O(d)$. Finally we provide a criterion for the choice of the refreshment rate of BPS.

math.PR↗

arXiv2018

On One-Dimensional Riccati Diffusions

This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. These one-dimensional stochastic differential equations exhibit a quadratic drift function and a non-Lipschitz continuous diffusion function. We present a novel approach, combining tangent process techniques, Feynman-Kac path integration, and exponential change of measures, to derive sharp exponential decays to equilibrium. We also provide uniform estimates with respect to the time horizon, quantifying with some precision the fluctuations of these diffusions around a limiting deterministic Riccati differential equation. These results provide a stronger and almost sure version of the conventional central limit theorem. We illustrate these results in the context of ensemble Kalman-Bucy filtering. To the best of our knowledge, the exponential stability and the fluctuation analysis developed in this work are the first results of this kind for this class of nonlinear diffusions.

math.PR↗

arXiv2017

A Multi-Index Markov Chain Monte Carlo Method

In this article we consider computing expectations w.r.t.~probability laws associated to a certain class of stochastic systems. In order to achieve such a task, one must not only resort to numerical approximation of the expectation, but also to a biased discretization of the associated probability. We are concerned with the situation for which the discretization is required in multiple dimensions, for instance in space and time. In such contexts, it is known that the multi-index Monte Carlo (MIMC) method can improve upon i.i.d.~sampling from the most accurate approximation of the probability law. Indeed by a non-trivial modification of the multilevel Monte Carlo (MLMC) method and it can reduce the work to obtain a given level of error, relative to the afore mentioned i.i.d.~sampling and relative even to MLMC. In this article we consider the case when such probability laws are too complex to sampled independently. We develop a modification of the MIMC method which allows one to use standard Markov chain Monte Carlo (MCMC) algorithms to replace independent and coupled sampling, in certain contexts. We prove a variance theorem which shows that using our MIMCMC method is preferable, in the sense above, to i.i.d.~sampling from the most accurate approximation, under assumptions. The method is numerically illustrated on a problem associated to a stochastic partial differential equation (SPDE).

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arXiv2017

Bayesian inference for Stable Levy driven Stochastic Differential Equations with high-frequency data

In this article we consider parametric Bayesian inference for stochastic differential equations (SDE) driven by a pure-jump stable Levy process, which is observed at high frequency. In most cases of practical interest, the likelihood function is not available, so we use a quasi-likelihood and place an associated prior on the unknown parameters. It is shown under regularity conditions that there is a Bernstein-von Mises theorem associated to the posterior. We then develop a Markov chain Monte Carlo (MCMC) algorithm for Bayesian inference and assisted by our theoretical results, we show how to scale Metropolis-Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio going to zero in the large data limit. Our algorithm is presented on numerical examples that help to verify our theoretical findings.

math.ST↗

arXiv2017

Bayesian Static Parameter Estimation for Partially Observed Diffusions via Multilevel Monte Carlo

In this article we consider static Bayesian parameter estimation for partially observed diffusions that are discretely observed. We work under the assumption that one must resort to discretizing the underlying diffusion process, for instance using the Euler-Maruyama method. Given this assumption, we show how one can use Markov chain Monte Carlo (MCMC) and particularly particle MCMC [Andrieu, C., Doucet, A. and Holenstein, R. (2010). Particle Markov chain Monte Carlo methods (with discussion). J. R. Statist. Soc. Ser. B, 72, 269--342] to implement a new approximation of the multilevel (ML) Monte Carlo (MC) collapsing sum identity. Our approach comprises constructing an approximate coupling of the posterior density of the joint distribution over parameter and hidden variables at two different discretization levels and then correcting by an importance sampling method. The variance of the weights are independent of the length of the observed data set. The utility of such a method is that, for a prescribed level of mean square error, the cost of this MLMC method is provably less than i.i.d. sampling from the posterior associated to the most precise discretization. However the method here comprises using only known and efficient simulation methodologies. The theoretical results are illustrated by inference of the parameters of two prototypical processes given noisy partial observations of the process: the first is an Ornstein Uhlenbeck process and the second is a more general Langevin equation.

stat.CO↗

arXiv2016

Rate optimality of Random walk Metropolis algorithm in high-dimension with heavy-tailed target distribution

The choice of the increment distribution is crucial for the random-walk Metropolis-Hastings (RWM) algorithm. In this paper we study the optimal choice in high-dimension setting among all possible increment distributions. The conclusion is rather counter intuitive, but the optimal rate of convergence is attained by the usual choice, the normal distribution as the increment distribution. In particular, no heavy-tailed increment distribution can improve the rate.

stat.ME↗

arXiv2016

Multilevel Particle Filters: Normalizing Constant Estimation

In this article we introduce two new estimates of the normalizing constant (or marginal likelihood) for partially observed diffusion (POD) processes, with discrete observations. One estimate is biased but non-negative and the other is unbiased but not almost surely non-negative. Our method uses the multilevel particle filter of Jasra et al (2015). We show that, under assumptions, for Euler discretized PODs and a given $\varepsilon>0$. in order to obtain a mean square error (MSE) of $\mathcal{O}(\varepsilon^2)$ one requires a work of $\mathcal{O}(\varepsilon^{-2.5})$ for our new estimates versus a standard particle filter that requires a work of $\mathcal{O}(\varepsilon^{-3})$. Our theoretical results are supported by numerical simulations.

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