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1. Computing transition path theory quantities with trajectory stratification

   https://doi.org/10.1063/5.0087058  

   Vani, Bodhi P.; Weare, Jonathan; Dinner, Aaron R. (July 2022, The Journal of Chemical Physics)

   Transition path theory computes statistics from ensembles of reactive trajectories. A common strategy for sampling reactive trajectories is to control the branching and pruning of trajectories so as to enhance the sampling of low probability segments. However, it can be challenging to apply transition path theory to data from such methods because determining whether configurations and trajectory segments are part of reactive trajectories requires looking backward and forward in time. Here, we show how this issue can be overcome efficiently by introducing simple data structures. We illustrate the approach in the context of nonequilibrium umbrella sampling, but the strategy is general and can be used to obtain transition path theory statistics from other methods that sample segments of unbiased trajectories. 

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2. A diffusion approach to Stein’s method on Riemannian manifolds

   https://doi.org/10.3150/23-BEJ1625  

   Le, Huiling; Lewis, Alexander; Bharath, Karthik; Fallaize, Christopher (May 2024, Bernoulli)

   We detail an approach to developing Stein’s method for bounding integral metrics on probability measures defined on a Riemannian manifold M. Our approach exploits the relationship between the generator of a diffusion on M having a target invariant measure and its characterising Stein operator. We consider a pair of such diffusions with different starting points, and through analysis of the distance process between the pair, derive Stein factors, which bound the solution to the Stein equation and its derivatives. The Stein factors contain curvature-dependent terms and reduce to those currently available for Rm, and moreover imply that the bounds for Rm remain valid when M is a flat manifold. 

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3. A Benchmark for the Bayesian Inversion of Coefficients in Partial Differential Equations

   https://doi.org/10.1137/21M1399464  

   Aristoff, David; Bangerth, Wolfgang (November 2023, SIAM Review)

   na (Ed.)

   Bayesian methods have been widely used in the last two decades to infer statistical proper- ties of spatially variable coefficients in partial differential equations from measurements of the solutions of these equations. Yet, in many cases the number of variables used to param- eterize these coefficients is large, and oobtaining meaningful statistics of their probability distributions is difficult using simple sampling methods such as the basic Metropolis– Hastings algorithm—in particular, if the inverse problem is ill-conditioned or ill-posed. As a consequence, many advanced sampling methods have been described in the literature that converge faster than Metropolis–Hastings, for example, by exploiting hierarchies of statistical models or hierarchies of discretizations of the underlying differential equation. At the same time, it remains difficult for the reader of the literature to quantify the advantages of these algorithms because there is no commonly used benchmark. This paper presents a benchmark Bayesian inverse problem—namely, the determination of a spatially variable coefficient, discretized by 64 values, in a Poisson equation, based on point mea- surements of the solution—that fills the gap between widely used simple test cases (such as superpositions of Gaussians) and real applications that are difficult to replicate for de- velopers of sampling algorithms. We provide a complete description of the test case and provide an open-source implementation that can serve as the basis for further experiments. We have also computed 2 × 10^11 samples, at a cost of some 30 CPU years, of the poste- rior probability distribution from which we have generated detailed and accurate statistics against which other sampling algorithms can be tested. 

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4. An Invitation to Sequential Monte Carlo Samplers

   https://doi.org/10.1080/01621459.2022.2087659  

   Dai, Chenguang; Heng, Jeremy; Jacob, Pierre E.; Whiteley, Nick (June 2022, Journal of the American Statistical Association)

   Statisticians often use Monte Carlo methods to approximate probability distributions, primarily with Markov chain Monte Carlo and importance sampling. Sequential Monte Carlo samplers are a class of algorithms that combine both techniques to approximate distributions of interest and their normalizing constants. These samplers originate from particle filtering for state space models and have become general and scalable sampling techniques. This article describes sequential Monte Carlo samplers and their possible implementations, arguing that they remain under-used in statistics, despite their ability to perform sequential inference and to leverage parallel processing resources among other potential benefits. Supplementary materials for this article are available online. 

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5. Stein’s method and approximating the multidimensional quantum harmonic oscillator

   https://doi.org/10.1017/jpr.2022.125  

   McKeague, Ian W; Swan, Yvik (September 2023, Journal of Applied Probability)

   Abstract Stein’s method is used to study discrete representations of multidimensional distributions that arise as approximations of states of quantum harmonic oscillators. These representations model how quantum effects result from the interaction of finitely many classical ‘worlds’, with the role of sample size played by the number of worlds. Each approximation arises as the ground state of a Hamiltonian involving a particular interworld potential function. Our approach, framed in terms of spherical coordinates, provides the rate of convergence of the discrete approximation to the ground state in terms of Wasserstein distance. Applying a novel Stein’s method technique to the radial component of the ground state solution, the fastest rate of convergence to the ground state is found to occur in three dimensions. 

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