arXiv subjects

Joshua C Chang

Publications and source records attributed to Joshua C Chang.

3 recordsLinked to original sources

arXiv2026

Operator splitting for exploiting linear-rate closure in solving infinite ODE hierarchies

We introduce an operator-splitting method for infinite hierarchies of linear ordinary differential equations (ODEs) indexed by nonnegative integers. When the coupling coefficients depend linearly on the count index, an exact transformation closes the equations on finite count-index windows without an upper-boundary value. For more general hierarchies, Strang splitting applies the linear-rate closure during the linear-rate substeps and a conventional capped solver to the remainder. We derive the closure from generating functions and the method of characteristics and extend it to multi-indexed systems. The derivation requires neither positivity nor mass conservation, so it applies to a wider class of systems than the examplar stochastic models presented here. We discuss branching processes, stochastic predator-prey dynamics, the Schl\"ogl chemical kinetics model, and a telegraph model for gene expression. Through numerical experiments and computational cost analyses we demonstrate that our operator splitting method is typically advantageous for solving large scale systems in terms of memory usage and computational time, while retaining accuracy competitive with finite state projection (FSP) methods.

math.NA↗

arXiv2026

Perturbative adaptive importance sampling for Bayesian LOO cross-validation

Importance sampling (IS) is an efficient stand-in for model refitting in performing (LOO) cross-validation (CV) on a Bayesian model. IS inverts the Bayesian update for a single observation by reweighting posterior samples. The so-called importance weights have high variance -- we resolve this issue through adaptation by transformation. We observe that removing a single observation perturbs the posterior by $\mathcal{O}(1/n)$, motivating bijective transformations of the form $T(θ)=θ+ h Q(θ)$ for $0<h\ll 1.$ We introduce several such transformations: partial moment matching, which generalizes prior work on affine moment-matching with a tunable step size; log-likelihood descent, which partially invert the Bayesian update for an observation; and gradient flow steps that minimize the KL divergence or IS variance. The gradient flow and likelihood descent transformations require Jacobian determinants, which are available via auto-differentiation; we additionally derive closed-form expressions for logistic regression and shallow ReLU networks. We tested the methodology on classification ($n\ll p$), count regression (Poisson and zero-inflated negative binomial), and survival analysis problems, finding that no single transformation dominates but their combination nearly eliminates the need to refit.

stat.ME↗

arXiv2014

A path-integral approach to Bayesian inference for inverse problems using the semiclassical approximation

We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse problems of recovering continuous (infinite dimensional) coefficient functions from ordinary or partial differential equations (ODE, PDE), a problem which is typically ill-posed. Regularization of these problems using $L^2$ function spaces (Tikhonov regularization) is equivalent to Bayesian probabilistic inference, using a Gaussian prior. The Bayesian interpretation of inverse problem regularization is useful since it allows one to quantify and characterize error and degree of precision in the solution of inverse problems, as well as examine assumptions made in solving the problem -- namely whether the subjective choice of regularization is compatible with prior knowledge. Using path-integral formalism, Bayesian inference can be explored through various perturbative techniques, such as the semiclassical approximation, which we use in this manuscript. Perturbative path-integral approaches, while offering alternatives to computational approaches like Markov-Chain-Monte-Carlo (MCMC), also provide natural starting points for MCMC methods that can be used to refine approximations. In this manuscript, we illustrate a path-integral formulation for inverse problems and demonstrate it on an inverse problem in membrane biophysics as well as inverse problems in potential theories involving the Poisson equation.

physics.data-an↗