Joshua C Chang

Joshua C Chang

Countably infinite systems of linear ODEs arise as forward equations for many continuous-time Markov processes. The standard recipe -- truncate to a finite cap N and exponentiate -- pays cubic cost in N and a time-growing boundary-feedback bias. We identify a structural condition on the rates, L_{n+r,n} = alpha_r n + beta_r ("linear-rate"), under which the generating function G(z,t) = sum_n x_n(t) z^n satisfies a first-order linear PDE in z, and the method of characteristics yields a composition-multiplier representation G(z,t) = K_t(z) G(Phi_t(z), 0). The Taylor coefficients of Phi_t and K_t on any output window {0,...,N} are determined exactly by a closed lower-triangular polynomial ODE on R^{2(N+1)}, independent of any coefficients above N. Truncation enters only through the support M_0 of the initial law, set independently of N. For binary birth-death the closure collapses to the geometric tail p_n(t) = p_1(t) rho(t)^{n-1} with rho(t) = lambda(1 - e^{-(mu-lambda)t})/(mu - lambda e^{-(mu-lambda)t}). The linear-rate class spans Markov branching with immigration, multi-type branching, matrix-valued telegraph and G/R elongation, and signed or non-stochastic hierarchies. When the generator decomposes as L = A + B with A linear-rate and B non-affine (Schlogl bistable, predator-prey, lattice reaction-diffusion), we pair the closure with Strang splitting on B; Richardson extrapolation lifts the time order to Delta-t^4 at ~3x wall clock. On the Schlogl problem at V=500, N=8,000, the split runs 6.3x faster than dense Pade and 20x faster than sparse Krylov expv. For the stationary regime, a closure-Strang power iteration extends the same machinery to multi-dimensional product-state-space generators where sparse LU hits OOM/OOT or boundary-projection bias at usable caps. Numerical experiments locate where each route wins and where it is dominated by standard tools.

Joshua C Chang, Xiangting Li, Shixin Xu et al.

Importance sampling (IS) allows one to approximate leave one out (LOO) cross-validation for a Bayesian model, without refitting, by inverting the Bayesian update equation to subtract a given data point from a model posterior. For each data point, one computes expectations under the corresponding LOO posterior by weighted averaging over the full data posterior. This task sometimes requires weight stabilization in the form of adapting the posterior distribution via transformation. So long as one is successful in finding a suitable transformation, one avoids refitting. To this end, we motivate the use of bijective perturbative transformations of the form $T(\boldsymbolθ)=\boldsymbolθ + h Q(\boldsymbolθ),$ for $0<h\ll 1,$ and introduce two classes of such transformations: 1) partial moment matching and 2) gradient flow evolution. The former extends prior literature on moment-matching under the recognition that adaptation for LOO is a small perturbation on the full data posterior. The latter class of methods define transformations based on relaxing various statistical objectives: in our case the variance of the IS estimator and the KL divergence between the transformed distribution and the statistics of the LOO fold. Being model-specific, the gradient flow transformations require evaluating Jacobian determinants. While these quantities are generally readily available through auto-differentiation, we derive closed-form expressions in the case of logistic regression and shallow ReLU activated neural networks. We tested the methodology on an $n\ll p$ dataset that is known to produce unstable LOO IS weights.