BIDSim: A Hybrid Discretized Chained Virtual Urns Simulator for High-Dimensional Categorical Bayesian Inference (Part 4/7)

Bayesian inference for high-dimensional multinomial problems (n > 3) has long been dominated by Markov Chain Monte Carlo (MCMC) methods, which suffer from heavy computational latency, strong sample autocorrelation (lag-1 > 0.90), burn-in losses, convergence diagnostic burdens, and opaque stochastic behavior. This paper introduces BIDSim, a novel stochastic simulation engine built on the Discretized Bayesian Induction (BID) framework. Operating as a two-phase hybrid algorithm, BIDSim combines deterministic precomputation with fast stochastic sampling. By grounding probability spaces in discrete virtual urns of configurable resolution (Popu = 1,000) and enforcing E.T. Jaynes's Principle of Maximum Entropy via the PEMPI algorithm, BIDSim replaces high-dimensional joint probability space integration with a sequence of 1D chained virtual urns under the Principle of Maximum Independence. BIDSim operates under a "compile-once, execute-forever" paradigm: Phase 1 is a deterministic quadratic compilation phase O((n-2) * Popu^2) in 50–180 ms that precalculates conditional cumulative distribution lookup tables via sequential 2D trinomial inductions (BID3). Phase 2 is a stochastic extraction phase executing linear-time O(t * n) chained urn sampling. Benchmark tests on a 5-candidate political opinion poll (N = 1,200, t = 100,000 draws) demonstrate that BIDSim delivers exact, I.I.D. draws with zero autocorrelation (lag-1 = 0.000) and zero burn-in requirement, achieving a 149.4x speedup (35.8 ms vs. 5,349.6 ms) over MCMC samplers (JAX-NUTS) while generating complete probability distributions for candidate scores, front-runner lead gaps, victory chances, and second-round runoff duel scenarios.