BUICU — Belief Updating for ICU Crowding Under Uncertainty

CS 109 Challenge Project

Overview

BUICU models ICU crowding as a stochastic process using Bayesian inference. The system performs sequential belief updating, propagates uncertainty via Monte Carlo simulation, compares models with proper scoring rules, decomposes uncertainty into epistemic vs. aleatoric components, and systematically analyzes failure modes — demonstrating 16 CS109 concepts in a single project.

Probabilistic Model

Random Variable	Distribution	Description
λ	Gamma(α₀, β₀)	Prior on arrival rate
N_t | λ	Poisson(λ · Δt)	Arrivals in window Δt
λ | data	Gamma(α₀+Σk, β₀+T)	Posterior (conjugate update)
N_future	NegBin(α_post, β_post/(β_post+Δt))	Posterior predictive
L	Empirical / LogNormal mixture	Length of stay
O_t	Monte Carlo simulation	Occupancy (random variable)

Usage

pip install -r requirements.txt
python main.py

All outputs (14 figures + writeup) are saved to output/.

14 Visualizations

#	Figure	CS109 Concept
01	Belief evolution + anomalies + KL	Bayesian updating, KL divergence
02	Posterior predictive check + Q-Q	Posterior predictive, calibration
03	Calibration (stationary vs windowed)	Coverage, PIT, model comparison
04	48h occupancy forecast fan chart	Monte Carlo, uncertainty intervals
05	Stationary vs windowed model	Non-stationarity, adaptive inference
06	Prior sensitivity convergence	Prior robustness, Bayesian consistency
07	Information gain + anomaly detection	KL divergence, hypothesis testing
08	LOS heavy-tail analysis	Distribution fitting, tail risk
09	Prior → Posterior transformation	Bayes' theorem visualization
10	Log predictive score comparison	Proper scoring rules
11	Sensitivity analysis tornado	Decision sensitivity, robustness
12	Variance decomposition	Law of total variance
13	MLE vs Bayesian comparison	Frequentist vs Bayesian, CLT
14	Full summary dashboard	All key results

CS109 Concepts Demonstrated (16)

1. Random Variables 2. Probability Distributions (Poisson, Gamma, NegBin, LogNormal)
2. Conditional Probability 4. Bayes' Theorem 5. Posterior Predictive
3. Conjugate Priors 7. Law of Total Variance 8. Monte Carlo Simulation
4. Maximum Likelihood Estimation 10. Central Limit Theorem
5. Information Theory (KL divergence) 12. Hypothesis Testing (p-values)
6. Model Comparison (proper scoring) 14. Calibration 15. Sensitivity Analysis
7. Prior Sensitivity

Key Results

• Posterior λ: 11.72 adm/day, 95% CI [11.23, 12.23]
• Model comparison: Windowed wins by 39.4 log-score units
• Variance decomposition: 99.4% stochastic, 0.6% parameter after 180 days
• Crowding forecast: 41.5% within 48h (from near-capacity scenario)
• Sensitivity: Capacity -20% → P(crowded) jumps from 9% to 96%
• 5 failure modes detected, combined CI widening ×2.27