Hierarchical vs. Pooled Bayesian Regression

This repository contains the materials for a term project completed for ELG5218 "Uncertainty Evaluation in Engineering Measurements and Machine Learning" (Winter 2022), taught by Prof. Miodrag Bolic at the University of Ottawa.

Course GitHub Repository: Health-Devices/Course-Uncertanty-Machine-Learning-2021

Project Overview

This project explores and compares different Bayesian regression models for predicting blood pressure from Pulse Arrival Time (PAT) measurements:

• Pooled Model: Assumes all observations have the same parameters (θ), ignoring group information
• Separate Model: Assumes personalized parameters for each observation
• Hierarchical Model: Assumes different parameters for observations of each group, while all parameters are sampled from a common distribution

Repository Contents

• Jupyter notebook with Julia code implementation of the models
• Slides explaining the theoretical background and results
• Three key model diagrams (displayed below):
  • h_model_all.png - Hierarchical model diagram
  • p_model.png - Pooled model diagram
  • separate_lr.png - Separate model diagram

Model Diagrams

Pooled Model

  

Separate Model

  

Hierarchical Model

Key Findings

• The hierarchical model provides a better fit for the training data, learning different parameters (α, β, σ) for different subjects
• The pooled model is too constrained to capture individual differences between subjects
• Posterior Predictive Checks (PPC) and Highest Density Intervals (HDI) show the hierarchical model's superior fit on training data
• However, for new subjects (test data), the hierarchical model's predictions have high uncertainty
• As the number of data points per subject increases, the hierarchical model better distinguishes between subjects

Use Case

This work demonstrates blood pressure prediction using PAT measurements, with the relationship:

BP = α + β * ln(PAT) + ε

The dataset contains multiple PAT and BP readings from 10 subjects, showcasing how personalized Bayesian models can improve predictions for individual subjects.

Metrics Used

The models were evaluated using:

• WAIC (Widely Applicable Information Criterion)
• LOO (Leave-One-Out cross-validation)
• MAE (Mean Absolute Error) on both training and test data