Bayesian Linear Regression for Sensor Calibration

Table of Contents

Overview #

This series provides a complete, from-first-principles guide to Bayesian Linear Regression with Automatic Relevance Determination — a powerful technique for sensor calibration that combines rigorous uncertainty quantification, automatic feature selection, and a closed-form solution efficient enough for embedded systems.

Whether you’re calibrating a Hall effect sensor, building a measurement device, or working on any system where you need both accurate predictions and quantified confidence, BLR+ARD offers a principled alternative to black-box machine learning.

The Series #

Part 1: Mathematical Foundations #

When Your Sensor Knows What It Doesn’t Know: The Math Behind Bayesian Linear Regression and Automatic Relevance Determination

Start here to understand:

Why Bayesian statistics are the right tool for calibration (compared to least squares, neural networks, etc.)
The Gaussian miracle: why Bayesian posteriors have closed-form solutions
Linear regression through the Bayesian lens: deriving the posterior covariance and mean
How predictions become probability distributions with both aleatoric and epistemic uncertainty
The role of hyperparameters $\alpha_j$ and $\beta$, and how ARD performs automatic feature selection

Duration: ~20 minutes | Prerequisites: Linear algebra, basic probability

Part 2: Production Implementation #

From Math to Silicon: Implementing BLR+ARD with Rust and faer

Once you understand the math, discover how to implement it efficiently:

The critical principle: never invert a matrix if you can solve a linear system instead
Cholesky decomposition: the SPD factorization that underpins all numerical stability
Computing Mahalanobis distances without matrix inversion
Log-determinants via Cholesky diagonals (avoiding overflow/underflow)
Two algebraically identical posterior update forms: observation-space vs. parameter-space
The Woodbury matrix identity: exact transformation between spaces
Code patterns from the production blr-core crate in Rust using the faer numerical library

Duration: ~25 minutes | Prerequisites: Part 1, familiarity with Rust helpful but not required

Key Concepts #

Concept	What It Does	Why It Matters
Posterior Distribution	Encodes both point estimate and uncertainty	You know not just the prediction, but how confident you should be
Automatic Relevance Determination	Each feature gets its own regularization strength $\alpha_j$	Irrelevant features are suppressed without manual feature selection
Cholesky Decomposition	$A = LL^T$ for symmetric positive-definite matrices	Enables solving systems instead of inverting, reducing numerical error by a factor of condition number
Epistemic vs. Aleatoric Uncertainty	Model uncertainty vs. measurement noise	Tells you where you need more data (high epistemic) vs. where noise dominates (high aleatoric)
Hyperparameter Learning	EM algorithm for $\alpha_j$ and $\beta$	Automatically tune regularization without cross-validation loops

Use Cases #

BLR+ARD shines when you have:

Limited data — Hall sensor calibration with tens of observations, not millions
Interpretability requirements — Each learned feature has a clear physical meaning
Uncertainty budgets — You need to know not just predictions, but confidence bands
Resource constraints — Embedded systems, WASM, or real-time inference loops
Active learning — You can request measurements where uncertainty is highest

It’s less suitable for:

Massive unstructured data (ImageNet scale)
Problems where black-box optimization is acceptable
Domains where you have no domain knowledge for feature engineering

Quick Start #

Just want the math? → Read Part 1
Want to implement it? → Read both parts, then check the blr-core crate on GitHub
Want to use it as a library? → Look for the published Rust crate (coming soon to crates.io)
Want it in WASM? → Part 2 discusses embedding in WebAssembly components

What’s Next? #

Upcoming articles in this space will cover:

Active Learning Strategies — Automatically selecting the next measurement point to maximize information gain
WASM Deployment — Packaging BLR+ARD as a WebAssembly component for browser and edge deployment
Hyperparameter Sensitivity — Understanding when ARD selection is stable and when to use stronger priors
Multi-Sensor Fusion — Combining multiple sensors with heterogeneous noise models
Real-Time Calibration Loops — Sequential Bayesian updating during sensor operation

References & Further Reading #

Textbook: Pattern Recognition and Machine Learning by Christopher Bishop (Chapter 3: Linear Models for Regression)
Video: Philipp Hennig’s Probabilistic ML course (especially lectures 3–6 on Gaussian inference)
Original ARD paper: MacKay, D. J. (1994). “Bayesian nonlinear modeling for the prediction competition.” ASHRAE Transactions
Matrix inversion lemma: Woodbury matrix identity — the algebraic key to efficient Bayesian updates

Ready to dive in? Start with Part 1: When Your Sensor Knows What It Doesn’t Know →