Bayesian Methods

Bayesian statistics provides a coherent framework for reasoning under uncertainty by combining prior knowledge with observed data:

Probabilistic Programming: Frameworks like PyMC, Stan, and TensorFlow Probability enable Bayesian modeling with automatic inference. These tools power applications from medical diagnosis systems that quantify uncertainty to marketing mix models that account for prior knowledge about advertising effectiveness.

Bayesian Neural Networks: By placing distributions over weights instead of point estimates, these networks quantify prediction uncertainty. Self-driving vehicles use these uncertainty estimates to make safer decisions in ambiguous situations. Medical imaging systems communicate confidence levels alongside diagnoses, helping doctors prioritize cases requiring further investigation.

Bayesian Optimization: This approach to hyperparameter tuning models the performance landscape using Gaussian Processes, efficiently identifying promising configurations. This technique accelerates drug discovery by optimizing molecular properties and improves manufacturing processes by finding optimal operating conditions with minimal experimentation.

Bayesian Model Averaging: Instead of selecting a single "best" model, this approach combines predictions from multiple models weighted by their posterior probabilities. In climate science, this produces more robust projections by integrating diverse models. For stock market prediction, it hedges against the risk of model misspecification.

Prior Knowledge Integration: Bayesian methods explicitly incorporate domain expertise through prior distributions. In robotics, priors encode physical constraints and laws of motion. In natural language processing, priors capture linguistic regularities. For few-shot learning applications, priors enable generalization from minimal examples.

The Bayesian approach fundamentally changes how we think about learning from data—instead of seeking point estimates, we aim to capture entire distributions of possibilities consistent with our observations and prior knowledge. This perspective is increasingly valuable as machine learning systems are deployed in high-stakes domains where quantifying uncertainty is essential for responsible decision-making.