Bayesian Statistics

Bayesian statistics represents a fundamentally different philosophical approach to uncertainty, treating probability as a measure of belief rather than just long-run frequency. This framework explicitly incorporates prior knowledge—formal mathematical representations of what was known before collecting new data—and updates these beliefs through Bayes' theorem as evidence accumulates. Unlike traditional approaches that treat parameters as fixed but unknown constants, Bayesian methods model parameters themselves as random variables with probability distributions.

This approach offers several distinct advantages: it provides direct probability statements about parameters ("There's a 95% probability the effect is between 0.2 and 0.5") rather than the more convoluted frequentist interpretations of confidence intervals; it naturally incorporates existing knowledge through prior distributions; and it handles uncertainty more transparently by producing full posterior distributions rather than point estimates. The Bayesian framework particularly shines in sequential decision-making scenarios where beliefs must be continuously updated as new information arrives—making it ideal for reinforcement learning, online algorithms, and adaptive clinical trials. It also excels when data is limited but prior information is strong, allowing more stable inference than purely data-driven approaches. While historically limited by computational challenges, modern computational methods like Markov Chain Monte Carlo (MCMC) and variational inference have made Bayesian approaches practical for increasingly complex models, leading to growing adoption across machine learning and data science applications.