Bayesiano

Bayesiano, in the context of statistics and data analysis, refers to approaches and concepts derived from Bayes' theorem and Bayesian inference. It describes a framework in which probabilities express degrees of belief about unknown quantities and are updated as new data arrive.

The core idea is to combine prior information with observed data to form a posterior belief. The posterior distribution P(θ|D) is proportional to the product of the likelihood P(D|θ), which describes how data are generated given θ, and the prior P(θ), which encodes previous knowledge: P(θ|D) ∝ P(D|θ) P(θ). The prior, likelihood, and posterior are central components, with the marginal likelihood P(D) acting as a normalizing constant. Conjugate priors are often used to obtain analytical posteriors in simpler models.

Inference in bayesiano frameworks can be analytical in cases with conjugate priors, but more commonly relies on numerical methods. Markov chain Monte Carlo (MCMC) and variational inference (VI) are standard tools for approximating posteriors in complex models. Bayesian methods also frequently use hierarchical modeling to share information across groups and to account for multiple levels of uncertainty.

Applications span many domains, including medicine, finance, epidemiology, machine learning, and natural language processing. They provide coherent uncertainty quantification and the ability to update beliefs sequentially as data accumulate, while allowing incorporation of prior knowledge.

Limitations include sensitivity to the choice of priors, computational cost for large or complex models, and potential model misspecification. The bayesiano paradigm has evolved from the work of Thomas Bayes and Laplace, with modern developments driven by advances in computation and data science.