Bayesian Statistics
Bayesian statistics combines a probabilistic model, prior information and observed data to obtain a posterior distribution. The skill covers model specification, computation and interpretation of uncertainty. It is especially useful when uncertainty must propagate through a decision, but conclusions remain conditional on the likelihood, priors and data-generation assumptions.
What it is
Bayes' rule updates a prior distribution over unknown quantities using the likelihood of the observed data. The posterior can describe parameters, latent variables and predictions, and hierarchical models share information across related groups. Posterior predictive distributions include uncertainty about parameters as well as variation in new outcomes. For complex models, computation may use Markov chain Monte Carlo or approximate inference. The Bayesian framework makes assumptions explicit; it does not eliminate them. A posterior probability has an interpretation conditional on the chosen model and prior, which differs from the repeated-sampling interpretation of a frequentist confidence interval.
What the work involves
Specify the outcome distribution and the meaning of each parameter, then choose priors on a scale that has substantive meaning. Use prior predictive checks to identify implausible implied data before fitting. After computation, assess diagnostics and posterior predictive checks, and compare conclusions under reasonable alternative priors or model structures. Present estimates and credible intervals alongside decision-relevant predictions. The finished analysis should include reproducible model code, assumptions and checks that distinguish a computationally converged fit from a model that actually describes the observed phenomenon.
Illustrative example
In an illustrative service-quality study, several small teams have only a few recorded incidents. A hierarchical model estimates a team-specific rate while allowing information to be shared across teams through a population distribution. The analyst checks whether the model reproduces the variation in observed counts and examines sensitivity to the population prior. Reporting a range of plausible rates is more informative than ranking teams by noisy raw percentages, especially when a future staffing decision depends on uncertainty.
Limits and common mistakes
A precise posterior can still be wrong when the likelihood excludes important variation or the data are biased. Poorly chosen priors can dominate sparse data, and weak identification can make results sensitive to parameterization. Sampling diagnostics do not prove model adequacy; approximate methods may understate uncertainty. Hierarchical shrinkage is not evidence that groups are truly interchangeable. Report sensitivity and predictive checks, and avoid treating a credible interval as a model-free statement or a computational warning as something that can be ignored after a long run.
Prerequisites
Bayesian inference is built on probability distributions, Bayes' theorem, and likelihood — all of which require probability theory and information theory as prerequisites
Practical statistics (hypothesis testing, estimation) provides the frequentist baseline that Bayesian methods extend and contrast with
Related skills
- → is subcategory of: Statistical Inference
Sources and further reading
- Stan User’s Guide
Bayesian model specification, hierarchical models and predictive checks.
- Seeing Theory
Probability, Bayesian inference and statistical uncertainty.
Last updated: 2026-10-10