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| Priors | |
|---|---|
| Name | Priors |
| Field | Statistics, Bayesian inference |
| Introduced | 18th century (concepts), 20th century (formalization) |
| Notable | Thomas Bayes, Pierre-Simon Laplace, Harold Jeffreys, Edwin T. Jaynes |
Priors Priors are probability distributions representing beliefs about parameters or hypotheses before observing current data. They play a central role in Bayesian inference, connecting historical knowledge, subjective judgment, and formal models to update uncertainty using observed evidence. Their choice affects estimation, prediction, model selection, and decision-making across science, engineering, and policy.
In Bayesian methodology, a prior combines with a likelihood to produce a posterior via Bayes's theorem, a formulation associated with Thomas Bayes and Pierre-Simon Laplace. Foundational contributors include Harold Jeffreys, Ronald A. Fisher (historical debates), and Edwin T. Jaynes, who linked inference to information theory and the Principle of Maximum Entropy. Priors range from informative constructions based on studies like those from World Health Organization and Centers for Disease Control and Prevention to noninformative proposals used in frequentist comparisons by figures such as Jerzy Neyman and Egon Pearson.
Common classes include conjugate priors exemplified in the Beta distribution for binomial models and the Normal distribution for Gaussian models; improper priors like uniform measures used by Laplace; hierarchical priors appearing in models developed by researchers at institutions such as Harvard University and Stanford University; shrinkage priors like the James–Stein estimator and the Lasso-related Laplace prior; sparsity-inducing priors inspired by work at Carnegie Mellon University and in compressed sensing by David Donoho; and objective priors including reference priors advanced by José M. Bernardo and invariant priors advocated by Harold Jeffreys. Mixture priors and spike-and-slab constructions have been applied in genetics consortia like 1000 Genomes Project and in machine learning work at Google DeepMind.
Elicitation techniques translate expert judgment from panels such as those convened by the National Academies or committees at World Bank into parametric or nonparametric priors. Methods include moment matching used in actuarial studies at Prudential Financial, quantile-based elicitation employed in clinical trials overseen by Food and Drug Administration, and structured elicitation protocols developed at RAND Corporation. Empirical Bayes approaches estimate priors from data in contexts like genomics at Broad Institute or astronomy at European Southern Observatory, while subjective Bayesian frameworks lean on normative arguments from philosophers like Bruno de Finetti.
Priors determine posterior concentration, shrinkage, and regularization effects seen in studies by teams at MIT and Columbia University. Properties of interest include conjugacy facilitating analytical updates in textbook examples, tail behavior affecting outlier robustness as analyzed in robust statistics literature from John Tukey and Peter Huber, and propriety ensuring posteriors are valid probability distributions in work by legal statisticians advising institutions like International Court of Justice. Sensitivity analysis, advocated in reports by Institute of Medicine, evaluates how posteriors vary with alternative priors; model averaging procedures such as Bayesian model averaging used in economic forecasting by Federal Reserve researchers mitigate prior-dependent conclusions.
Computational strategies for working with priors include Markov chain Monte Carlo algorithms developed at Los Alamos National Laboratory and refined in software from teams at Stan Development Team and Hugh Salim's collaborators; variational inference popularized in industry labs like DeepMind and universities including UC Berkeley; importance sampling used in particle filter systems at NASA; and Laplace approximations rooted in asymptotic theory by Henri Poincaré. Software implementations span packages maintained by groups at R Consortium and Python Software Foundation, enabling hierarchical and nonparametric priors such as Gaussian processes from research at Google.
Priors are used in clinical trial design at National Institutes of Health, risk assessment at World Bank, signal processing at Bell Labs, cosmology at NASA and European Space Agency, ecology studies by Smithsonian Institution researchers, natural language processing at OpenAI, and legal forensics examined in reports by American Statistical Association. In genetics, priors guide association mapping in projects like Human Genome Project; in finance, risk models at Goldman Sachs and central banks incorporate shrinkage priors; in engineering, reliability analyses cite standards from IEEE.
Critiques focus on subjectivity and potential misuse highlighted by commentators in journals such as Nature and Science, legal challenges considered in cases before United States Supreme Court panels, and reproducibility concerns raised by initiatives at Center for Open Science. Alternatives include frequentist estimators promoted by practitioners at American Statistical Association, likelihoodist approaches advocated by scholars at London School of Economics, and robust, nonparametric methods developed by teams at INRIA.