Nonformative

Nonformative, in statistics, refers to priors that are intended to convey little or no substantive information about a parameter before observing the data. In Bayesian analysis, a noninformative (or objective) prior is chosen so that the posterior distribution is driven primarily by the data, especially when prior knowledge is weak or disputed. Noninformative priors can be implemented as improper priors that are not normalizable but yield a proper posterior, or as proper priors designed to be as uninformative as possible.

Constructions and examples include uniform priors on a bounded parameter space, such as theta ~ Uniform(0,1) for a probability, representing maximal ignorance within that interval. For an unbounded location parameter, a flat prior p(mu) ∝ constant on the real line is common, though improper. Jeffreys priors use the square root of the Fisher information, yielding priors that are invariant under reparameterization; for a Bernoulli parameter theta this gives a Beta(1/2, 1/2), and for a normal mean with known variance a flat prior on mu. Reference priors, developed by Bernardo and colleagues, are designed to maximize information from the data for a given model.

Limitations and considerations include that nonformative priors are not uniquely defined and may depend on parameterization; improper priors can lead to improper posteriors in some models. They can also influence results, especially in small samples or complex models. In practice, researchers may use weakly informative priors to maintain objectivity while improving stability and robustness.