Normalprioras

Normalprioras is a class of prior distributions used in Bayesian statistics to express beliefs about a parameter vector that are centered around a normal model. The term combines 'normal' and 'prior' to indicate its base form as a multivariate normal distribution.

In standard form, a Normalprioras prior assigns θ ∼ N(μ, Σ). Because the parameters μ and Σ can be unknown, a Normalprioras framework commonly employs a hierarchical specification, such as μ ∼ N(μ0, Λ) and Σ ∼ Inverse-Wishart(Ψ, ν), or their scalable variants. This structure allows the data to inform the location and the scale of θ while preserving a prior belief in approximate normality.

Its practical properties include facilitating analytical tractability in some linear Gaussian models with known Σ, where the posterior for θ is Gaussian. When Σ is unknown, the joint posterior can be sampled using Gibbs steps or related collapsed schemes; marginalizing over μ or Σ often yields heavier-tailed behavior akin to a Student-t distribution. Extensions incorporate scale mixtures of normals to obtain robust priors, and can employ block-diagonal or structured Σ to encode correlations. Sparsity-inducing adaptations may be built by combining with shrinkage priors on the precision matrix.

Applications of Normalprioras appear in Bayesian linear regression, factor analysis, multivariate calibration, and various hierarchical modeling contexts. The framework provides a natural baseline for comparing non-Gaussian priors and can serve as a component in more complex priors, such as hierarchical or Bayesian shrinkage models.

See also: normal distribution, prior distribution, Bayesian statistics, hierarchical modeling.