A family of prior distributions chosen so that, after updating beliefs with a given likelihood function, the resulting updated belief over parameters belongs to the same parametric family as the prior.

Principle

Closure under the Bayesian updating rule: prior × likelihood yields a function of the same parametric form as the prior, permitting posterior parameters to be obtained by simple algebraic updates of hyperparameters.

Demonstration

For binary-trial data with counts of successes and failures, selecting a Beta prior over the success probability produces an updated distribution with Beta form whose shape parameters are incremented by observed counts.

Misapplication

Selecting a conjugate prior solely for algebraic convenience when its implied prior information contradicts substantive knowledge, thereby biasing updated beliefs inappropriately.

Consequence

Provides analytic tractability: closed-form updates of hyperparameters, reduced computational cost, and interpretable parameter updates that facilitate sequential or real-time updating.

Reversal

Using a nonconjugate prior yields an updated belief outside the prior family, typically requiring numerical integration or sampling to represent the updated distribution.

Boundary

Conjugacy depends on the likelihood family; many models lack simple conjugate priors, and conjugate choices may be improper or incompatible with regularity needs unless care is taken.

Semantic Tension

Tension with 'uninformative' or reference prior strategies: conjugate priors prioritize algebraic closure while reference priors prioritize minimal subjective influence; the two aims can conflict.

Synthesis

A conjugate prior is a prior family selected so that belief-updating with a specified likelihood preserves the family's form, enabling closed-form hyperparameter updates and analytical convenience.