ParameterSet

Parameter set, or parameter space, refers to the collection of all values that the parameters of a model may assume. It defines the domain for the parameter vector and, together with the model equations, determines the system's behavior. The set may be continuous or discrete and can have constraints that reflect prior knowledge or theoretical limits.

In statistics and probabilistic modeling, the parameter set Θ is the domain of the model’s unknown quantities. For a parametric family, θ ∈ Θ specifies the distribution or function. The choice of Θ affects identifiability, the interpretation of parameters, and the feasibility of estimation. Common cases include Θ = R^k for real-valued parameters or Θ ⊂ R^k with bounds.

Constraints may arise from physical limits, prior information, or model structure. Bound constraints, equality or inequality constraints, or monotonicity can define a feasible parameter set. Identifiability requires that different θ in Θ produce distinct distributions; poor identifiability can invalidate estimation.

Estimation and inference operate over the parameter set. Optimization seeks a θ in Θ that optimizes a criterion such as the likelihood or a loss function. The geometry of Θ can influence convergence, regularization, and the existence of solutions; reparameterization can sometimes simplify the problem.

Examples include linear regression, where Θ = R^{p+1}, and logistic regression, where Θ = R^{p+1}. In Bayesian settings, the prior is defined over Θ and the posterior is computed for θ.