parameterspace

In statistics and machine learning, the parameter space is the set of all possible values that a model's parameters can take. For a model with n parameters, the parameter space is typically a subset of R^n; each point in the space specifies a complete configuration of the model. The domain can be continuous, discrete, or a mixture, and may include constraints such as bounds or equality and inequality relationships.

The parameter space is distinct from the data space. An estimator or learning algorithm searches the parameter space to find values that optimize a criterion, such as a loss or likelihood function. In Bayesian inference, the parameter vector is treated as a random variable with a prior distribution over the parameter space and a posterior distribution after observing data.

The parameter space can be high-dimensional, which brings challenges such as the curse of dimensionality, identifiability, and overparameterization. Methods to explore it include grid search, random search, gradient-based optimization, evolutionary algorithms, and Bayesian optimization. Markov chain Monte Carlo (MCMC) techniques sample from the posterior distribution over the parameter space.

Hyperparameters occupy a related but distinct concept. Hyperparameters control the learning process or model structure and are not typically learned from the data within the primary modeling framework; they define a separate hyperparameter space. Understanding the parameter space is central to model specification, estimation, inference, and optimization across statistical and machine learning domains.