PDtheta

PDtheta is a term used in probability theory and statistics to denote a parametric family of probability distributions indexed by a parameter vector theta. In this usage the collection {P_theta : theta in Theta} assigns to each theta a probability measure P_theta on a given measurable space (X, F). The subscript theta indicates the dependence of the distribution on the parameter; PDtheta can be read as the PD for theta, or as a concise shorthand for a parametric distribution.

When a density exists with respect to a base measure mu, one writes p_theta(x) for the density of P_theta. Regularity conditions often assume that theta -> p_theta(x) is differentiable, enabling the definitions of the score function, Fisher information, and likelihood-based methods. The Fisher information matrix is I(theta) = E_theta[(∇_theta log p_theta(X)) (∇_theta log p_theta(X))^T], provided expectations are finite.

Examples: the family of normal distributions with fixed variance sigma^2 and varying mean mu is a PDtheta with theta = mu. The exponential family forms a broad class of PDtheta with densities of the form p_theta(x) = h(x) exp(η(theta)·T(x) - A(theta)).

Applications include parameter estimation via maximum likelihood, construction of confidence sets, hypothesis testing, and Bayesian inference using a prior over theta. The notation varies by author, with alternative symbols such as P_theta, f_theta, or p(x|theta) commonly used.

See also: probability distribution, parametric family, likelihood, Fisher information, exponential family.