Ptheta

Ptheta is a symbolic name used in probability theory and statistics to denote a parametric family of probability measures, typically written Pθ. The parameter θ encodes the characteristics of the distribution and may be a scalar or a vector. When a random variable X is said to have law Pθ, its distribution is determined by θ. The notation emphasizes that θ governs the distribution's location, scale, shape, or other features.

In statistical inference, Pθ serves as the basis for the likelihood function. Given data x1, ..., xn assumed to be independent with Xi ~ Pθ, the likelihood is L(θ) = ∏ fθ(xi). Estimation aims to select θ̂ that maximizes L or its logarithm. Bayesian methods place a prior on θ and update it with data to obtain a posterior distribution over θ.

Common examples include the normal family where Pθ is N(μ, σ^2) with θ = (μ, σ^2); the Bernoulli family with θ = p; and the exponential family, which can be parameterized by a natural parameter η and sufficient statistics T(X). In information geometry, Pθ forms a statistical manifold equipped with the Fisher information metric.

Limitations include identifiability issues and boundary behavior of the parameter space. The concept is foundational across parametric statistics and machine learning, serving as a formal anchor for inference, estimation, and learning algorithms.

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