hatXt

hatXt is the notation used in estimation theory to denote the estimate of the hidden state X_t in a discrete-time dynamical system. Given a model that evolves X_t and produces observations Y_t, hatXt represents the best available guess of X_t based on information up to time t. In Bayesian terms, hatXt often denotes the conditional expectation E[X_t | Y_1, ..., Y_t], or a value that minimizes a chosen loss function such as squared error. It is common to distinguish between the posterior estimate x̂_t|t (after observing Y_1 through Y_t) and the prior or predicted estimate x̂_t|t-1 (based on information up to time t-1).

Estimation methods for hatXt include a range of algorithms. In linear Gaussian settings, the Kalman filter computes x̂_t|t as the posterior mean with a well-defined prediction-update recursion. For nonlinear models, extensions such as the extended Kalman filter and the unscented Kalman filter apply, while particle filters use a set of samples to approximate the posterior distribution of X_t. The choice of method depends on model structure, noise characteristics, and computational constraints.

Applications for hatXt are broad. It is used in robotics and navigation for state tracking, in signal processing for denoising and localization, and in finance and econometrics for latent-factor estimation. The accuracy of hatXt is evaluated via estimation error e_t = X_t − x̂_t, with performance influenced by model fidelity, noise statistics, and the estimation algorithm.

See also state estimation, Kalman filter, Bayesian estimation, and particle filter.