21.4 Predictive distributions
An elegant way of using the a posteriori distribution π(·|(x1, y1),…,(xn, yn)) is the predictive distribution of Yx based on the observed data (x1, y1),…,(xn, yn). This predictive distribution is calculated as the marginal distribution of Yx of the stochastic vector (Yx, ).
For continuous variables Yx and continuous parameter θ with probability density fx(…|θ) of Yx and a posteriori density π(·|(x1, y1),…,(xn, yn)) of the joint density g(·, ·) of (Yx,) is given by its values
Therefore the predictive distribution of Yx is given by its probability density px(·|(x1, y1),…,(xn, yn)) which is the marginal density of Yx, given by its values
The density px(·|(x1, y1),…,(xn, yn)) is denoted as the predictive density of Yx.
The predictive distribution of Yx can be used to calculate Bayesian predictive intervals for Yx.
Based on the predictive density a predictive interval [a; b] for Yx with probability of coverage 1 − α is defined by
Special predictive intervals ...