Stat.Minimax.Squared­Loss.Finite

Given state-specific observation likelihoods, a real target for each state, a bounded randomized procedure, and a state, the statewise squared-loss risk is the procedure-design-weighted sum of likelihood-weighted squared differences between the selected action and that state's target.

Given a set of design indices and an observation space for each design index, the ambient coordinate space consists of one real design-weight vector and one real action coordinate for every design-index and observation pair.

Given a family of finite observation spaces indexed by design, a lower action bound, and an upper action bound, the feasible procedure-coordinate set contains exactly those ambient coordinates whose design weights form a probability distribution and whose action coordinates all lie in the closed interval from the lower to the upper bound.

Given a bounded finite randomized procedure, its ambient Euclidean coordinates are its design probabilities together with its action at every design-index and observation pair.

Given an ambient coordinate point that belongs to the feasible procedure-coordinate set, the corresponding bounded finite procedure uses its design-weight coordinates as design probabilities and its action coordinates as bounded decisions.

Given state-specific observation likelihoods, a real target for each state, an ambient coordinate point, and a state, the raw squared-loss risk is the design-coordinate-weighted sum of likelihood-weighted squared differences between action coordinates and that state's target.

Given state-specific observation likelihoods, a real target for each state, and an ambient coordinate point, the risk vector assigns to every state its raw squared-loss risk at that coordinate point.

Given a vector of weights over the finite state space that belongs to the standard probability simplex, the corresponding finite design is the probability design whose mass function is that vector.

Given a real-valued risk vector over states and a vector of prior weights, the finite Bayes payoff is the sum over states of each prior weight times its risk-vector value.

Given a finite set of design points, a finite observation set at each design point, ordered action bounds, a mixing weight no smaller than zero, the same weight no larger than one, and two feasible randomized procedures, the mixed feasible randomized procedure assigns the convex combination of their design masses at each design point. At a point with positive mixed mass it averages their actions using the corresponding posterior mixture weights, and at a point with zero mixed mass it uses the lower action bound.

Given a parameter space, a finite set of design points, a finite observation set at each design point, likelihood coefficients, a target value for each parameter, and two action bounds, the set of dominated risk vectors consists of exactly those real-valued functions on the parameter space for which there exists a feasible finite randomized procedure whose squared-loss risk is no greater at every parameter.

Given a dependent finite experiment, a prior probability design over states, a design index, and an observation in that design's observation space, the prior predictive mass is the sum over states of prior probability times the conditional probability of that observation.

Given a dependent finite experiment, a prior probability design over states, a design index, and an observation in that design's observation space, the predictive target numerator is the sum over states of prior probability times observation likelihood times the state's target.

Given a dependent finite experiment, a prior probability design over states, a design index, and an observation in that design's observation space, the guarded posterior mean is the predictive target numerator divided by predictive mass when that mass is nonzero, and is zero when it is zero.

Given a dependent finite experiment, a prior probability design over states, and a design index, the posterior residual is the sum, over states and observations, of prior probability times observation likelihood times squared deviation of the guarded posterior mean from the state's target.

Given a probability design over design indices, grid weights for each design index, observation, and grid action, real grid actions, a default action, a design index, and an observation in that design's observation space, the guarded conditional barycenter is the weighted average of grid actions divided by that design's probability when it is positive, and is the default action when it is zero.

The barycenter procedure combines a probability design with nonnegative grid weights whose fiber totals equal the design probabilities. Its decision rule averages grid actions lying in the closed action interval, using an in-range default action on zero-mass fibers.