This book defines the observable object, the counterfactual target, and the information a representation must preserve. It connects exact identification, local repair, approximate loss, measurement across environments, timing error, administrative transmission, and model comparison through one target-directed language.
A counterfactual question becomes precise only after the structural state, observation rule, policy set, horizon, target, loss, and admissible records are fixed. The chapter treats that list as a research contract and separates within-model policy comparisons from comparisons across model families.
Linear algebra supplies the first exact language for lost information. Observation and target maps are compared through kernels, quotient spaces, projections, singular values, and weighted inner products. The rank of the target restricted to discarded directions measures the number of missing local coordinates.
Exact sufficiency requires target constancy on every observation fiber. A derivative condition sees only tangent directions at one regular point. Constant-rank coordinates can turn that condition into a local factorization, while disconnected fibers and separate branches can still support different target levels.
Missing dimension gives a lower bound on repair, while data collection imposes a second feasibility problem. Oracle coordinates chosen after seeing the target can differ sharply from records available before the policy or environment is known. Additivity, cost, timing, and measurement constraints determine attainable repairs.
A representation budget forces approximation. Worst-direction error and distribution-weighted mean-square loss answer different questions and produce different spectral summaries. Continuity restrictions govern nonlinear encoders, while Taylor remainders connect a local spectral frontier to finite policy and state perturbations.
Conditional means minimize squared reconstruction loss, and conditional variance measures the remaining risk. Fiberwise derivatives control variation only after geometric and probabilistic conditions are supplied. Poincare inequalities, component means, generator residuals, and graph approximation error form a complete risk certificate.
Economic objects are often valued under changing prices, conventions, or environments. Scalar transport works only when attained ranges, rankings, and composition remain compatible. Mixed derivatives and bilinear representations measure environment-dependent information that a single recorded value fails to carry.
Record selection begins after conditioning on the source information already retained. Conditional covariance measures residual target risk, while a feasible field dictionary turns free coordinate choice into a combinatorial selection problem. Greedy and exchange procedures require explicit benchmarks and leakage controls.
Recorded time W can differ from event time G through displacement U. A decision may require only selected moments of that displacement. Binomial moment recursions recover latent temporal moments, and polynomial losses reveal exactly which clock moments determine an action and its residual loss.
An inverse problem can identify a payoff query without recovering the entire latent response. Adjoint equations characterize exact bridges, closures characterize approximable queries, and stability determines estimability. Multiple admissible clocks require a common bridge or a transparent range of decision values.
Policies pass through channels that transform an upstream experiment and may generate new information downstream. Score projection describes information contraction under a Markov handoff. Delay, arrival, measurement, and causal mediation remain separate objects in the resulting stage ledger.
Singular Fisher information permits some targets to remain estimable while structural directions are unidentified. Kernel and range conditions answer identification; pseudoinverse variance answers precision. Gaussian repair and feasible PSD contractions translate those distinctions into record design.
Information design allocates scarce retention and acquisition resources against target-specific decision loss. Local quadratic regret defines the objective, semidefinite constraints describe feasible information, and alternative release schemes carry different likelihoods and cost measures.
An archive accumulates directional precision through repeated reports. The number of observations alone can conceal persistent blind directions. Prior precision, data information, target-weighted posterior loss, state persistence, and field labels determine what the archive teaches.
Institutions balance current operating loss, the cost of changing a reporting system, and the value of information for later targets. Reporting directions can evolve gradually, react to observed content, and create archive externalities for future decision makers.
Model comparison requires a common response experiment: the same shock, normalization, observed response, target, and support. Family-entry and sign-reversal radii answer different questions. Coverage failure, within-response ambiguity, and held-out prediction failure provide distinct diagnoses.