Decision Theory Ontology

flow "DecisionTheoryOntology" v4.0

# DECISION THEORY ONTOLOGY
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/risk-decision-mathematics/decision-theory-ontology-2.html"
  title: "Decision Theory Ontology"
  domain: "Probability and Statistics"
  category: "Risk & Decision Mathematics"
  entity: DECISION_THEORY
  entity_type: RISK_DECISION_ONTOLOGY

DEFINE:
  DECISION_THEORY :=
    FORMAL_FRAMEWORK(
      STATES,
      ACTIONS,
      OUTCOMES,
      PROBABILITIES,
      LOSSES,
      UTILITIES,
      INFORMATION,
      DECISION_RULES
    )

PURPOSE:
  REPRESENT:
    RATIONAL_ACTION_SELECTION
    DECISIONS_UNDER_UNCERTAINTY
    PREFERENCE_STRUCTURE
    LOSS_MINIMISATION
    UTILITY_MAXIMISATION
    INFORMATION_VALUE

FOUNDATION:
  STATE_SPACE: Theta
  ACTION_SPACE: A
  OBSERVATION_SPACE: X
  STATE: theta IN Theta
  ACTION: a IN A
  LOSS_FUNCTION: L(theta,a)
  UTILITY_FUNCTION: U(theta,a)
  DECISION_RULE: delta(X) -> A

ENTITIES:
  DECISION: ACTION_SELECTION
  STATE: POSSIBLE_CONDITION_OF_WORLD
  ACTION: AVAILABLE_CHOICE
  OUTCOME: CONSEQUENCE_OF_ACTION_AND_STATE
  PROBABILITY: UNCERTAINTY_MEASURE
  LOSS: UNDESIRABLE_CONSEQUENCE_VALUE
  UTILITY: PREFERENCE_VALUE
  DECISION_RULE: MAPPING_FROM_INFORMATION_TO_ACTION
  POLICY: RULE_FOR_SEQUENTIAL_ACTION_SELECTION
  EXPECTED_LOSS: PROBABILITY_WEIGHTED_LOSS
  EXPECTED_UTILITY: PROBABILITY_WEIGHTED_UTILITY
  REGRET: LOSS_RELATIVE_TO_STATEWISE_BEST_ACTION
  INFORMATION: EVIDENCE_AVAILABLE_BEFORE_DECISION

RELATIONSHIPS:
  ACTION -> COMBINES_WITH -> STATE
  ACTION_STATE_PAIR -> PRODUCES -> OUTCOME
  OUTCOME -> HAS -> LOSS
  OUTCOME -> HAS -> UTILITY
  PROBABILITY -> WEIGHTS -> STATES_OR_OUTCOMES
  INFORMATION -> UPDATES -> BELIEFS
  DECISION_RULE -> MAPS -> INFORMATION_TO_ACTION
  EXPECTED_LOSS -> SUPPORTS -> ACTION_SELECTION
  EXPECTED_UTILITY -> SUPPORTS -> ACTION_SELECTION

DECISION_UNDER_CERTAINTY:
  state: KNOWN
  rule: SELECT_ACTION_WITH_PREFERRED_KNOWN_OUTCOME

DECISION_UNDER_RISK:
  state: UNCERTAIN
  probabilities: SPECIFIED_OR_ESTIMATED
  rule: OPTIMISE_EXPECTED_OR_RISK_ADJUSTED_CRITERION

DECISION_UNDER_UNCERTAINTY:
  probabilities: UNKNOWN_OR_INCOMPLETELY_SPECIFIED
  criteria:
    - MAXIMIN
    - MAXIMAX
    - MINIMAX_REGRET
    - HURWICZ
    - LAPLACE
    - ROBUST_OPTIMISATION

EXPECTED_LOSS:
  R(a) := E[L(theta,a)]
  optimal_action: a_star := argmin_a R(a)

EXPECTED_UTILITY:
  EU(a) := E[U(theta,a)]
  optimal_action: a_star := argmax_a EU(a)

LOSS_UTILITY_RELATION:
  common_mapping: U(theta,a) := -L(theta,a)
  caution: UTILITY_NEED_NOT_BE_LITERAL_NEGATIVE_LOSS

EXPECTED_UTILITY_THEORY:
  objective: MAXIMISE E[U(X)]
  classical_axioms:
    - COMPLETENESS
    - TRANSITIVITY
    - CONTINUITY
    - INDEPENDENCE

RISK_ATTITUDE:
  RISK_NEUTRAL: LINEAR_UTILITY
  RISK_AVERSE: CONCAVE_UTILITY
  RISK_SEEKING: CONVEX_UTILITY

CERTAINTY_EQUIVALENT:
  CE: U(CE) = E[U(X)]

RISK_PREMIUM:
  RP := E[X]-CE

LOSS_FUNCTIONS:
  examples:
    - ZERO_ONE_LOSS
    - SQUARED_ERROR_LOSS
    - ABSOLUTE_ERROR_LOSS
    - ASYMMETRIC_LOSS
    - COST_SENSITIVE_LOSS

SQUARED_ERROR_LOSS:
  L(theta,a) := (theta-a)^2
  Bayes_action: POSTERIOR_MEAN

ABSOLUTE_ERROR_LOSS:
  L(theta,a) := |theta-a|
  Bayes_action: POSTERIOR_MEDIAN

STATISTICAL_DECISION_RULE:
  frequentist_risk:
    R(theta,delta) := E_theta[L(theta,delta(X))]

BAYES_RULE:
  prior: pi(theta)
  Bayes_risk:
    r(pi,delta) := INTEGRAL R(theta,delta)pi(theta)dtheta
  Bayes_rule:
    delta_pi := argmin_delta r(pi,delta)

POSTERIOR_DECISION:
  posterior: p(theta|x)
  posterior_expected_loss:
    rho(a|x) := E[L(theta,a)|x]
  Bayes_action:
    a_star(x) := argmin_a rho(a|x)

MINIMAX_RULE:
  delta_star := argmin_delta sup_theta R(theta,delta)

REGRET:
  best_loss: L_star(theta) := min_a L(theta,a)
  regret: Regret(theta,a) := L(theta,a)-L_star(theta)

MINIMAX_REGRET:
  action: a_star := argmin_a max_theta Regret(theta,a)

MAXIMIN:
  action: a_star := argmax_a min_theta U(theta,a)

MAXIMAX:
  action: a_star := argmax_a max_theta U(theta,a)

HURWICZ_CRITERION:
  optimism_parameter: alpha IN [0,1]
  score: alpha*BEST_OUTCOME+(1-alpha)*WORST_OUTCOME

LAPLACE_CRITERION:
  assumption: STATES_TREATED_AS_EQUALLY_LIKELY
  caution: EQUAL_PROBABILITY_ASSUMPTION_REQUIRES_JUSTIFICATION

DECISION_MATRIX:
  rows: ACTIONS
  columns: STATES
  cells: OUTCOMES_LOSSES_OR_UTILITIES

DECISION_TREE:
  node_types:
    - DECISION_NODE
    - CHANCE_NODE
    - TERMINAL_NODE
  evaluation: BACKWARD_INDUCTION

BACKWARD_INDUCTION:
  flow:
    EVALUATE_TERMINAL_NODES
      -> COMPUTE_EXPECTED_VALUES_AT_CHANCE_NODES
      -> OPTIMISE_AT_DECISION_NODES
      -> PROPAGATE_BACKWARD
      -> RETURN_POLICY

BAYESIAN_UPDATING:
  prior: p(theta)
  likelihood: p(x|theta)
  posterior: p(theta|x) proportional_to p(x|theta)p(theta)
  consequence: NEW_INFORMATION_MAY_CHANGE_OPTIMAL_ACTION

VALUE_OF_INFORMATION:
  current_value: OPTIMAL_EXPECTED_VALUE_WITH_CURRENT_INFORMATION
  informed_value: OPTIMAL_EXPECTED_VALUE_WITH_ADDITIONAL_INFORMATION
  VOI: informed_value-current_value

EXPECTED_VALUE_OF_PERFECT_INFORMATION:
  alias: EVPI
  definition: EXPECTED_VALUE_WITH_PERFECT_INFORMATION_MINUS_CURRENT_INFORMATION_VALUE

EXPECTED_VALUE_OF_SAMPLE_INFORMATION:
  alias: EVSI
  definition: EXPECTED_VALUE_WITH_SAMPLE_INFORMATION_MINUS_CURRENT_INFORMATION_VALUE
  relation: EVSI <= EVPI UNDER_STANDARD_CONDITIONS

NET_VALUE_OF_INFORMATION:
  NVI := VALUE_OF_INFORMATION-COST_OF_INFORMATION

SEQUENTIAL_DECISION:
  state: S_t
  action: A_t
  information: I_t
  policy: pi(A_t|I_t)
  objective: OPTIMISE_EXPECTED_CUMULATIVE_UTILITY

MARKOV_DECISION_PROCESS_CONNECTION:
  components:
    - STATE
    - ACTION
    - TRANSITION_PROBABILITY
    - REWARD
    - DISCOUNT_FACTOR
  Bellman_equation:
    V_star(s) := max_a[R(s,a)+gamma*SUM_s_prime P(s_prime|s,a)V_star(s_prime)]

PARTIALLY_OBSERVED_DECISION:
  hidden_state: S_t
  observation: O_t
  belief_state: b_t(s) := P(S_t=s|HISTORY)
  action: SELECTED_USING_BELIEF_STATE

MULTI_CRITERIA_DECISION:
  criteria: c_1,...,c_m
  values: v_j(a)
  weights: w_j
  weighted_score: V(a) := SUM_j w_j v_j(a)
  caution: WEIGHTS_ENCODE_PREFERENCES_AND_REQUIRE_JUSTIFICATION

PARETO_FRONTIER:
  contains: NONDOMINATED_ACTIONS

DECISION_THRESHOLD:
  probability: p
  threshold: t
  rule:
    IF p >= t:
      ACT
    ELSE:
      DO_NOT_ACT
  optimal_t:
    DEPENDS_ON:
      FALSE_POSITIVE_COST
      FALSE_NEGATIVE_COST
      BENEFITS
      PREVALENCE
      CALIBRATION

CLASSIFICATION_DECISION_CONNECTION:
  model_output: p(Y=1|X)
  expected_loss: COMPUTED_FROM_PROBABILITY_AND_CLASSIFICATION_COSTS
  rule: HIGHEST_ACCURACY_THRESHOLD != MINIMUM_EXPECTED_COST_THRESHOLD_BY_DEFAULT

PROBABILITY_CALIBRATION_CONNECTION:
  calibrated_probability:
    SUPPORTS:
      EXPECTED_LOSS
      EXPECTED_UTILITY
      THRESHOLD_SELECTION
  rule: DISCRIMINATION != CALIBRATION

ROC_CONNECTION:
  ROC_CURVE: REPRESENTS_TPR_FPR_TRADEOFF_ACROSS_THRESHOLDS
  decision_theory:
    ADDS:
      PREVALENCE
      COSTS
      BENEFITS

PRECISION_RECALL_CONNECTION:
  PR_CURVE: REPRESENTS_PRECISION_RECALL_TRADEOFF
  decision_theory: SELECTS_OPERATIONAL_POINT_USING_UTILITY_OR_COST

RISK_ANALYSIS_CONNECTION:
  risk_estimate:
    PROVIDES:
      PROBABILITIES
      CONSEQUENCES
      LOSS_DISTRIBUTIONS
  decision_theory: USES_RISK_INFORMATION_TO_SELECT_ACTION

ROBUST_DECISION_MAKING:
  methods:
    - MINIMAX
    - DISTRIBUTIONALLY_ROBUST_OPTIMISATION
    - MINIMAX_REGRET
    - SENSITIVITY_ANALYSIS

DISTRIBUTIONALLY_ROBUST_DECISION:
  ambiguity_set: P_set
  objective: min_a sup_{P IN P_set} E_P[L(a,Y)]

SENSITIVITY_ANALYSIS:
  vary:
    - PROBABILITIES
    - UTILITIES
    - LOSSES
    - MODEL_PARAMETERS
    - DISCOUNT_FACTOR
  observe:
    - OPTIMAL_ACTION
    - EXPECTED_LOSS
    - EXPECTED_UTILITY
    - REGRET

SCENARIO_ANALYSIS:
  purpose: EVALUATE_ACTIONS_ACROSS_PLAUSIBLE_STATES_OR_FUTURES

DECISION_BOUNDARY:
  definition: SET_OF_INFORMATION_STATES_WHERE_COMPETING_ACTIONS_HAVE_EQUAL_EXPECTED_VALUE

POLICY:
  mapping: INFORMATION_STATE -> ACTION

RANDOMISED_DECISION_RULE:
  action_distribution: delta(a|x)
  requirement: SUM_a delta(a|x) = 1

GAME_THEORY_CONNECTION:
  distinction: SINGLE_AGENT_DECISION_THEORY != STRATEGIC_MULTI_AGENT_GAME_BY_DEFAULT

BEHAVIOURAL_DECISION_THEORY:
  studies: ACTUAL_HUMAN_DECISION_BEHAVIOUR
  phenomena:
    - LOSS_AVERSION
    - FRAMING
    - PROBABILITY_WEIGHTING
    - REFERENCE_DEPENDENCE
  distinction: NORMATIVE_DECISION_THEORY != DESCRIPTIVE_BEHAVIOURAL_MODEL

APPLICATIONS:
  DECISION_THEORY -> SUPPORTS -> STATISTICAL_DECISION_THEORY
  DECISION_THEORY -> SUPPORTS -> BAYESIAN_DECISION_MAKING
  DECISION_THEORY -> SUPPORTS -> RISK_MANAGEMENT
  DECISION_THEORY -> SUPPORTS -> MACHINE_LEARNING
  DECISION_THEORY -> SUPPORTS -> OPERATIONS_RESEARCH
  DECISION_THEORY -> SUPPORTS -> STOCHASTIC_CONTROL
  DECISION_THEORY -> SUPPORTS -> RESOURCE_ALLOCATION

MODEL_CONNECTIONS:
  DECISION_THEORY -> CONNECTS_TO -> RISK_ANALYSIS
  DECISION_THEORY -> CONNECTS_TO -> PROBABILITY_THEORY
  DECISION_THEORY -> CONNECTS_TO -> BAYESIAN_STATISTICS
  DECISION_THEORY -> CONNECTS_TO -> UTILITY_THEORY
  DECISION_THEORY -> CONNECTS_TO -> GAME_THEORY
  DECISION_THEORY -> CONNECTS_TO -> OPTIMISATION
  DECISION_THEORY -> CONNECTS_TO -> MARKOV_DECISION_PROCESSES

SEMANTIC_EQUIVALENCE:
  "decision theory" ~= "formal analysis of action selection under uncertainty"
  "Bayes action" ~= "action minimising posterior expected loss"
  "expected utility" ~= "probability-weighted utility"
  "regret" ~= "loss relative to statewise optimal action"

  EXCLUDE_EQUIVALENCE:
    DECISION != PREDICTION
    LOSS != PROBABILITY
    UTILITY != PROBABILITY
    BAYES_RISK != FREQUENTIST_RISK
    REGRET != LOSS
    DISCRIMINATION != CALIBRATION
    NORMATIVE_DECISION_THEORY != BEHAVIOURAL_DECISION_THEORY
    ASSOCIATION != CAUSATION

VALIDATION_RULES:
  RULE_1: STATE_ACTION_OUTCOME_AND_INFORMATION_SPACES_MUST_BE_DEFINED
  RULE_2: PROBABILITIES_MUST_BE_VALID_AND_NORMALISED
  RULE_3: LOSS_OR_UTILITY_ORIENTATION_MUST_BE_EXPLICIT
  RULE_4: DECISION_RULE_MUST_USE_ONLY_INFORMATION_AVAILABLE_AT_DECISION_TIME
  RULE_5: VALUE_OF_INFORMATION_MUST_USE_CONSISTENT_BASELINE_AND_INFORMED_DECISION_MODELS
  RULE_6: MULTI_CRITERIA_WEIGHTS_AND_VALUE_FUNCTIONS_MUST_BE_EXPLICIT
  RULE_7: ROBUST_DECISION_ASSUMPTIONS_AND_AMBIGUITY_SETS_MUST_BE_DEFINED
  RULE_8: PREDICTIVE_ASSOCIATIONS_MUST_NOT_BE_INTERPRETED_AS_CAUSAL_EFFECTS_WITHOUT_CAUSAL_ASSUMPTIONS

CONFIDENCE:
  ontology_identity: 0.99
  expected_loss_structure: 0.99
  expected_utility_structure: 0.99
  bayesian_decision_structure: 0.99
  value_of_information_structure: 0.99
  sequential_decision_structure: 0.99
  semantic_distinctions: 0.99

UPDATE_MODEL:
  IF: NEW_INFORMATION_ARRIVES
  THEN:
    UPDATE:
      BELIEFS
      POSTERIOR_DISTRIBUTIONS
      EXPECTED_LOSSES
      EXPECTED_UTILITIES
    RECOMPUTE: OPTIMAL_ACTION

  IF: LOSS_OR_UTILITY_CHANGES
  THEN:
    UPDATE: PREFERENCE_MODEL
    RECOMPUTE:
      DECISION_RULE
      REGRET
      VALUE_OF_INFORMATION

  IF: ACTION_SET_CHANGES
  THEN:
    UPDATE: ACTION_SPACE
    REASSESS:
      DOMINANCE
      PARETO_FRONTIER
      OPTIMAL_ACTION

  PRESERVE:
    STATE_DEFINITIONS
    ACTION_DEFINITIONS
    OUTCOME_DEFINITIONS
    PROBABILITY_MODEL
    LOSS_FUNCTION
    UTILITY_FUNCTION
    INFORMATION_STRUCTURE
    TIME_HORIZON
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_STATES_ACTIONS_AND_OUTCOMES
    - IDENTIFY_AVAILABLE_INFORMATION
    - IDENTIFY_PROBABILITY_MODEL
    - IDENTIFY_LOSS_OR_UTILITY_FUNCTION
    - COMPUTE_EXPECTED_LOSS_OR_UTILITY
    - ELIMINATE_DOMINATED_ACTIONS_WHEN_VALID
    - ASSESS_REGRET_AND_ROBUSTNESS
    - UPDATE_BELIEFS_IF_NEW_INFORMATION_EXISTS
    - COMPUTE_VALUE_OF_INFORMATION_IF_RELEVANT
    - SOLVE_SEQUENTIAL_STRUCTURE_IF_PRESENT
    - RETURN_DECISION_THEORETIC_INTERPRETATION

  constraints:
    - PRESERVE_STATE_ACTION_OUTCOME_SEMANTICS
    - PRESERVE_INFORMATION_TIMING
    - PRESERVE_PROBABILITY_NORMALISATION
    - DISTINGUISH_LOSS_FROM_UTILITY
    - DISTINGUISH_DECISION_FROM_PREDICTION
    - DISTINGUISH_NORMATIVE_FROM_DESCRIPTIVE_MODELS
    - DISTINGUISH_ASSOCIATION_FROM_CAUSATION
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "decision-theory-ontology-2"
  namespace: "probability-statistics/risk-decision-mathematics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - STATES_ACTIONS_OUTCOMES
    - EXPECTED_LOSS_UTILITY
    - BAYES_DECISION_RULE
    - VALUE_OF_INFORMATION

END_FLOW