Risk & Decision Mathematics

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

flow "RiskDecisionMathematics" v4.0

# RISK & DECISION MATHEMATICS
# 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.html"
  title: "Risk & Decision Mathematics"
  domain: "Probability and Statistics"
  category: "Risk & Decision Mathematics"
  entity: RISK_DECISION_MATHEMATICS
  entity_type: MATHEMATICAL_DECISION_FRAMEWORK

DEFINE:
  RISK_DECISION_MATHEMATICS :=
    FORMAL_FRAMEWORK(
      UNCERTAINTY,
      ACTIONS,
      STATES,
      OUTCOMES,
      PROBABILITIES,
      LOSSES,
      UTILITIES,
      DECISION_RULES
    )

PURPOSE:
  REPRESENT:
    DECISIONS_UNDER_UNCERTAINTY
    RISK_QUANTIFICATION
    LOSS_AND_UTILITY
    EXPECTED_CONSEQUENCES
    OPTIMAL_ACTION_SELECTION
    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:
  RISK: UNCERTAIN_LOSS_OR_ADVERSE_OUTCOME_MEASURE
  DECISION: ACTION_SELECTION
  STATE: POSSIBLE_WORLD_OR_PARAMETER_STATE
  ACTION: AVAILABLE_DECISION
  OUTCOME: CONSEQUENCE_OF_ACTION_AND_STATE
  PROBABILITY: UNCERTAINTY_MEASURE
  LOSS: COST_OR_UNDESIRABLE_CONSEQUENCE
  UTILITY: PREFERENCE_VALUE
  EXPECTED_LOSS: PROBABILITY_WEIGHTED_LOSS
  EXPECTED_UTILITY: PROBABILITY_WEIGHTED_UTILITY
  BAYES_RISK: PRIOR_OR_POSTERIOR_EXPECTED_LOSS
  REGRET: LOSS_RELATIVE_TO_BEST_ACTION_IN_STATE
  VALUE_OF_INFORMATION: BENEFIT_OF_ADDITIONAL_INFORMATION

RELATIONSHIPS:
  STATE -> INFLUENCES -> OUTCOME
  ACTION -> INFLUENCES -> OUTCOME
  OUTCOME -> DETERMINES -> LOSS
  OUTCOME -> DETERMINES -> UTILITY
  PROBABILITY -> WEIGHTS -> POSSIBLE_OUTCOMES
  DECISION_RULE -> SELECTS -> ACTION
  INFORMATION -> UPDATES -> BELIEFS
  UPDATED_BELIEFS -> MAY_CHANGE -> OPTIMAL_ACTION

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

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

LOSS_UTILITY_DUALITY:
  common_relation:
    U(theta,a) := -L(theta,a)
  caution:
    UTILITY_NEED_NOT_ALWAYS_BE_NEGATIVE_LOSS

DECISION_UNDER_RISK:
  states: UNCERTAIN
  probabilities: KNOWN_OR_MODELLED
  decision:
    OPTIMISE_EXPECTED_OR_RISK_ADJUSTED_CRITERION

DECISION_UNDER_UNCERTAINTY:
  probabilities: NOT_FULLY_SPECIFIED
  methods:
    - MINIMAX
    - MAXIMIN
    - MINIMAX_REGRET
    - ROBUST_OPTIMISATION
    - IMPRECISE_PROBABILITY_METHODS

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

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

MINIMAX_DECISION:
  criterion:
    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:
  decision:
    a_star := argmin_a max_theta Regret(theta,a)


EXPECTED_UTILITY_THEORY:
  preference_representation:
    E[U(X)]
  assumptions_commonly_include:
    - COMPLETENESS
    - TRANSITIVITY
    - CONTINUITY
    - INDEPENDENCE

RISK_AVERSION:
  utility:
    CONCAVE_IN_WEALTH

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

RISK_PREMIUM:
  RP := E[X]-CE

ABSOLUTE_RISK_AVERSION:
  A(w) := -U_double_prime(w)/U_prime(w)

RELATIVE_RISK_AVERSION:
  R(w) := -w*U_double_prime(w)/U_prime(w)

EXPECTED_MONETARY_VALUE:
  EMV(a) := SUM_s P(s)V(a,s)

VALUE_AT_RISK:
  alias: VaR
  confidence: alpha
  VaR_alpha(L) :=
    inf{l: P(L <= l) >= alpha}
  caution:
    DOES_NOT_MEASURE_TAIL_SEVERITY_BEYOND_QUANTILE
    MAY_FAIL_COHERENCE_PROPERTIES_IN_GENERAL

EXPECTED_SHORTFALL:
  aliases:
    ES
    CVaR
  role:
    AVERAGE_UPPER_TAIL_LOSS_BEYOND_VAR_THRESHOLD
  caution:
    PRECISE_DEFINITION_REQUIRES_CARE_WITH_ATOMS_AND_QUANTILES

COHERENT_RISK_MEASURE:
  properties:
    - MONOTONICITY
    - SUBADDITIVITY
    - POSITIVE_HOMOGENEITY
    - TRANSLATION_INVARIANCE

MEAN_VARIANCE_DECISION:
  objective:
    E[R] - lambda*Var(R)
  caution:
    MEAN_AND_VARIANCE_DO_NOT_FULLY_DESCRIBE_GENERAL_DISTRIBUTIONS

DECISION_TREE:
  nodes:
    - DECISION_NODE
    - CHANCE_NODE
    - TERMINAL_NODE
  evaluation:
    ROLLBACK_EXPECTED_VALUES_OR_UTILITIES

SEQUENTIAL_DECISION:
  state: S_t
  action: A_t
  transition:
    P(S_next|S_t,A_t)
  reward: R_t
  objective:
    MAXIMISE_EXPECTED_CUMULATIVE_UTILITY_OR_REWARD

MARKOV_DECISION_PROCESS:
  alias: MDP
  components:
    - STATE_SPACE
    - ACTION_SPACE
    - TRANSITION_MODEL
    - REWARD_FUNCTION
    - DISCOUNT_FACTOR
  Bellman_optimality:
    V_star(s) :=
      max_a[
        R(s,a)
        + gamma*SUM_s_next P(s_next|s,a)V_star(s_next)
      ]

DECISION_THRESHOLD:
  score: q
  rule:
    IF q >= t:
      ACTION_POSITIVE
    ELSE:
      ACTION_NEGATIVE
  optimal_threshold:
    DEPENDS_ON:
      COSTS
      BENEFITS
      PROBABILITIES
      CALIBRATION
      DECISION_CONTEXT

BINARY_CLASSIFICATION_COST:
  expected_cost:
    DEPENDS_ON:
      FALSE_POSITIVE_COST
      FALSE_NEGATIVE_COST
      FPR
      FNR
      PREVALENCE
  rule:
    ACCURACY_ALONE_DOES_NOT_DEFINE_OPTIMAL_DECISION

PROBABILITY_CALIBRATION_CONNECTION:
  calibrated_probability:
    SUPPORTS:
      EXPECTED_COST_CALCULATION
      EXPECTED_UTILITY_CALCULATION
  caution:
    RANKING_QUALITY_DOES_NOT_GUARANTEE_CALIBRATED_PROBABILITIES

SENSITIVITY_ANALYSIS:
  vary:
    - PROBABILITIES
    - COSTS
    - UTILITIES
    - MODEL_PARAMETERS
    - DISCOUNT_RATE
  observe:
    - OPTIMAL_ACTION
    - EXPECTED_LOSS
    - EXPECTED_UTILITY
    - RISK_MEASURE

SCENARIO_ANALYSIS:
  purpose:
    EXPLORE_DECISIONS_ACROSS_PLAUSIBLE_FUTURES

STRESS_TESTING:
  scenarios:
    EXTREME_BUT_PLAUSIBLE
  purpose:
    ASSESS_TAIL_RISK_ROBUSTNESS_AND_FAILURE_MODES

ROBUST_DECISION:
  uncertainty_set: U
  forms:
    - WORST_CASE_OPTIMISATION
    - DISTRIBUTIONALLY_ROBUST_OPTIMISATION
    - MINIMAX_REGRET

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


VALUE_OF_INFORMATION:
  baseline:
    OPTIMAL_EXPECTED_VALUE_WITH_CURRENT_INFORMATION
  informed:
    OPTIMAL_EXPECTED_VALUE_WITH_ADDITIONAL_INFORMATION
  VOI:
    informed - baseline

EXPECTED_VALUE_OF_PERFECT_INFORMATION:
  alias: EVPI
  definition:
    EXPECTED_VALUE_WITH_PERFECT_INFORMATION
    -
    EXPECTED_VALUE_WITH_CURRENT_INFORMATION

EXPECTED_VALUE_OF_SAMPLE_INFORMATION:
  alias: EVSI
  definition:
    EXPECTED_GAIN_FROM_SPECIFIC_ADDITIONAL_SAMPLE_INFORMATION

EXPECTED_NET_BENEFIT_OF_SAMPLING:
  ENBS := EVSI - COST_OF_INFORMATION

BAYESIAN_UPDATING:
  prior: p(theta)
  evidence: x
  posterior: p(theta|x)
  decision_effect:
    POSTERIOR_CHANGES_EXPECTED_LOSS_EXPECTED_UTILITY_AND_OPTIMAL_ACTION

MULTI_CRITERIA_DECISION:
  criteria:
    c_1,...,c_m
  weights:
    w_1,...,w_m
  aggregate:
    MAY_USE_SUM_j w_j v_j(a)
  caution:
    WEIGHTS_AND_VALUE_FUNCTIONS_ENCODE_PREFERENCES

PARETO_OPTIMALITY:
  action:
    PARETO_OPTIMAL_IF_NO_OTHER_ACTION_IMPROVES_ONE_CRITERION_WITHOUT_WORSENING_ANOTHER

RISK_COMMUNICATION:
  preserve:
    - ABSOLUTE_RISK
    - RELATIVE_RISK
    - BASE_RATE
    - TIME_HORIZON
    - UNCERTAINTY
    - OUTCOME_DEFINITION

ABSOLUTE_RISK:
  definition:
    P(EVENT)
    OVER:
      SPECIFIED_TIME_HORIZON_AND_POPULATION

RELATIVE_RISK:
  RR := Risk_exposed/Risk_reference
  caution:
    DOES_NOT_ENCODE_ABSOLUTE_EVENT_FREQUENCY

RISK_DIFFERENCE:
  RD := Risk_exposed-Risk_reference

ODDS:
  Odds := p/(1-p)

ODDS_RATIO:
  OR := Odds_group1/Odds_group0
  caution:
    ODDS_RATIO != RISK_RATIO

UNCERTAINTY_PROPAGATION:
  uncertain_inputs: X
  model:
    Y := g(X)
  methods:
    - ANALYTIC_PROPAGATION
    - MONTE_CARLO_SIMULATION
    - DELTA_METHOD
    - BAYESIAN_POSTERIOR_PREDICTIVE

MONTE_CARLO_RISK_ANALYSIS:
  flow:
    SAMPLE_UNCERTAIN_INPUTS
      -> COMPUTE_OUTCOME
      -> REPEAT
      -> ESTIMATE_OUTCOME_DISTRIBUTION
      -> COMPUTE_RISK_MEASURES

APPLICATIONS:
  RISK_DECISION_MATHEMATICS -> SUPPORTS -> STATISTICAL_DECISION_THEORY
  RISK_DECISION_MATHEMATICS -> SUPPORTS -> BAYESIAN_DECISION_MAKING
  RISK_DECISION_MATHEMATICS -> SUPPORTS -> MACHINE_LEARNING
  RISK_DECISION_MATHEMATICS -> SUPPORTS -> OPERATIONS_RESEARCH
  RISK_DECISION_MATHEMATICS -> SUPPORTS -> RISK_MANAGEMENT
  RISK_DECISION_MATHEMATICS -> SUPPORTS -> EXPERIMENT_DESIGN
  RISK_DECISION_MATHEMATICS -> SUPPORTS -> RESOURCE_ALLOCATION

MODEL_CONNECTIONS:
  RISK_DECISION_MATHEMATICS -> CONNECTS_TO -> PROBABILITY_THEORY
  RISK_DECISION_MATHEMATICS -> CONNECTS_TO -> BAYESIAN_STATISTICS
  RISK_DECISION_MATHEMATICS -> CONNECTS_TO -> EXPECTED_UTILITY
  RISK_DECISION_MATHEMATICS -> CONNECTS_TO -> GAME_THEORY
  RISK_DECISION_MATHEMATICS -> CONNECTS_TO -> OPTIMISATION
  RISK_DECISION_MATHEMATICS -> CONNECTS_TO -> MARKOV_DECISION_PROCESSES
  RISK_DECISION_MATHEMATICS -> CONNECTS_TO -> INFORMATION_THEORY

SEMANTIC_EQUIVALENCE:
  "expected loss"
    ~= "probability-weighted average loss"

  "expected utility"
    ~= "probability-weighted average utility"

  "regret"
    ~= "loss relative to best action for realised state"

  "value of information"
    ~= "expected improvement in decision value from additional information"

  EXCLUDE_EQUIVALENCE:
    RISK != UNCERTAINTY
    LOSS != PROBABILITY
    UTILITY != PROBABILITY
    EXPECTED_VALUE != CERTAINTY
    VAR != EXPECTED_SHORTFALL
    ODDS_RATIO != RISK_RATIO
    PREDICTIVE_ACCURACY != DECISION_UTILITY
    ASSOCIATION != CAUSATION

VALIDATION_RULES:
  RULE_1:
    STATE_ACTION_OUTCOME_AND_TIME_HORIZON_MUST_BE_DEFINED
  RULE_2:
    PROBABILITIES_MUST_BE_VALID_AND_NORMALISED
  RULE_3:
    LOSS_OR_UTILITY_ORIENTATION_MUST_BE_EXPLICIT
  RULE_4:
    DECISION_CRITERION_MUST_MATCH_RISK_PREFERENCE_AND_CONTEXT
  RULE_5:
    VALUE_OF_INFORMATION_CALCULATIONS_MUST_USE_CONSISTENT_DECISION_MODEL
  RULE_6:
    RISK_MEASURE_DEFINITION_AND_CONFIDENCE_LEVEL_MUST_BE_EXPLICIT
  RULE_7:
    SENSITIVITY_TO_PROBABILITIES_COSTS_AND_MODEL_ASSUMPTIONS_SHOULD_BE_ASSESSED
  RULE_8:
    ASSOCIATIONAL_RESULTS_MUST_NOT_BE_INTERPRETED_AS_CAUSAL_WITHOUT_CAUSAL_ASSUMPTIONS

CONFIDENCE:
  ontology_identity: 0.99
  expected_loss_structure: 0.99
  utility_structure: 0.99
  bayesian_decision_structure: 0.99
  risk_measure_structure: 0.99
  value_of_information_structure: 0.99
  semantic_distinctions: 0.99


UPDATE_MODEL:
  IF:
    NEW_EVIDENCE_ARRIVES
  THEN:
    UPDATE:
      STATE_PROBABILITIES
      POSTERIOR_BELIEFS
      EXPECTED_LOSSES
      EXPECTED_UTILITIES
    RECOMPUTE:
      OPTIMAL_ACTION

  IF:
    COSTS_OR_UTILITIES_CHANGE
  THEN:
    UPDATE:
      LOSS_FUNCTION
      UTILITY_FUNCTION
    RECOMPUTE:
      DECISION_RULE
      RISK_MEASURES

  IF:
    ACTION_SET_CHANGES
  THEN:
    UPDATE:
      ACTION_SPACE
    REEVALUATE:
      DOMINANCE
      EXPECTED_LOSS
      EXPECTED_UTILITY
      REGRET

  PRESERVE:
    STATE_DEFINITIONS
    ACTION_DEFINITIONS
    OUTCOME_DEFINITIONS
    PROBABILITY_MODEL
    LOSS_FUNCTION
    UTILITY_FUNCTION
    TIME_HORIZON
    DECISION_CRITERION
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_STATES_ACTIONS_AND_OUTCOMES
    - IDENTIFY_PROBABILITY_MODEL
    - IDENTIFY_LOSS_OR_UTILITY_FUNCTION
    - COMPUTE_EXPECTED_LOSS_OR_EXPECTED_UTILITY
    - COMPUTE_RELEVANT_RISK_MEASURES
    - ASSESS_REGRET_AND_DOMINANCE
    - UPDATE_BELIEFS_IF_EVIDENCE_AVAILABLE
    - COMPUTE_VALUE_OF_INFORMATION_IF_RELEVANT
    - PERFORM_SENSITIVITY_AND_STRESS_ANALYSIS
    - SELECT_ACTION_UNDER_SPECIFIED_DECISION_CRITERION
    - RETURN_RISK_DECISION_INTERPRETATION

  constraints:
    - PRESERVE_STATE_ACTION_OUTCOME_SEMANTICS
    - PRESERVE_PROBABILITY_NORMALISATION
    - DISTINGUISH_RISK_FROM_UNCERTAINTY
    - DISTINGUISH_LOSS_FROM_UTILITY
    - DISTINGUISH_VAR_FROM_EXPECTED_SHORTFALL
    - DISTINGUISH_PREDICTION_FROM_DECISION_UTILITY
    - DISTINGUISH_ASSOCIATION_FROM_CAUSATION
    - PRESERVE_DATA_PROVENANCE

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

END_FLOW