Bayesian Network Ontology

flow "BayesianNetworkOntology" v4.0

# ============================================================
# BAYESIAN NETWORK ONTOLOGY
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)
# ============================================================

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/bayesian-statistics/bayesian-network-ontology.html"
  title: "Bayesian Network Ontology"
  domain: "Probability and Statistics"
  category: "Bayesian Statistics"
  entity: BAYESIAN_NETWORK
  entity_type: PROBABILISTIC_GRAPHICAL_MODEL

DEFINE BAYESIAN_NETWORK:
  BAYESIAN_NETWORK :=
    DIRECTED_ACYCLIC_GRAPH(
      RANDOM_VARIABLES,
      CONDITIONAL_DEPENDENCIES,
      CONDITIONAL_PROBABILITY_DISTRIBUTIONS
    )

CORE_STRUCTURE:
  GRAPH:
    type: DIRECTED_ACYCLIC_GRAPH
    symbol: G
    components:
      - NODES
      - DIRECTED_EDGES

  NODE:
    represents: RANDOM_VARIABLE

  DIRECTED_EDGE:
    represents:
      DIRECT_PROBABILISTIC_DEPENDENCY
      OR
      CONDITIONAL_DEPENDENCY

  ACYCLICITY:
    REQUIRE:
      NO_DIRECTED_CYCLE

ENTITIES:
  RANDOM_VARIABLE:
    type: PROBABILISTIC_ENTITY
    symbol: X_i

  NODE:
    type: GRAPH_ENTITY
    represents: RANDOM_VARIABLE

  EDGE:
    type: DIRECTED_RELATION
    form:
      X_i -> X_j

  PARENT:
    type: GRAPH_RELATION
    notation:
      Pa(X_i)

  CHILD:
    type: GRAPH_RELATION

  CONDITIONAL_PROBABILITY_DISTRIBUTION:
    type: PROBABILITY_DISTRIBUTION
    definition:
      P(X_i | Pa(X_i))

  JOINT_PROBABILITY_DISTRIBUTION:
    type: PROBABILITY_DISTRIBUTION

  EVIDENCE:
    type: OBSERVED_VARIABLE_STATE

  QUERY_VARIABLE:
    type: INFERENCE_TARGET

  LATENT_VARIABLE:
    type: UNOBSERVED_RANDOM_VARIABLE

RELATIONSHIPS:
  BAYESIAN_NETWORK
    -> CONSISTS_OF
    -> NODES

  BAYESIAN_NETWORK
    -> CONSISTS_OF
    -> DIRECTED_EDGES

  NODE
    -> REPRESENTS
    -> RANDOM_VARIABLE

  DIRECTED_EDGE
    -> CONNECTS
    -> PARENT_TO_CHILD

  CHILD_NODE
    -> CONDITIONED_ON
    -> PARENT_NODES

  RANDOM_VARIABLE
    -> ASSOCIATED_WITH
    -> CONDITIONAL_PROBABILITY_DISTRIBUTION

  EVIDENCE
    -> CONDITIONS
    -> INFERENCE

  BAYESIAN_NETWORK
    -> REPRESENTS
    -> JOINT_PROBABILITY_DISTRIBUTION

FACTORIZATION:
  GIVEN:
    VARIABLES := {X_1, X_2, ..., X_n}
    GRAPH := G

  JOINT_DISTRIBUTION:
    P(X_1, X_2, ..., X_n) :=
      PRODUCT[i=1..n] P(X_i | Pa(X_i))

  REQUIRE:
    GRAPH_IS_DAG = TRUE

CONDITIONAL_INDEPENDENCE:
  LOCAL_MARKOV_PROPERTY:
    EACH_NODE:
      CONDITIONALLY_INDEPENDENT_OF(
        NON_DESCENDANTS
      )
      GIVEN(
        PARENTS
      )

  GRAPHICAL_SEPARATION:
    method: D_SEPARATION

  D_SEPARATION:
    purpose:
      DETERMINE_CONDITIONAL_INDEPENDENCE(
        VARIABLE_SETS,
        GIVEN_EVIDENCE
      )

INFERENCE:
  INPUT:
    BAYESIAN_NETWORK
    EVIDENCE
    QUERY_VARIABLES

  PROCESS:
    CONDITION_ON(EVIDENCE)
    -> PROPAGATE_PROBABILISTIC_INFORMATION
    -> COMPUTE_POSTERIOR_DISTRIBUTION

  OUTPUT:
    P(QUERY_VARIABLES | EVIDENCE)

  METHODS:
    EXACT:
      - VARIABLE_ELIMINATION
      - BELIEF_PROPAGATION
      - JUNCTION_TREE

    APPROXIMATE:
      - MONTE_CARLO_SAMPLING
      - IMPORTANCE_SAMPLING
      - GIBBS_SAMPLING

BAYESIAN_UPDATE:
  PRIOR_BELIEFS:
    encoded_by:
      NETWORK_PARAMETERS
      CONDITIONAL_PROBABILITY_DISTRIBUTIONS

  OBSERVE:
    EVIDENCE

  UPDATE:
    PRIOR_BELIEFS
      -> CONDITION_ON(EVIDENCE)
      -> POSTERIOR_BELIEFS

LEARNING:
  STRUCTURE_LEARNING:
    goal:
      INFER_GRAPH_STRUCTURE_FROM_DATA

    approaches:
      - SCORE_BASED
      - CONSTRAINT_BASED
      - HYBRID

  PARAMETER_LEARNING:
    goal:
      ESTIMATE_CONDITIONAL_PROBABILITY_PARAMETERS

    methods:
      - MAXIMUM_LIKELIHOOD
      - BAYESIAN_ESTIMATION

  MISSING_OR_LATENT_DATA:
    methods:
      - EXPECTATION_MAXIMISATION
      - BAYESIAN_INFERENCE

CAUSAL_INTERPRETATION:
  DEFAULT:
    BAYESIAN_NETWORK_EDGES:
      REPRESENT_PROBABILISTIC_DEPENDENCIES

  CAUSAL_NETWORK:
    REQUIRE:
      ADDITIONAL_CAUSAL_ASSUMPTIONS

  RULE:
    DIRECTED_EDGE
      DOES_NOT_AUTOMATICALLY_IMPLY
      CAUSATION

  IF:
    CAUSAL_ASSUMPTIONS_VALID

  THEN:
    NETWORK_MAY_SUPPORT:
      - INTERVENTIONAL_REASONING
      - CAUSAL_INFERENCE
      - COUNTERFACTUAL_EXTENSIONS

NETWORK_PATTERNS:
  CHAIN:
    structure:
      X -> Y -> Z

  FORK:
    structure:
      X <- Y -> Z

  COLLIDER:
    structure:
      X -> Y <- Z

  PURPOSE:
    DETERMINE:
      DEPENDENCE
      CONDITIONAL_INDEPENDENCE
      EVIDENCE_FLOW

APPLICATIONS:
  BAYESIAN_NETWORK
    -> SUPPORTS
    -> DIAGNOSIS

  BAYESIAN_NETWORK
    -> SUPPORTS
    -> PREDICTION

  BAYESIAN_NETWORK
    -> SUPPORTS
    -> CLASSIFICATION

  BAYESIAN_NETWORK
    -> SUPPORTS
    -> DECISION_SUPPORT

  BAYESIAN_NETWORK
    -> SUPPORTS
    -> UNCERTAINTY_REASONING

  BAYESIAN_NETWORK
    -> SUPPORTS
    -> KNOWLEDGE_REPRESENTATION

MODEL_CONNECTIONS:
  BAYESIAN_NETWORK
    -> IS_A
    -> PROBABILISTIC_GRAPHICAL_MODEL

  BAYESIAN_NETWORK
    -> USES
    -> CONDITIONAL_PROBABILITY

  BAYESIAN_NETWORK
    -> USES
    -> BAYES_THEOREM

  BAYESIAN_NETWORK
    -> CONNECTS_TO
    -> CONDITIONAL_INDEPENDENCE

  BAYESIAN_NETWORK
    -> CONNECTS_TO
    -> GRAPH_THEORY

  BAYESIAN_NETWORK
    -> CONNECTS_TO
    -> MACHINE_LEARNING

  BAYESIAN_NETWORK
    -> CONNECTS_TO
    -> ARTIFICIAL_INTELLIGENCE

SEMANTIC_EQUIVALENCE:
  "Bayesian network"
    ~= "Bayes network"

  "Bayesian network"
    ~= "belief network"

  "Bayesian network"
    ~= "directed probabilistic graphical model"

  "conditional probability table"
    ~= "discrete conditional probability representation"

  EXCLUDE_EQUIVALENCE:
    BAYESIAN_NETWORK != GENERIC_DIRECTED_GRAPH
    BAYESIAN_NETWORK != MARKOV_RANDOM_FIELD
    PROBABILISTIC_EDGE != NECESSARILY_CAUSAL_EDGE

VALIDATION_RULES:
  RULE_1:
    GRAPH_MUST_BE:
      DIRECTED
      ACYCLIC

  RULE_2:
    EACH_NODE_MUST_HAVE:
      VALID_PROBABILITY_MODEL(
        CONDITIONED_ON_PARENTS
      )

  RULE_3:
    JOINT_DISTRIBUTION_MUST_FACTORISE:
      ACCORDING_TO_GRAPH_STRUCTURE

  RULE_4:
    CONDITIONAL_INDEPENDENCE_CLAIMS:
      MUST_BE_CONSISTENT_WITH
      GRAPH_AND_D_SEPARATION

  RULE_5:
    POSTERIOR_INFERENCE:
      MUST_CONDITION_ON
      OBSERVED_EVIDENCE

CONFIDENCE:
  ontology_identity: 0.99
  graphical_model_definition: 0.99
  factorization_structure: 0.99
  conditional_independence_structure: 0.99
  inference_structure: 0.98
  causal_distinction: 0.99

UPDATE_MODEL:
  IF:
    NEW_EVIDENCE_ARRIVES

  THEN:
    ADD_TO:
      EVIDENCE_SET

    RECOMPUTE:
      POSTERIOR_DISTRIBUTIONS

  IF:
    NEW_TRAINING_DATA_ARRIVES

  THEN:
    OPTIONAL_UPDATE:
      NETWORK_PARAMETERS
      NETWORK_STRUCTURE

  PRESERVE:
    DAG_CONSTRAINT
    VARIABLE_DEFINITIONS
    CONDITIONAL_DEPENDENCY_SEMANTICS
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_RANDOM_VARIABLES
    - IDENTIFY_GRAPH_STRUCTURE
    - VALIDATE_ACYCLICITY
    - IDENTIFY_PARENT_CHILD_RELATIONSHIPS
    - LOAD_CONDITIONAL_PROBABILITY_MODELS
    - IDENTIFY_EVIDENCE
    - IDENTIFY_QUERY_VARIABLES
    - APPLY_CONDITIONAL_INDEPENDENCE
    - PERFORM_PROBABILISTIC_INFERENCE
    - RETURN_POSTERIOR_BELIEFS

  constraints:
    - DO_NOT_CREATE_DIRECTED_CYCLES
    - DO_NOT_EQUATE_DEPENDENCY_WITH_CAUSATION
    - PRESERVE_CONDITIONAL_PROBABILITY_STRUCTURE
    - DISTINGUISH_OBSERVED_AND_LATENT_VARIABLES
    - EXPOSE_INFERENCE_ASSUMPTIONS

FLOW_HASH:
  canonical_entity: "bayesian-network"
  namespace: "probability-statistics/bayesian-statistics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - DAG_STRUCTURE
    - CONDITIONAL_PROBABILITY_RELATIONSHIPS
    - JOINT_FACTORIZATION

END_FLOW