Likelihood Ontology

flow "LikelihoodOntology" v4.0

# ============================================================
# LIKELIHOOD 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/likelihood-ontology.html"
  title: "Likelihood Ontology"
  domain: "Probability and Statistics"
  category: "Bayesian Statistics"
  entity: LIKELIHOOD
  entity_type: STATISTICAL_CONCEPT

DEFINE LIKELIHOOD:
  LIKELIHOOD :=
    DEGREE_TO_WHICH(
      OBSERVED_DATA
      SUPPORTS(
        PARAMETER_VALUE
        OR
        STATISTICAL_MODEL
        OR
        HYPOTHESIS
      )
    )

CORE_DISTINCTION:
  PROBABILITY:
    evaluates: DATA_GIVEN_PARAMETERS
    expression: P(DATA | PARAMETER)

  LIKELIHOOD:
    evaluates: PARAMETERS_GIVEN_FIXED_DATA
    proportional_expression: L(PARAMETER | DATA)
    canonical_mapping:
      L(θ | x) := f(x | θ)

  REQUIRE:
    OBSERVED_DATA = FIXED
    PARAMETER = VARIABLE

ENTITIES:
  OBSERVED_DATA:
    type: DATA
    symbol: x

  PARAMETER:
    type: MODEL_PARAMETER
    symbol: θ

  STATISTICAL_MODEL:
    type: PROBABILITY_MODEL

  LIKELIHOOD_FUNCTION:
    type: FUNCTION
    mapping:
      θ -> f(x | θ)

  LOG_LIKELIHOOD:
    type: FUNCTION
    definition:
      ℓ(θ | x) := log(L(θ | x))

  MAXIMUM_LIKELIHOOD_ESTIMATE:
    type: PARAMETER_ESTIMATE
    definition:
      θ_hat := argmax_θ L(θ | x)

RELATIONSHIPS:
  OBSERVED_DATA
    -> EVALUATED_UNDER
    -> STATISTICAL_MODEL

  STATISTICAL_MODEL
    -> PARAMETERISED_BY
    -> PARAMETER

  OBSERVED_DATA
    + PARAMETER
    -> DEFINE
    -> LIKELIHOOD_FUNCTION

  LIKELIHOOD_FUNCTION
    -> LOG_TRANSFORM
    -> LOG_LIKELIHOOD

  LIKELIHOOD_FUNCTION
    -> MAXIMISED_BY
    -> MAXIMUM_LIKELIHOOD_ESTIMATE

  PRIOR_PROBABILITY
    + LIKELIHOOD
    -> BAYESIAN_UPDATE
    -> POSTERIOR_PROBABILITY

BAYESIAN_RELATION:
  BAYES_THEOREM:
    P(θ | x) =
      [P(x | θ) * P(θ)] / P(x)

  MAP:
    P(x | θ) -> LIKELIHOOD
    P(θ)     -> PRIOR
    P(θ | x) -> POSTERIOR
    P(x)     -> EVIDENCE

  INFERENCE_CHAIN:
    PRIOR
      -> COMBINE_WITH(LIKELIHOOD)
      -> NORMALISE_BY(EVIDENCE)
      -> POSTERIOR

LIKELIHOOD_FOR_INDEPENDENT_OBSERVATIONS:
  GIVEN:
    DATA := {x_1, x_2, ..., x_n}
    CONDITIONAL_INDEPENDENCE := TRUE

  COMPUTE:
    L(θ | DATA) :=
      PRODUCT[i=1..n] f(x_i | θ)

  LOG_FORM:
    ℓ(θ | DATA) :=
      SUM[i=1..n] log(f(x_i | θ))

INTERPRETATION_RULES:
  RULE_1:
    IF L(θ_a | x) > L(θ_b | x)
    THEN:
      DATA_SUPPORTS(θ_a) MORE_THAN DATA_SUPPORTS(θ_b)
      WITHIN_SAME_MODEL_AND_DATA

  RULE_2:
    LIKELIHOOD_IS_NOT:
      POSTERIOR_PROBABILITY_OF_PARAMETER

  RULE_3:
    LIKELIHOOD_NEED_NOT:
      SUM_TO_ONE_OVER_PARAMETERS

  RULE_4:
    LIKELIHOOD_COMPARISON_REQUIRES:
      SAME_OBSERVED_DATA
      COMPATIBLE_MODEL_CONTEXT

  RULE_5:
    MONOTONIC_TRANSFORMATION:
      MAXIMISE(LIKELIHOOD)
      EQUIVALENT_FOR_ARGMAX_TO
      MAXIMISE(LOG_LIKELIHOOD)

ESTIMATION:
  MAXIMUM_LIKELIHOOD:
    INPUT:
      OBSERVED_DATA
      PARAMETRIC_MODEL

    PROCESS:
      CONSTRUCT(LIKELIHOOD_FUNCTION)
      -> OPTIONAL(LOG_TRANSFORM)
      -> OPTIMISE_OVER(PARAMETER_SPACE)
      -> RETURN(MAXIMUM_LIKELIHOOD_ESTIMATE)

  OUTPUT:
    θ_hat_MLE

COMPARATIVE_LIKELIHOOD:
  LIKELIHOOD_RATIO:
    definition:
      LR(θ_1, θ_2 | x) :=
        L(θ_1 | x) / L(θ_2 | x)

    interpretation:
      COMPARES_RELATIVE_SUPPORT(
        θ_1,
        θ_2,
        GIVEN_FIXED_DATA
      )

MODEL_CONNECTIONS:
  LIKELIHOOD
    -> SUPPORTS
    -> PARAMETER_ESTIMATION

  LIKELIHOOD
    -> SUPPORTS
    -> BAYESIAN_INFERENCE

  LIKELIHOOD
    -> SUPPORTS
    -> MODEL_COMPARISON

  LIKELIHOOD
    -> CONNECTS_TO
    -> INFORMATION_CRITERIA

  LOG_LIKELIHOOD
    -> CONNECTS_TO
    -> OPTIMISATION

SEMANTIC_EQUIVALENCE:
  "likelihood function"
    ~= "data support function over parameter values"

  "maximum likelihood"
    ~= "parameter selection by likelihood maximisation"

  "log-likelihood"
    ~= "logarithm of likelihood"

  EXCLUDE_EQUIVALENCE:
    LIKELIHOOD != PROBABILITY_OF_PARAMETER
    LIKELIHOOD != POSTERIOR
    LIKELIHOOD != PRIOR

CONFIDENCE:
  ontology_identity: 0.99
  mathematical_definition: 0.99
  bayesian_relationship: 0.99
  semantic_distinction_probability_likelihood: 0.99
  inference_structure: 0.98

UPDATE_MODEL:
  IF:
    NEW_DATA_ARRIVES

  THEN:
    OBSERVED_DATA :=
      OBSERVED_DATA UNION NEW_DATA

    RECOMPUTE:
      LIKELIHOOD_FUNCTION
      LOG_LIKELIHOOD

    IF BAYESIAN_CONTEXT:
      UPDATE:
        PRIOR
        -> POSTERIOR

  PRESERVE:
    MODEL_ASSUMPTIONS
    PARAMETER_DEFINITIONS
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_OBSERVED_DATA
    - IDENTIFY_STATISTICAL_MODEL
    - IDENTIFY_PARAMETERS
    - CONSTRUCT_LIKELIHOOD
    - DISTINGUISH_LIKELIHOOD_FROM_PROBABILITY
    - APPLY_ESTIMATION_OR_BAYESIAN_UPDATE
    - VALIDATE_ASSUMPTIONS
    - RETURN_INTERPRETATION

  constraints:
    - DO_NOT_TREAT_LIKELIHOOD_AS_POSTERIOR_PROBABILITY
    - KEEP_OBSERVED_DATA_FIXED_DURING_PARAMETER_COMPARISON
    - PRESERVE_CONDITIONAL_STRUCTURE
    - EXPOSE_MODEL_ASSUMPTIONS

FLOW_HASH:
  canonical_entity: "likelihood"
  namespace: "probability-statistics/bayesian-statistics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - CORE_RELATIONSHIPS
    - BAYESIAN_RELATION

END_FLOW