Markov Process Ontology

flow "MarkovProcessOntology2" v4.0

# ============================================================
# MARKOV PROCESS ONTOLOGY 2
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)
# ============================================================

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/stochastic-mathematics/markov-process-ontology-2.html"
  title: "Markov Process Ontology 2"
  domain: "Probability and Statistics"
  category: "Stochastic Mathematics"
  entity: MARKOV_PROCESS
  entity_type: STOCHASTIC_PROCESS

DEFINE_MARKOV_PROCESS:
  MARKOV_PROCESS :=
    STOCHASTIC_PROCESS(
      STATE_SPACE,
      TRANSITION_STRUCTURE,
      MARKOV_PROPERTY
    )

CORE_PROPERTY:
  MARKOV_PROPERTY:
    FUTURE:
      CONDITIONALLY_INDEPENDENT_OF:
        PAST
      GIVEN:
        PRESENT_STATE

  symbolic:
    P(
      X_t IN A
      |
      HISTORY_UP_TO_s
    )
    =
    P(
      X_t IN A
      |
      X_s
    )
    FOR:
      s < t

ENTITIES:
  MARKOV_PROCESS:
    type: STOCHASTIC_PROCESS

  STATE:
    type: PROCESS_CONFIGURATION

  STATE_SPACE:
    type: SET_OF_POSSIBLE_STATES

  TRANSITION_PROBABILITY:
    type: CONDITIONAL_PROBABILITY

  TRANSITION_KERNEL:
    type: PROBABILITY_KERNEL

  TRANSITION_MATRIX:
    type: STOCHASTIC_MATRIX

  INITIAL_DISTRIBUTION:
    type: PROBABILITY_DISTRIBUTION

  STATIONARY_DISTRIBUTION:
    type: INVARIANT_DISTRIBUTION

  SEMIGROUP:
    type: TRANSITION_OPERATOR_FAMILY

  GENERATOR:
    type: INFINITESIMAL_OPERATOR

RELATIONSHIPS:
  MARKOV_PROCESS
    -> IS_A
    -> STOCHASTIC_PROCESS

  MARKOV_PROCESS
    -> HAS
    -> STATE_SPACE

  MARKOV_PROCESS
    -> HAS
    -> TRANSITION_STRUCTURE

  CURRENT_STATE
    -> DETERMINES
    -> CONDITIONAL_FUTURE_DISTRIBUTION

  TRANSITION_KERNEL
    -> MAPS
    -> CURRENT_STATE_TO_FUTURE_DISTRIBUTION

  INITIAL_DISTRIBUTION
    + TRANSITION_STRUCTURE
    -> DETERMINE
    -> PROCESS_DISTRIBUTIONS

DISCRETE_TIME_MARKOV_PROCESS:
  index:
    n := 0,1,2,...

  property:
    P(X_(n+1) IN A | X_0,...,X_n)
      =
    P(X_(n+1) IN A | X_n)

  transition_kernel:
    K(x,A) :=
      P(X_(n+1) IN A | X_n=x)

MARKOV_CHAIN:
  special_case:
    DISCRETE_STATE_MARKOV_PROCESS
    OR:
    DISCRETE_TIME_MARKOV_PROCESS
    DEPENDING_ON_CONVENTION

  countable_state_transition:
    P_ij :=
      P(X_(n+1)=j | X_n=i)

  matrix:
    P := [P_ij]

  require:
    P_ij >= 0
    SUM_j P_ij = 1

CONTINUOUS_TIME_MARKOV_PROCESS:
  index:
    t >= 0

  transition_kernel:
    P_t(x,A)
    FOR:
      TIME_HOMOGENEOUS_CASE

  property:
    FUTURE_GIVEN_PRESENT:
      INDEPENDENT_OF_PAST

TIME_HOMOGENEITY:
  DEFINE:
    TRANSITION_LAW:
      DEPENDS_ON:
        ELAPSED_TIME
      NOT:
        ABSOLUTE_TIME

  notation:
    P_t(x,A)

  relation:
    P(s,x;t,A)
      =
    P_(t-s)(x,A)

CHAPMAN_KOLMOGOROV:
  time_homogeneous_form:
    P_(s+t)(x,A)
      =
    INTEGRAL
      P_s(x,dy)
      P_t(y,A)

  role:
    COMPOSE:
      TRANSITIONS_OVER_SUCCESSIVE_INTERVALS

TRANSITION_SEMIGROUP:
  operators:
    P_t f(x) :=
      E_x[f(X_t)]

  properties:
    P_0 := IDENTITY
    P_(s+t) := P_s P_t

  role:
    REPRESENT:
      TIME_EVOLUTION_OF_EXPECTATIONS

GENERATOR:
  continuous_time:
    L f :=
      LIMIT[t->0+]
      (
        P_t f - f
      )
      /
      t

  role:
    DESCRIBE:
      INFINITESIMAL_PROCESS_DYNAMICS

  examples:
    CONTINUOUS_TIME_MARKOV_CHAIN:
      GENERATOR_MATRIX_Q

    DIFFUSION:
      DIFFERENTIAL_OPERATOR

CONTINUOUS_TIME_MARKOV_CHAIN:
  alias:
    CTMC

  state_space:
    FINITE_OR_COUNTABLE

  generator_matrix:
    Q := [q_ij]

  off_diagonal:
    q_ij >= 0
    FOR:
      i != j

  diagonal:
    q_ii :=
      -SUM[j != i] q_ij

  row_sum:
    SUM_j q_ij = 0

  transition_matrix:
    P(t) :=
      exp(tQ)
    UNDER:
      STANDARD_FINITE_STATE_TIME_HOMOGENEOUS_CASE

KOLMOGOROV_FORWARD_EQUATION:
  finite_state:
    dP(t)/dt :=
      P(t) Q

KOLMOGOROV_BACKWARD_EQUATION:
  finite_state:
    dP(t)/dt :=
      Q P(t)

INITIAL_DISTRIBUTION:
  notation:
    mu_0

  evolution:
    mu_t :=
      mu_0 P_t

  discrete_matrix_case:
    pi_n :=
      pi_0 P^n

STATIONARY_DISTRIBUTION:
  continuous_or_general:
    pi P_t :=
      pi
    FOR:
      ALL t >= 0

  discrete_time:
    pi P :=
      pi

  finite_ctmc:
    pi Q :=
      0

  normalisation:
    SUM_i pi_i = 1
    pi_i >= 0

INVARIANT_MEASURE:
  DEFINE:
    MEASURE_mu:
      SATISFIES:
        mu P_t = mu

  note:
    MAY_NOT_BE:
      NORMALISED_PROBABILITY_DISTRIBUTION

STATE_CLASSIFICATION:
  ACCESSIBILITY:
    STATE_y:
      REACHABLE_FROM:
        STATE_x
      WITH:
        POSITIVE_PROBABILITY

  COMMUNICATION:
    x COMMUNICATES_WITH y
    IF:
      x ACCESSIBLE_FROM y
      AND:
      y ACCESSIBLE_FROM x

  IRREDUCIBILITY:
    ALL_RELEVANT_STATES:
      COMMUNICATE

  RECURRENCE:
    STATE:
      RETURNS_WITH_PROBABILITY_ONE
      UNDER:
        RELEVANT_DEFINITION

  TRANSIENCE:
    STATE:
      MAY_FAIL_TO_RETURN

  ABSORBING_STATE:
    ONCE_ENTERED:
      PROCESS_REMAINS_THERE

PERIODICITY:
  primarily_discrete_time:
    PERIOD:
      GCD_OF_POSSIBLE_RETURN_TIMES

  APERIODIC:
    PERIOD = 1

  relevance:
    CONVERGENCE_TO_STATIONARITY

ERGODICITY:
  general_role:
    ENSURE:
      LONG_RUN_BEHAVIOUR
      FORGETS_INITIAL_CONDITION
    UNDER:
      APPROPRIATE_CONDITIONS

  possible_consequences:
    - CONVERGENCE_TO_STATIONARY_DISTRIBUTION
    - LONG_RUN_TIME_AVERAGE_RESULTS

  caution:
    EXACT_CONDITIONS:
      DEPEND_ON:
        STATE_SPACE
        TIME_STRUCTURE
        PROCESS_CLASS

REVERSIBILITY:
  stationary_distribution:
    pi

  detailed_balance:
    pi(dx) P_t(x,dy)
      =
    pi(dy) P_t(y,dx)

  discrete_case:
    pi_i P_ij :=
      pi_j P_ji

  consequence:
    DETAILED_BALANCE
      -> STATIONARITY
    UNDER:
      VALID_NORMALISATION

STRONG_MARKOV_PROPERTY:
  DEFINE:
    MARKOV_PROPERTY_EXTENDED_TO:
      SUITABLE_STOPPING_TIMES

  form:
    FUTURE_AFTER_STOPPING_TIME:
      DEPENDS_ON:
        STATE_AT_STOPPING_TIME
      NOT:
        PRE_STOPPING_HISTORY
    UNDER:
      STRONG_MARKOV_CONDITIONS

  examples:
    - BROWNIAN_MOTION
    - MANY_DIFFUSIONS
    - STANDARD_MARKOV_CHAINS

STOPPING_TIME:
  tau:
    random_time

  requirement:
    {tau <= t}:
      F_t_MEASURABLE

HITTING_TIME:
  tau_A :=
    inf{
      t >= 0 :
      X_t IN A
    }

  role:
    FIRST_ENTRY_TIME_INTO_SET_A

FIRST_PASSAGE_ANALYSIS:
  targets:
    - HITTING_PROBABILITY
    - EXPECTED_HITTING_TIME
    - EXIT_TIME
    - ABSORPTION_PROBABILITY

BROWNIAN_MOTION:
  type:
    CONTINUOUS_STATE
    CONTINUOUS_TIME
    MARKOV_PROCESS

  transition:
    W_t | W_s=x
      ~
    NORMAL(
      x,
      t-s
    )

  property:
    STRONG_MARKOV = TRUE

DIFFUSION_PROCESS:
  SDE:
    dX_t :=
      b(X_t) dt
      + sigma(X_t) dW_t

  generator:
    Lf(x) :=
      b(x) f_prime(x)
      + 0.5 sigma(x)^2 f_double_prime(x)
    IN:
      ONE_DIMENSION

  markov_property:
    HOLDS_UNDER:
      APPROPRIATE_EXISTENCE_UNIQUENESS_AND_COEFFICIENT_CONDITIONS

ORNSTEIN_UHLENBECK_PROCESS:
  SDE:
    dX_t :=
      theta*(mu-X_t) dt
      + sigma dW_t

  type:
    GAUSSIAN_MARKOV_PROCESS

  behaviour:
    MEAN_REVERTING

  stationary_distribution:
    NORMAL(
      mu,
      sigma^2/(2*theta)
    )
    FOR:
      theta > 0

POISSON_PROCESS:
  type:
    COUNTING_MARKOV_PROCESS

  transition:
    N_t-N_s
      ~
    POISSON(lambda*(t-s))

  state:
    NONNEGATIVE_INTEGER_COUNT

HIDDEN_MARKOV_MODEL_CONNECTION:
  HIDDEN_STATE_PROCESS:
    MARKOV

  OBSERVATIONS:
    CONDITIONALLY_DEPEND_ON:
      CURRENT_HIDDEN_STATE

  distinction:
    OBSERVED_SEQUENCE:
      NEED_NOT_BE:
        MARKOV

MARKOV_DECISION_PROCESS_CONNECTION:
  state:
    MARKOVIAN

  action:
    INFLUENCES:
      TRANSITION_DISTRIBUTION
      REWARD

  transition:
    P(s_next | s_current, action)

MARKOV_RANDOM_FIELD_DISTINCTION:
  MARKOV_PROCESS:
    DIRECTED_TEMPORAL_OR_INDEXED_TRANSITION_STRUCTURE

  MARKOV_RANDOM_FIELD:
    UNDIRECTED_CONDITIONAL_DEPENDENCE_STRUCTURE

  rule:
    MARKOV_PROCESS != MARKOV_RANDOM_FIELD

SIMULATION:
  DISCRETE_TIME:
    SET X_0
      -> FOR n:
           SAMPLE X_(n+1) FROM K(X_n, .)
           RECORD STATE

  CONTINUOUS_TIME_CHAIN:
    SET INITIAL_STATE
      -> SAMPLE_HOLDING_TIME
      -> SAMPLE_NEXT_STATE
      -> UPDATE_TIME
      -> REPEAT

  DIFFUSION:
    APPROXIMATE_USING:
      EULER_MARUYAMA
      OR:
      HIGHER_ORDER_METHOD
    WHEN:
      EXACT_SIMULATION_UNAVAILABLE

INFERENCE:
  targets:
    - TRANSITION_PROBABILITIES
    - GENERATOR_RATES
    - DRIFT_PARAMETERS
    - DIFFUSION_PARAMETERS
    - STATIONARY_DISTRIBUTION
    - HITTING_PROBABILITIES

  data:
    - FULL_PATH
    - DISCRETE_OBSERVATIONS
    - PARTIALLY_OBSERVED_PATH

MODEL_CONNECTIONS:
  MARKOV_PROCESS
    -> CONNECTS_TO
    -> STOCHASTIC_PROCESS

  MARKOV_PROCESS
    -> CONNECTS_TO
    -> MARKOV_CHAIN

  MARKOV_PROCESS
    -> CONNECTS_TO
    -> BROWNIAN_MOTION

  MARKOV_PROCESS
    -> CONNECTS_TO
    -> DIFFUSION_PROCESS

  MARKOV_PROCESS
    -> CONNECTS_TO
    -> POISSON_PROCESS

  MARKOV_PROCESS
    -> CONNECTS_TO
    -> HIDDEN_MARKOV_MODEL

  MARKOV_PROCESS
    -> CONNECTS_TO
    -> MARKOV_DECISION_PROCESS

APPLICATIONS:
  MARKOV_PROCESS
    -> SUPPORTS
    -> QUEUEING_THEORY

  MARKOV_PROCESS
    -> SUPPORTS
    -> RELIABILITY

  MARKOV_PROCESS
    -> SUPPORTS
    -> FINANCIAL_MODELLING

  MARKOV_PROCESS
    -> SUPPORTS
    -> POPULATION_MODELLING

  MARKOV_PROCESS
    -> SUPPORTS
    -> MACHINE_LEARNING

  MARKOV_PROCESS
    -> SUPPORTS
    -> PHYSICAL_SYSTEMS

SEMANTIC_EQUIVALENCE:
  "Markov process"
    ~= "stochastic process satisfying the Markov property"

  "transition kernel"
    ~= "conditional probability kernel governing state evolution"

  "stationary distribution"
    ~= "invariant probability distribution"

  "strong Markov property"
    ~= "Markov property at suitable stopping times"

  EXCLUDE_EQUIVALENCE:
    MARKOV_PROCESS != INDEPENDENT_PROCESS
    MARKOV_PROPERTY != MEMORYLESS_HOLDING_TIME_BY_DEFAULT
    MARKOV_PROCESS != MARKOV_RANDOM_FIELD
    STATIONARY_DISTRIBUTION != INITIAL_DISTRIBUTION_BY_DEFAULT
    TIME_HOMOGENEOUS != STATIONARY_PROCESS
    REVERSIBILITY != MARKOV_PROPERTY

VALIDATION_RULES:
  RULE_1:
    STATE_SPACE:
      MUST_BE:
        DEFINED

  RULE_2:
    TRANSITION_KERNEL:
      MUST_DEFINE:
        VALID_PROBABILITY_DISTRIBUTION
      FOR:
        EACH_CURRENT_STATE

  RULE_3:
    MARKOV_PROPERTY:
      MUST_BE:
        JUSTIFIED_OR_ASSUMED_EXPLICITLY

  RULE_4:
    TIME_HOMOGENEITY:
      MUST_NOT_BE:
        ASSUMED_WITHOUT_MODEL_SUPPORT

  RULE_5:
    STATIONARY_DISTRIBUTION:
      MUST_SATISFY:
        INVARIANCE_RELATION

  RULE_6:
    GENERATOR_MATRIX:
      MUST_HAVE:
        NONNEGATIVE_OFF_DIAGONALS
        ZERO_ROW_SUMS

  RULE_7:
    CONVERGENCE_TO_STATIONARITY:
      MUST_REQUIRE:
        APPROPRIATE_ERGODIC_CONDITIONS

  RULE_8:
    MARKOV_PROPERTY:
      MUST_NOT_BE:
        INTERPRETED_AS_FULL_INDEPENDENCE

CONFIDENCE:
  ontology_identity: 0.99
  markov_property: 0.99
  transition_kernel_structure: 0.99
  semigroup_structure: 0.99
  generator_structure: 0.99
  stationary_distribution_structure: 0.99
  strong_markov_structure: 0.98

UPDATE_MODEL:
  IF:
    NEW_STATE_OBSERVATION_ARRIVES

  THEN:
    APPEND:
      PROCESS_HISTORY

    UPDATE_IF_REQUIRED:
      TRANSITION_ESTIMATES
      GENERATOR_ESTIMATES
      STATE_ESTIMATE

  IF:
    TRANSITION_MODEL_CHANGES

  THEN:
    UPDATE:
      TRANSITION_KERNEL
      SEMIGROUP
      GENERATOR_IF_APPLICABLE

    REASSESS:
      STATIONARITY
      ERGODICITY
      HITTING_PROPERTIES

  IF:
    STATE_SPACE_CHANGES

  THEN:
    UPDATE:
      STATE_DEFINITIONS
      TRANSITION_STRUCTURE

    REVALIDATE:
      NORMALISATION
      ACCESSIBILITY
      COMMUNICATION_CLASSES

  PRESERVE:
    PROCESS_IDENTITY
    STATE_IDENTITIES
    TIME_ORDER
    TRANSITION_HISTORY
    MODEL_PARAMETERS
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_STATE_SPACE
    - IDENTIFY_TIME_INDEX
    - IDENTIFY_INITIAL_DISTRIBUTION
    - IDENTIFY_TRANSITION_KERNEL
    - VERIFY_MARKOV_PROPERTY
    - DETERMINE_TIME_HOMOGENEITY
    - APPLY_CHAPMAN_KOLMOGOROV
    - IDENTIFY_SEMIGROUP_AND_GENERATOR_IF_APPLICABLE
    - ANALYSE_STATE_CLASSIFICATION
    - TEST_STATIONARITY_OR_REVERSIBILITY_IF_REQUIRED
    - COMPUTE_HITTING_QUANTITIES_IF_REQUIRED
    - SIMULATE_OR_PREDICT
    - RETURN_MARKOV_PROCESS_INTERPRETATION

  constraints:
    - PRESERVE_STATE_SPACE
    - PRESERVE_TRANSITION_STRUCTURE
    - DO_NOT_EQUATE_MARKOV_PROPERTY_WITH_INDEPENDENCE
    - DO_NOT_ASSUME_TIME_HOMOGENEITY
    - DO_NOT_ASSUME_STATIONARITY
    - REQUIRE_VALID_ERGODIC_CONDITIONS_FOR_LONG_RUN_CONVERGENCE
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "markov-process-ontology-2"
  namespace: "probability-statistics/stochastic-mathematics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - MARKOV_PROPERTY
    - TRANSITION_KERNEL
    - STATE_SPACE

END_FLOW