bayesStep_is_update

formal source

  40    rw [← Finset.sum_div]
  41    exact div_self (ne_of_gt (evidence_pos prior L o)) }
  42
  43theorem bayesStep_is_update (prior : ProbDist ι) (L : Likelihood ι Obs) (o : Obs) :
  44    ∀ i, (bayesStep prior L o).prob i =
  45      prior.prob i * L.like o i / evidence prior L o := by
  46  intro i
  47  rfl
  48
  49def iterFilter (L : Likelihood ι Obs) : List Obs → ProbDist ι → ProbDist ι
  50  | [], p => p
  51  | o :: os, p => iterFilter L os (bayesStep p L o)
  52
  53theorem iterFilter_nil (L : Likelihood ι Obs) (p : ProbDist ι) :
  54    iterFilter L [] p = p := rfl
  55
  56theorem iterFilter_cons (L : Likelihood ι Obs) (o : Obs) (os : List Obs) (p : ProbDist ι) :
  57    iterFilter L (o :: os) p = iterFilter L os (bayesStep p L o) := rfl
  58
  59structure BayesianFilteringCert where
  60  evidence_positive : ∀ (prior : ProbDist ι) (L : Likelihood ι Obs) (o : Obs),
  61    0 < evidence prior L o
  62  bayes_step_update : ∀ (prior : ProbDist ι) (L : Likelihood ι Obs) (o : Obs) (i : ι),
  63    (bayesStep prior L o).prob i =
  64      prior.prob i * L.like o i / evidence prior L o
  65
  66theorem bayesianFilteringCert_holds : BayesianFilteringCert (ι := ι) (Obs := Obs) :=
  67{ evidence_positive := evidence_pos
  68  bayes_step_update := fun prior L o i => bayesStep_is_update prior L o i }
  69
  70end
  71
  72end IndisputableMonolith.Statistics.BayesianFilteringFromVFE