| import Mathlib |
| import OAI.Probability.Perceptron.Variational.Shape |
|
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| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Classical ENNReal NNReal BigOperators Topology |
| namespace SphericalPerceptronFreeEnergy |
| variable {X S : Type} [MeasurableSpace X] [MeasurableSpace S] [Nonempty S] |
|
|
| def decoratedFocusedProbability (ν : ProbabilityMeasure S) (step : X×S → X) : |
| (n : ℕ) → (Fin n → ℝ) → (Fin n → X×S → ℝ) → ℕ → |
| CascadeVisitShape n → X×DecoratedCascade S n → ℝ≥0∞ |
| | n, z, F, 0, s, p => decoratedShapeProbability ν step n z F (trivialDecoratedShape X n s) p |
| | 0, _, _, _+1, _, _ => 0 |
| | _+1, _, _, _+1, [], _ => 0 |
| | n+1, z, F, d+1, s::ss, p => markedBlockProbability |
| ((cascadeVisitCount n s+1, fun t : S×DecoratedCascade S n => |
| decoratedFocusedProbability ν step n (fun i => z i.succ) (fun i => F i.succ) d s |
| (step (p.1,t.1),t.2)) :: |
| ss.map (fun t => (cascadeVisitCount n t,fun u : S×DecoratedCascade S n => |
| decoratedShapeProbability ν step n (fun i => z i.succ) (fun i => F i.succ) |
| (trivialDecoratedShape X n t) (step (p.1,u.1),u.2)))) |
| ((markedStableCountKernel p.2).map (fun t : ℝ×(S×DecoratedCascade S n) => |
| (t.1+decoratedCenteredRoot ν step n z F (p.1,t.2),t.2))) (Fin.elim0 : Fin 0 → ℝ) |
|
|
| lemma decoratedFocusedProbability_measurable (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin n → ℝ) |
| (F : Fin n → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (d : ℕ) (s : CascadeVisitShape n) (hv : s.Valid n) : |
| Measurable (decoratedFocusedProbability ν step n z F d s) := by |
| induction n generalizing d with |
| | zero => |
| cases d |
| · exact decoratedShapeProbability_measurable ν step hs 0 z F hF _ |
| (trivialDecoratedShape_valid X 0 s hv) |
| · exact measurable_const |
| | succ n ih => |
| cases d with |
| | zero => |
| exact decoratedShapeProbability_measurable ν step hs (n+1) z F hF _ |
| (trivialDecoratedShape_valid X (n+1) s hv) |
| | succ d => |
| cases s with |
| | nil => exact measurable_const |
| | cons s ss => |
| let ns : List (ℕ×(X×(S×DecoratedCascade S n) → ℝ≥0∞)) := |
| (cascadeVisitCount n s+1,fun p => |
| decoratedFocusedProbability ν step n (fun i => z i.succ) (fun i => F i.succ) d s |
| (step (p.1,p.2.1),p.2.2)) :: |
| ss.map (fun t => (cascadeVisitCount n t, fun p => |
| decoratedShapeProbability ν step n (fun i => z i.succ) (fun i => F i.succ) |
| (trivialDecoratedShape X n t) (step (p.1,p.2.1),p.2.2))) |
| have hm : ∀ nf ∈ ns, Measurable nf.2 := by |
| intro nf hnf |
| rcases List.mem_cons.mp hnf with rfl | ht |
| · exact (ih _ _ (fun i => hF i.succ) d s (hv.2 s (by simp))).comp |
| ((hs.comp (measurable_fst.prodMk measurable_snd.fst)).prodMk measurable_snd.snd) |
| · obtain ⟨t,hmem,rfl⟩ := List.mem_map.mp ht |
| exact (decoratedShapeProbability_measurable ν step hs n _ _ (fun i => hF i.succ) _ |
| (trivialDecoratedShape_valid X n t (hv.2 t (by simp [hmem])))).comp |
| ((hs.comp (measurable_fst.prodMk measurable_snd.fst)).prodMk measurable_snd.snd) |
| have hmap := countKernel_parameter_map_measurable |
| (fun p : X×(ℝ×(S×DecoratedCascade S n)) => |
| (p.2.1+decoratedCenteredRoot ν step n z F (p.1,p.2.2),p.2.2)) |
| ((measurable_snd.fst.add ((decoratedCenteredRoot_measurable ν step hs n z F hF).comp |
| (measurable_fst.prodMk measurable_snd.snd))).prodMk measurable_snd.snd) |
| have hh := (markedBlockProbability_parameter_measurable ns hm 0).comp |
| (measurable_fst.prodMk (hmap.prodMk (measurable_const (a := (Fin.elim0 : Fin 0 → ℝ))))) |
| convert hh using 1 |
| funext p |
| simp only [decoratedFocusedProbability,ns,List.map_cons,List.map_map,Function.comp_def] |
|
|
| lemma trivialDecoratedShape_integral (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin n → ℝ) (hz : StrictMono z) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) |
| (F : Fin n → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (hI : ∀ i x, Integrable (fun s => Real.exp (z i*F i (x,s))) ν) |
| (hM : ∀ i x, (∫ s, Real.exp (z i*F i (x,s)) ∂ν) = 1) |
| (a : ℝ) (ha : ∀ i, a < z i) (s : CascadeVisitShape n) (hv : s.Valid n) (x : X) : |
| (∫⁻ η, decoratedShapeProbability ν step n z F (trivialDecoratedShape X n s) (x,η) |
| ∂decoratedBiasedLaw ν step n z F x a) = cascadeShapeLikelihood n z a s := by |
| rw [decoratedShapeProbability_integral ν step hs n z hz hz0 hz1 F hF hI hM a ha _ |
| (trivialDecoratedShape_valid X n s hv) x, |
| trivialDecoratedShape_bare,trivialDecoratedShape_markValue ν step n z F hI hM s x,mul_one] |
|
|
| omit [Nonempty S] in |
| lemma exponentialMarkTilt_probability_of_one (ν : ProbabilityMeasure S) {b : ℝ} {G : S → ℝ} |
| (hI : Integrable (fun s => Real.exp (b*G s)) ν) |
| (hM : (∫ s, Real.exp (b*G s) ∂ν) = 1) : |
| IsProbabilityMeasure (exponentialMarkTilt ν b G) := by |
| rw [exponentialMarkTilt_eq_density_of_one (ν : Measure S) (b := b) (F := G) hI hM] |
| exact normalized_mark_density_probability (ν : Measure S) (b := b) (F := G) hI hM |
|
|
| lemma decoratedFocusedProbability_integral_succ_eq (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin (n+1) → ℝ) (hz : StrictMono z) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) |
| (F : Fin (n+1) → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (hI : ∀ i x, Integrable (fun s => Real.exp (z i*F i (x,s))) ν) |
| (hM : ∀ i x, (∫ s, Real.exp (z i*F i (x,s)) ∂ν) = 1) |
| (a : ℝ) (ha : ∀ i, a < z i) (d : ℕ) |
| (s : CascadeVisitShape n) (ss : List (CascadeVisitShape n)) |
| (hv : CascadeVisitShape.Valid (n+1) (s::ss)) (x : X) |
| (hchild : ∀ y, (∫⁻ η, decoratedFocusedProbability ν step n (fun i => z i.succ) |
| (fun i => F i.succ) d s (y,η) ∂decoratedBiasedLaw ν step n (fun i => z i.succ) |
| (fun i => F i.succ) y (z 0)) = ∫⁻ η, focusedShapeProbability n (fun i => z i.succ) |
| d s η ∂cascadeBiasedLaw n (fun i => z i.succ) (z 0)) : |
| (∫⁻ η, decoratedFocusedProbability ν step (n+1) z F (d+1) (s::ss) (x,η) |
| ∂decoratedBiasedLaw ν step (n+1) z F x a) = |
| ∫⁻ η, focusedShapeProbability (n+1) z (d+1) (s::ss) η |
| ∂cascadeBiasedLaw (n+1) z a := by |
| have htail : StrictMono (fun i : Fin n => z i.succ) := |
| fun i j hij => hz (Fin.succ_lt_succ_iff.mpr hij) |
| have har : ∀ i : Fin n, z 0 < z i.succ := fun i => hz (by simp) |
| let ns : List (ℕ×(S×DecoratedCascade S n → ℝ≥0∞)) := |
| (cascadeVisitCount n s+1,fun t => decoratedFocusedProbability ν step n |
| (fun i => z i.succ) (fun i => F i.succ) d s (step (x,t.1),t.2)) :: |
| ss.map (fun t => (cascadeVisitCount n t,fun u => decoratedShapeProbability ν step n |
| (fun i => z i.succ) (fun i => F i.succ) (trivialDecoratedShape X n t) |
| (step (x,u.1),u.2))) |
| let ps : List (ℕ×(StableCascade n → ℝ≥0∞)) := |
| (cascadeVisitCount n s+1,focusedShapeProbability n (fun i => z i.succ) d s):: |
| ss.map (fun t => (cascadeVisitCount n t,cascadeShapeProbability n (fun i => z i.succ) t)) |
| have hn : ∀ nf ∈ ns, 1 ≤ nf.1 := by |
| intro nf hnf |
| rcases List.mem_cons.mp hnf with rfl | ht |
| · simp |
| · obtain ⟨t,hmem,rfl⟩ := List.mem_map.mp ht |
| exact cascadeVisitCount_pos n t (hv.2 t (by simp [hmem])) |
| have hpn : ∀ nf ∈ ps, 1 ≤ nf.1 := by |
| intro nf hnf |
| rcases List.mem_cons.mp hnf with rfl | ht |
| · simp |
| · obtain ⟨t,hmem,rfl⟩ := List.mem_map.mp ht |
| exact cascadeVisitCount_pos n t (hv.2 t (by simp [hmem])) |
| have hm : ∀ nf ∈ ns, Measurable nf.2 := by |
| intro nf hnf |
| rcases List.mem_cons.mp hnf with rfl | ht |
| · exact (decoratedFocusedProbability_measurable ν step hs n _ _ |
| (fun i => hF i.succ) d s (hv.2 s (by simp))).comp |
| ((hs.comp (measurable_const.prodMk measurable_fst)).prodMk measurable_snd) |
| · obtain ⟨t,hmem,rfl⟩ := List.mem_map.mp ht |
| exact (decoratedShapeProbability_measurable ν step hs n _ _ (fun i => hF i.succ) _ |
| (trivialDecoratedShape_valid X n t (hv.2 t (by simp [hmem])))).comp |
| ((hs.comp (measurable_const.prodMk measurable_fst)).prodMk measurable_snd) |
| have hpm : ∀ nf ∈ ps, Measurable nf.2 := by |
| intro nf hnf |
| rcases List.mem_cons.mp hnf with rfl | ht |
| · exact focusedShapeProbability_measurable n _ d s |
| · obtain ⟨t,hmem,rfl⟩ := List.mem_map.mp ht |
| exact cascadeShapeProbability_measurable n _ t |
| change (∫⁻ η, markedBlockProbability ns |
| ((markedStableCountKernel η).map (logMarkShift |
| (centeredLogMark ((ν : Measure S).prod |
| (decoratedCascadeLaw ν n (fun i => z i.succ))) (z 0) |
| (fun p => decoratedRootPotential step n F (x,p))))) |
| (Fin.elim0 : Fin 0 → ℝ) ∂decoratedBiasedLaw ν step (n+1) z F x a) = |
| ∫⁻ η, markedBlockProbability ps _ (Fin.elim0 : Fin 0 → ℝ) ∂cascadeBiasedLaw (n+1) z a |
| rw [decoratedRootBlock_factor ν step hs n z hz hz0 hz1 F hF hI hM x a (ha 0) |
| ns (by simp [ns]) hn hm, |
| cascadeRootBlock_factor z hz hz0 hz1 (ha 0) ps (by simp [ps]) hpn hpm] |
| let := exponentialMarkTilt_probability_of_one ν (hI 0 x) (hM 0 x) |
| have he : ns.map Prod.fst = ps.map Prod.fst := by |
| simp only [ns,ps,List.map_cons,List.map_map,Function.comp_def] |
| rw [he] |
| apply congrArg (fun q : ℝ≥0∞ => q * _) |
| simp only [ns,ps,List.map_cons,List.prod_cons] |
| apply congrArg₂ (· * ·) |
| · have hh : ∀ u, (∫⁻ C, decoratedFocusedProbability ν step n (fun i => z i.succ) |
| (fun i => F i.succ) d s (step (x,u),C) ∂decoratedBiasedLaw ν step n |
| (fun i => z i.succ) (fun i => F i.succ) (step (x,u)) (z 0)) = |
| ∫⁻ C, focusedShapeProbability n (fun i => z i.succ) d s C |
| ∂cascadeBiasedLaw n (fun i => z i.succ) (z 0) := by |
| intro u |
| exact hchild (step (x,u)) |
| simp_rw [hh] |
| simp |
| · congr 1 |
| simp only [List.map_map,Function.comp_def] |
| apply List.map_congr_left |
| intro t ht |
| simp_rw [trivialDecoratedShape_integral ν step hs n _ htail |
| (fun i => hz0 i.succ) (fun i => hz1 i.succ) _ (fun i => hF i.succ) |
| (fun i => hI i.succ) (fun i => hM i.succ) (z 0) har t (hv.2 t (by simp [ht])), |
| cascadeShapeProbability_integral n _ htail (fun i => hz0 i.succ) |
| (fun i => hz1 i.succ) (z 0) har t (hv.2 t (by simp [ht]))] |
| simp |
|
|
| lemma decoratedFocusedProbability_integral_eq (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin n → ℝ) (hz : StrictMono z) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) |
| (F : Fin n → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (hI : ∀ i x, Integrable (fun s => Real.exp (z i*F i (x,s))) ν) |
| (hM : ∀ i x, (∫ s, Real.exp (z i*F i (x,s)) ∂ν) = 1) |
| (a : ℝ) (ha : ∀ i, a < z i) (d : ℕ) (hd : d ≤ n) |
| (s : CascadeVisitShape n) (hv : s.Valid n) (x : X) : |
| (∫⁻ η, decoratedFocusedProbability ν step n z F d s (x,η) |
| ∂decoratedBiasedLaw ν step n z F x a) = |
| ∫⁻ η, focusedShapeProbability n z d s η ∂cascadeBiasedLaw n z a := by |
| induction d generalizing n a x with |
| | zero => |
| simp only [decoratedFocusedProbability,focusedShapeProbability] |
| rw [trivialDecoratedShape_integral ν step hs n z hz hz0 hz1 F hF hI hM a ha s hv x, |
| cascadeShapeProbability_integral n z hz hz0 hz1 a ha s hv] |
| | succ d ih => |
| cases n with |
| | zero => omega |
| | succ n => |
| cases s with |
| | nil => exact (hv.1 rfl).elim |
| | cons s ss => |
| apply decoratedFocusedProbability_integral_succ_eq ν step hs n z hz hz0 hz1 F hF hI hM |
| a ha d s ss hv x |
| intro y |
| exact ih n _ (fun i j hij => hz (Fin.succ_lt_succ_iff.mpr hij)) |
| (fun i => hz0 i.succ) (fun i => hz1 i.succ) _ |
| (fun i => hF i.succ) (fun i => hI i.succ) (fun i => hM i.succ) |
| (z 0) (fun i => hz (by simp)) (by omega) s (hv.2 s (by simp)) y |
|
|
| lemma decoratedFocusedProbability_integral (ν : ProbabilityMeasure S) (step : X×S → X) |
| (hs : Measurable step) (n : ℕ) (z : Fin n → ℝ) (hz : StrictMono z) |
| (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) |
| (F : Fin n → X×S → ℝ) (hF : ∀ i, Measurable (F i)) |
| (hI : ∀ i x, Integrable (fun s => Real.exp (z i*F i (x,s))) ν) |
| (hM : ∀ i x, (∫ s, Real.exp (z i*F i (x,s)) ∂ν) = 1) |
| (a : ℝ) (ha1 : a < 1) (ha : ∀ i, a < z i) (d : ℕ) (hd : d ≤ n) |
| (s : CascadeVisitShape n) (hv : s.Valid n) (x : X) : |
| (∫⁻ η, decoratedFocusedProbability ν step n z F d s (x,η) |
| ∂decoratedBiasedLaw ν step n z F x a) = |
| cascadeShapeLikelihood n z a s * |
| ENNReal.ofReal (focusedShapeNumerator n z a d s/((cascadeVisitCount n s:ℝ)-a)) := by |
| rw [decoratedFocusedProbability_integral_eq ν step hs n z hz hz0 hz1 F hF hI hM a ha d hd s hv x, |
| focusedShapeProbability_integral n z hz hz0 hz1 a ha1 ha d hd s hv] |
|
|
| end SphericalPerceptronFreeEnergy |
|
|
| end |
|
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| end OAI |
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