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import OAI.Probability.Perceptron.Variational.Shape
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
end OAI
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