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import OAI.Probability.Perceptron.Variational.FocusedSampleLaw
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped Classical ENNReal NNReal BigOperators Topology
namespace SphericalPerceptronFreeEnergy
def indexedBlockSample {I : Type} [Fintype I] (k : ℕ)
(A : I → I → Fin (k+1)) (b : IndexedCascadeBase k) : ℝ≥0∞ :=
∑' xs : I → IndexedLeaf k, (∏ i, (indexedLeafProbability k b) {xs i}) *
if (fun i j => indexedCommonDepth k (xs i) (xs j)) = A then 1 else 0
def indexedBlockFocused {I : Type} [Fintype I] (k d : ℕ)
(A : I → I → Fin (k+1)) (i₀ : I) (b : IndexedCascadeBase k) : ℝ≥0∞ :=
∑' xs : I → IndexedLeaf k, ((∏ i, (indexedLeafProbability k b) {xs i}) *
if (fun i j => indexedCommonDepth k (xs i) (xs j)) = A then 1 else 0) *
∑' l : IndexedLeaf k, (indexedLeafProbability k b) {l} *
if d ≤ (indexedCommonDepth k (xs i₀) l).val then 1 else 0
lemma reindexed_shape_matches {I : Type} [Fintype I] (k : ℕ)
(s : CascadeVisitShape k) (e : I ≃ CascadeShapeReplicas k s)
(A : I → I → Fin (k+1))
(hA : ∀ i j, A i j = indexedCommonDepth k (cascadeShapeLeaves k s (e i))
(cascadeShapeLeaves k s (e j))) (ys : CascadeShapeReplicas k s → IndexedLeaf k) :
(fun i j => indexedCommonDepth k (ys (e i)) (ys (e j))) = A ↔
CascadeShapeMatches k s ys := by
rw [cascadeShapeMatches_iff]
constructor
· intro h i j
obtain ⟨i,rfl⟩ := e.surjective i
obtain ⟨j,rfl⟩ := e.surjective j
exact congrFun (congrFun h i) j |>.trans (hA i j)
· intro h
funext i j
exact (h (e i) (e j)).trans (hA i j).symm
lemma indexedBlockSample_shape {I : Type} [Fintype I] (k : ℕ)
(s : CascadeVisitShape k) (e : I ≃ CascadeShapeReplicas k s)
(A : I → I → Fin (k+1))
(hA : ∀ i j, A i j = indexedCommonDepth k (cascadeShapeLeaves k s (e i))
(cascadeShapeLeaves k s (e j))) (b : IndexedCascadeBase k) :
indexedBlockSample k A b = indexedSampleShape k s b := by
unfold indexedBlockSample indexedSampleShape
rw [← (Equiv.arrowCongr e.symm (Equiv.refl (IndexedLeaf k))).tsum_eq]
apply tsum_congr
intro ys
change ((∏ i, (indexedLeafProbability k b) {ys (e i)}) *
if (fun i j => indexedCommonDepth k (ys (e i)) (ys (e j))) = A then 1 else 0) = _
rw [e.prod_comp (fun i => (indexedLeafProbability k b) {ys i})]
simp only [reindexed_shape_matches k s e A hA ys]
lemma indexedBlockFocused_shape {I : Type} [Fintype I] (k d : ℕ)
(s : CascadeVisitShape k) (hs : s.Valid k) (e : I ≃ CascadeShapeReplicas k s)
(A : I → I → Fin (k+1)) (i₀ : I) (hi₀ : e i₀ = cascadeShapeFocus k s hs)
(hA : ∀ i j, A i j = indexedCommonDepth k (cascadeShapeLeaves k s (e i))
(cascadeShapeLeaves k s (e j))) (b : IndexedCascadeBase k) :
indexedBlockFocused k d A i₀ b = indexedSampleFocused k d s hs b := by
unfold indexedBlockFocused indexedSampleFocused
rw [← (Equiv.arrowCongr e.symm (Equiv.refl (IndexedLeaf k))).tsum_eq]
apply tsum_congr
intro ys
change (((∏ i, (indexedLeafProbability k b) {ys (e i)}) *
if (fun i j => indexedCommonDepth k (ys (e i)) (ys (e j))) = A then 1 else 0) *
∑' l, (indexedLeafProbability k b) {l} * if d ≤ (indexedCommonDepth k (ys (e i₀)) l).val then 1 else 0) = _
rw [e.prod_comp (fun i => (indexedLeafProbability k b) {ys i})]
simp only [reindexed_shape_matches k s e A hA ys,hi₀]
lemma indexedBlockSample_measurable {I : Type} [Fintype I] (k : ℕ)
(A : I → I → Fin (k+1)) : Measurable (indexedBlockSample k A) := by
unfold indexedBlockSample
refine Measurable.tsum (fun xs => ?_)
exact (Finset.measurable_prod Finset.univ (fun i _ =>
(Measure.measurable_coe (measurableSet_singleton (xs i))).comp
(indexedLeafProbability_measurable k))).mul_const _
lemma indexedBlockFocused_measurable {I : Type} [Fintype I] (k d : ℕ)
(A : I → I → Fin (k+1)) (i₀ : I) : Measurable (indexedBlockFocused k d A i₀) := by
unfold indexedBlockFocused
refine Measurable.tsum (fun xs => ?_)
exact ((Finset.measurable_prod Finset.univ (fun i _ =>
(Measure.measurable_coe (measurableSet_singleton (xs i))).comp
(indexedLeafProbability_measurable k))).mul_const _).mul
(Measurable.tsum (fun l => ((Measure.measurable_coe (measurableSet_singleton l)).comp
(indexedLeafProbability_measurable k)).mul_const _))
lemma indexedBlockSample_eq_measure {I : Type} [Fintype I] (k : ℕ)
(A : I → I → Fin (k+1)) (b : IndexedCascadeBase k) :
indexedBlockSample k A b = (Measure.pi (fun _ : I => indexedLeafProbability k b))
{xs | (fun i j => indexedCommonDepth k (xs i) (xs j)) = A} := by
rw [← lintegral_indicator_one ((Set.to_countable _).measurableSet),lintegral_countable']
unfold indexedBlockSample
apply tsum_congr
intro xs
rw [Measure.pi_singleton]
simp only [Set.indicator,Set.mem_ofPred_eq,Pi.one_apply]
split_ifs <;> simp
lemma indexedBlockFocused_eq_measure {I : Type} [Fintype I] (k d : ℕ)
(A : I → I → Fin (k+1)) (i₀ : I) (b : IndexedCascadeBase k) :
indexedBlockFocused k d A i₀ b =
((Measure.pi (fun _ : I => indexedLeafProbability k b)).prod (indexedLeafProbability k b))
{p | (fun i j => indexedCommonDepth k (p.1 i) (p.1 j)) = A ∧
d ≤ (indexedCommonDepth k (p.1 i₀) p.2).val} := by
rw [← lintegral_indicator_one ((Set.to_countable _).measurableSet),lintegral_countable',
ENNReal.tsum_prod']
unfold indexedBlockFocused
apply tsum_congr
intro xs
rw [← ENNReal.tsum_mul_left]
apply tsum_congr
intro l
rw [← Set.singleton_prod_singleton,Measure.prod_prod,Measure.pi_singleton]
simp only [Set.indicator,Set.mem_ofPred_eq,Pi.one_apply]
split_ifs <;> simp_all
theorem indexedBlockFocused_integral {I : Type} [Fintype I] (k : ℕ)
(z : Fin k → ℝ) (hz : StrictMono z) (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1)
(d : ℕ) (hd : d ≤ k) (A : I → I → Fin (k+1)) (i₀ : I) :
(∫⁻ b, indexedBlockFocused k d A i₀ b ∂indexedCascadeBaseLaw k z) =
(∫⁻ b, indexedBlockSample k A b ∂indexedCascadeBaseLaw k z) *
ENNReal.ofReal (((∑ j, if d ≤ (A i₀ j).val then (1:ℝ) else 0) -
(if h : d = 0 then 0 else z ⟨d-1,by omega⟩)) / Fintype.card I) := by
by_cases hxs : ∃ xs : I → IndexedLeaf k, (fun i j => indexedCommonDepth k (xs i) (xs j)) = A
· obtain ⟨xs,hxs⟩ := hxs
obtain ⟨s,hs,e,he,hm⟩ := finite_leaf_tuple_pointed_shape k I xs i₀
have hA (i j : I) : A i j = indexedCommonDepth k (cascadeShapeLeaves k s (e i))
(cascadeShapeLeaves k s (e j)) := (congrFun (congrFun hxs i) j).symm.trans (hm i j)
simp_rw [indexedBlockFocused_shape k d s hs e A i₀ he hA,
indexedBlockSample_shape k s e A hA]
rw [indexedSampleFocused_integral k z hz hz0 hz1 d hd s hs,
indexedSampleShape_integral k z hz hz0 hz1 s hs,
focusedShapeNumerator_eq_count k z 0 d hd s hs]
have hc : Fintype.card I = cascadeVisitCount k s :=
(Fintype.card_congr e).trans (cascadeShapeReplicas_card k s)
rw [hc]
congr 3
unfold cascadeFocusCount
rw [← e.sum_comp]
apply congrArg (fun a : ℝ => a - (if h : d = 0 then 0 else z ⟨d-1,by omega⟩))
apply Finset.sum_congr rfl
intro j hj
rw [hA,he]
· have ha (xs : I → IndexedLeaf k) :
¬ (fun i j => indexedCommonDepth k (xs i) (xs j)) = A := fun h => hxs ⟨xs,h⟩
simp [indexedBlockFocused,indexedBlockSample,ha]
end SphericalPerceptronFreeEnergy
end
end OAI
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