| import Mathlib |
| import OAI.Probability.Perceptron.Cascade.WeightedTrialLaw |
| import OAI.Probability.Perceptron.Variational.QuantileUpperTail |
| import OAI.Probability.Perceptron.Variational.GlobalContactLimit |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology ENNReal NNReal BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| variable (k : ℕ) (P : Measure BrownianPath) (f : ℝ →ᵇ ℝ) (p d : ℕ→ℕ) |
| variable (w : Fin (k+1)→ℝ) (q : Fin (k+1)→Time) |
| variable (hw : ∀ i, 0<w i) (hw1 : ∑ i, w i=1) (hq : Monotone q) |
| variable (δ : ℝ) (α : ℝ≥0) (H : ℝ) |
| variable (c : (n : ℕ)→SourceContactParameter n k) |
| variable (hc : ∀ n, c n∈sourceContactCompactDomain n k α H) |
| variable (hmin : ∀ n, IsMinOn (sourceStepContactObjective n k P f |
| (fun i => p i.val) (fun i => d i.val) w q (fun i => (hw i).le) hw1 δ) |
| (sourceContactCompactDomain n k α H) (c n)) |
| variable (hcap : ∀ n, (c n).2.1 (Fin.last k)<H) |
| variable (hH : 0≤H) (hcover : ∀ a b : ℕ, 1≤a+b → ∃ j, p j=a ∧ d j=b) |
| variable {ν : ProbabilityMeasure (CompactArray CompactJointOverlap)} {s : ℕ→ℕ} (hs : StrictMono s) |
| variable (hlim : Tendsto (fun n => sourceGibbsArrayLaw (s n) k f (fun i => p i.val) (fun i => d i.val) |
| (c (s n)).2.1 (c (s n)).2.2 (stepCumulative w) (c (s n)).1) atTop (𝓝 ν)) |
|
|
| include hc hmin hcap hH hcover hs hlim hq in |
| lemma source_global_contact_limit_stoploss (r : ℝ) : |
| (∑ i, w i*max ((q i:ℝ)-r) 0)≤ |
| ∫ x : Time, max (x.val-r) 0 ∂(compactPositivePairLaw ν).map Prod.fst := by |
| have hm n := source_global_contact_perturbation_min n k P f (fun i => p i.val) (fun i => d i.val) |
| w q (fun i => (hw i).le) hw1 δ α H (hc n) (hmin n) |
| obtain ⟨hn,hl,-⟩ := source_contact_limit_quantile k f p d w hw hw1 |
| (fun n => (c n).1) (fun n => (c n).2.1) (fun n => (hc n).2.1.1.1) |
| (fun n => (hc n).2.1.2) (fun n => (c n).2.2) (fun n => (hc n).2.2) hm hH |
| (fun n => (hc n).2.1.1.2 0) (T := (α:ℝ)) (fun n => (hc n).1.2) hcover hs hlim |
| obtain ⟨hr,ht⟩ := sourceGibbsArray_limit_gram k f (fun _ i => p i.val) (fun _ i => d i.val) |
| (fun n => (c n).2.1) (fun n => (c n).2.2) (stepCumulative w) (fun n => (c n).1) s hlim |
| apply source_field_stoploss w q (fun i => (hw i).le) hq (compactPositivePairLaw ν) hl |
| intro a ha0 ha |
| rw [← compactPositivePairLaw_field_integral ν hr ht hn a] |
| exact source_global_contact_limit_field k P f p d w q hw hw1 δ α H c hc hmin hcap hlim ha0 ha |
|
|
| include hc hmin hcap hH hcover hs hlim hq in |
|
|
| theorem source_global_contact_limit_tail_order : |
| quantileTrial (boundedQuantile (weightedTrialLaw w q))= |
| weightedStepTrial w q (fun i => (hw i).le) hw1 ∧ |
| ∀ t : Time, (∫ u : Time in Ici t, (boundedQuantile (weightedTrialLaw w q) u:ℝ))≤ |
| ∫ u : Time in Ici t, (boundedQuantile ((compactPositivePairLaw ν).map Prod.fst) u:ℝ) := by |
| have := weightedTrialLaw_probability w q (fun i => (hw i).le) hw1 |
| have : IsProbabilityMeasure ((compactPositivePairLaw ν).map Prod.fst) := |
| (Measure.isProbabilityMeasure_map_iff measurable_fst.aemeasurable).2 inferInstance |
| refine ⟨quantileTrial_weightedTrialLaw w q (fun i => (hw i).le) hw1,?_⟩ |
| apply upper_tail_order_of_stoploss (boundedQuantile_measurable _) |
| (boundedQuantile_monotone _) |
| intro r |
| rw [boundedQuantile_stoploss,boundedQuantile_stoploss,weightedTrialLaw_integral w q (fun i => (hw i).le)] |
| exact source_global_contact_limit_stoploss k P f p d w q hw hw1 hq δ α H c hc hmin hcap hH hcover hs hlim r |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
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| end OAI |
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