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a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 | import Mathlib
import OAI.Probability.Perceptron.Interpolation.ReplicaMeanConst
import OAI.Probability.Perceptron.Cascade.ReweightedMomentComparison
import OAI.Probability.Perceptron.Cavity.CavityBoundedContinuous
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped ENNReal NNReal Topology BigOperators BoundedContinuousFunction
namespace SphericalPerceptronFreeEnergy
lemma expectedPressure_terminal_scaling (α β : ℝ) (φ : ℝ→ᵇℝ) (n : ℕ) :
expectedPressure α β φ (n+1)=expectedPressure α 1 (β • φ) (n+1) := by
unfold expectedPressure
apply integral_congr_ae
filter_upwards [] with g
rw [←normalizedPatternLog_source α β φ n g,
←normalizedPatternLog_source α 1 (β • φ) n g,one_smul]
theorem main (P : Measure BrownianPath) [IsProbabilityMeasure P]
(hB : IsBrownianReal brownianEval P)
(α β : ℝ) (hα : 0 < α) (_hβ : 0 < β) (φ : ℝ →ᵇ ℝ) :
∃ p : ℝ, variationalValue P α β φ = (p : EReal) ∧
Tendsto (fun N : ℕ => expectedPressure α β φ (N + 1)) atTop (𝓝 p) ∧
∀ ε : ℝ, 0 < ε →
Tendsto (fun N : ℕ => patternLaw α (N + 1)
{g | ε < |pressure α β φ (N + 1) g - p|}) atTop (𝓝 0) := by
let p:=(terminalVariational P α (β • φ)).toReal
have hp : Tendsto (fun N : ℕ=>expectedPressure α β φ (N+1)) atTop (𝓝 p) := by
have heq : (fun N : ℕ=>expectedPressure α β φ (N+1)) =
(fun N : ℕ=>expectedPressure α 1 (β • φ) (N+1)) :=
funext (expectedPressure_terminal_scaling α β φ)
rw [heq]
exact expectedPressure_limit_terminal P hB hα (β • φ)
refine ⟨p,?_,hp,?_⟩
· rw [variationalValue_eq_terminal]
exact (terminalVariational_coe_toReal P α hα.le (β • φ)).symm
· exact pressure_probability_of_expected_limit α β hα.le φ p hp
end SphericalPerceptronFreeEnergy
end
end OAI
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