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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.FinitePosterior
variable {S V : Type*} [Fintype S] [Nonempty S] [Fintype V] [DecidableEq V]
theorem integral_fieldEntropyGradient_mul_noise
(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
(hspin : ∀ s, observable s = -1 ∨ observable s = 1)
(hneg : ∃ s, observable s = -1) (hpos : ∃ s, observable s = 1)
(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1)
(signal : V → ℝ) (t : ℝ) (v : V) :
(∫ z, fieldEntropyGradient energy spin observable
(gaussianObservationCurve signal z t) v * z v
∂Measure.pi (fun _ : V => gaussianReal 0 1)) =
Real.sqrt t * ∫ z, fieldEntropyHessianDiagonal energy spin observable
(gaussianObservationCurve signal z t) v
∂Measure.pi (fun _ : V => gaussianReal 0 1) := by
have h := Problem367.gaussian_pi_integral_mul_affine_of_bounded_deriv
(fun j => t * signal j) (fun _ => Real.sqrt t) v
(continuous_fieldEntropyGradient energy spin observable v hspin hneg hpos)
(continuous_fieldEntropyHessianDiagonal energy spin observable v hspin hneg hpos)
(fun z r => hasDerivAt_fieldEntropyGradient_update energy spin observable z v r
(fun s => hcoords s v) hspin hneg hpos)
(fun z => abs_fieldEntropyGradient_le energy spin observable z v hspin hneg hpos
(fun s => (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) s).le))
(fun z => abs_fieldEntropyHessianDiagonal_le energy spin observable z v hspin hneg hpos
(fun s => (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) s).le))
simpa only [gaussianObservationCurve, Pi.add_def, Pi.smul_def, smul_eq_mul,
mul_comm] using h
section Planted
variable [MeasurableSpace S] [MeasurableSingletonClass S]
omit [DecidableEq V] in
private theorem integrable_bounded_entropy_test
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
(X : Ω → V → ℝ) (hX : Measurable X) (f : (V → ℝ) → ℝ)
(hf : Continuous f) (C : ℝ) (hbound : ∀ z, |f z| ≤ C) :
Integrable (fun x => f (X x)) μ := by
apply Integrable.of_bound (hf.measurable.comp hX).aestronglyMeasurable C
exact Filter.Eventually.of_forall fun x => by
simpa only [Real.norm_eq_abs, Function.comp_def] using hbound (X x)
omit [DecidableEq V] in
theorem integrable_finiteStandardEntropyGradient (μ : Measure S) [IsFiniteMeasure μ]
(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
(hspin : ∀ s, observable s = -1 ∨ observable s = 1)
(hneg : ∃ s, observable s = -1) (hpos : ∃ s, observable s = 1)
(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) (v : V) :
Integrable (fun x => fieldEntropyGradient energy spin observable
(finiteStandardObservation spin t x) v)
(μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) :=
integrable_bounded_entropy_test _ _ (measurable_finiteStandardObservation spin t) _
(continuous_fieldEntropyGradient energy spin observable v hspin hneg hpos) 2
(fun z => abs_fieldEntropyGradient_le energy spin observable z v hspin hneg hpos
(fun s => (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) s).le))
omit [DecidableEq V] in
theorem integrable_finiteStandardEntropyHessian (μ : Measure S) [IsFiniteMeasure μ]
(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
(hspin : ∀ s, observable s = -1 ∨ observable s = 1)
(hneg : ∃ s, observable s = -1) (hpos : ∃ s, observable s = 1)
(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) (v : V) :
Integrable (fun x => fieldEntropyHessianDiagonal energy spin observable
(finiteStandardObservation spin t x) v)
(μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) :=
integrable_bounded_entropy_test _ _ (measurable_finiteStandardObservation spin t) _
(continuous_fieldEntropyHessianDiagonal energy spin observable v hspin hneg hpos) 8
(fun z => abs_fieldEntropyHessianDiagonal_le energy spin observable z v hspin hneg hpos
(fun s => (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) s).le))
omit [DecidableEq V] in
theorem integrable_finiteStandardEntropyGradient_mul_noise
(μ : Measure S) [IsFiniteMeasure μ]
(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
(hspin : ∀ s, observable s = -1 ∨ observable s = 1)
(hneg : ∃ s, observable s = -1) (hpos : ∃ s, observable s = 1)
(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) (v : V) :
Integrable (fun x => fieldEntropyGradient energy spin observable
(finiteStandardObservation spin t x) v * x.2 v)
(μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) := by
have hg := integrable_finiteStandardEntropyGradient μ energy spin observable hspin hneg hpos hcoords t v
have h := (integrable_finiteStandardNoise μ v).mul_bdd hg.aestronglyMeasurable
(Filter.Eventually.of_forall fun x => by
simpa only [Real.norm_eq_abs, Function.comp_def] using abs_fieldEntropyGradient_le energy spin observable
(finiteStandardObservation spin t x) v hspin hneg hpos
(fun s => (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) s).le))
simpa only [mul_comm] using h
theorem integral_finiteStandardEntropyGradient_mul_noise
(μ : Measure S) [IsFiniteMeasure μ]
(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
(hspin : ∀ s, observable s = -1 ∨ observable s = 1)
(hneg : ∃ s, observable s = -1) (hpos : ∃ s, observable s = 1)
(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) (v : V) :
(∫ x, fieldEntropyGradient energy spin observable
(finiteStandardObservation spin t x) v * x.2 v
∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) =
Real.sqrt t * ∫ x, fieldEntropyHessianDiagonal energy spin observable
(finiteStandardObservation spin t x) v
∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1)) := by
rw [integral_prod _ (integrable_finiteStandardEntropyGradient_mul_noise
μ energy spin observable hspin hneg hpos hcoords t v),
integral_prod _ (integrable_finiteStandardEntropyHessian
μ energy spin observable hspin hneg hpos hcoords t v), ← integral_const_mul]
apply integral_congr_ae
exact Filter.Eventually.of_forall fun s =>
integral_fieldEntropyGradient_mul_noise energy spin observable hspin hneg hpos hcoords (spin s) t v
theorem integral_finiteStandardEntropyVelocity [StandardBorelSpace S]
(μ : Measure S) [IsProbabilityMeasure μ] (energy : S → ℝ)
(hprior : ∀ s, μ.real {s} = Real.exp (energy s) / ∑ a, Real.exp (energy a))
(spin : S → V → ℝ) (observable : S → ℝ)
(hspin : ∀ s, observable s = -1 ∨ observable s = 1)
(hneg : ∃ s, observable s = -1) (hpos : ∃ s, observable s = 1)
(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1)
{t : ℝ} (ht : 0 < t) :
(∫ x, ∑ v, fieldEntropyGradient energy spin observable
(finiteStandardObservation spin t x) v *
gaussianObservationVelocity (spin x.1) x.2 t v
∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) =
∫ x, (1 + fieldMean energy spin observable (finiteStandardObservation spin t x) ^ 2) *
∑ v, normalizedFieldCovariance energy spin observable
(finiteStandardObservation spin t x) v ^ 2
∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1)) := by
have hposterior (v : V) :
(∫ x, fieldEntropyGradient energy spin observable
(finiteStandardObservation spin t x) v * spin x.1 v
∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) =
∫ x, fieldEntropyGradient energy spin observable
(finiteStandardObservation spin t x) v *
fieldMean energy spin (fun s => spin s v) (finiteStandardObservation spin t x)
∂μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1)) := by
have hsq : ∀ s i, spin s i ^ 2 = 1 := by
intro s i
rcases hcoords s i with h | h <;> simp [h]
have htest (z : V → ℝ) : ‖fieldEntropyGradient energy spin observable z v‖ ≤ 2 := by
rw [Real.norm_eq_abs]
exact abs_fieldEntropyGradient_le energy spin observable z v hspin hneg hpos
(fun s => (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) s).le)
exact Problem367.finiteStandardGaussianPlanted_integral_test_fieldMean μ energy hprior t ht.le
spin hsq (fun s => spin s v) (fun z => fieldEntropyGradient energy spin observable z v)
(continuous_fieldEntropyGradient energy spin observable v hspin hneg hpos).measurable 2 htest
let ρ := μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))
let Y := finiteStandardObservation spin t
let g := fun x v => fieldEntropyGradient energy spin observable (Y x) v
let H := fun x v => fieldEntropyHessianDiagonal energy spin observable (Y x) v
let p := fun x v => fieldMean energy spin (fun s => spin s v) (Y x)
let c := 1 / (2 * Real.sqrt t)
have hy : Measurable Y := measurable_finiteStandardObservation spin t
have hg (v : V) : Integrable (fun x => g x v) ρ :=
integrable_finiteStandardEntropyGradient μ energy spin observable hspin hneg hpos hcoords t v
have hH (v : V) : Integrable (fun x => H x v) ρ :=
integrable_finiteStandardEntropyHessian μ energy spin observable hspin hneg hpos hcoords t v
have hn (v : V) : Integrable (fun x => g x v * x.2 v) ρ :=
integrable_finiteStandardEntropyGradient_mul_noise μ energy spin observable hspin hneg hpos hcoords t v
have hs (v : V) : Integrable (fun x => g x v * spin x.1 v) ρ := by
apply (hg v).mul_bdd
(((measurable_of_finite (fun s => spin s v)).comp measurable_fst).aestronglyMeasurable)
exact Filter.Eventually.of_forall fun x => by
simpa only [Real.norm_eq_abs, Function.comp_def] using
(abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) x.1).le
have hp (v : V) : Integrable (fun x => g x v * p x v) ρ := by
apply (hg v).mul_bdd
(((continuous_fieldMean energy spin (fun s => spin s v)).measurable.comp hy).aestronglyMeasurable)
exact Filter.Eventually.of_forall fun x => by
simpa only [Real.norm_eq_abs, Function.comp_def] using abs_fieldMean_le energy spin (fun s => spin s v)
(Y x) 1 (fun s => (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) s).le)
have hvEq (v : V) :
(fun x : S × (V → ℝ) => g x v * gaussianObservationVelocity (spin x.1) x.2 t v) =
(fun x => g x v * spin x.1 v + c * (g x v * x.2 v)) := by
funext x
change g x v * (spin x.1 v + c * x.2 v) = _
ring
have hv (v : V) : Integrable
(fun x => g x v * gaussianObservationVelocity (spin x.1) x.2 t v) ρ := by
rw [hvEq]
exact (hs v).add ((hn v).const_mul c)
have hcancel (x : S × (V → ℝ)) (v : V) :
g x v * p x v + (1 / 2 : ℝ) * H x v =
(1 + fieldMean energy spin observable (Y x) ^ 2) *
normalizedFieldCovariance energy spin observable (Y x) v ^ 2 := by
dsimp [g, H, p, fieldEntropyGradient, fieldEntropyHessianDiagonal]
ring
have hc (v : V) : Integrable
(fun x => (1 + fieldMean energy spin observable (Y x) ^ 2) *
normalizedFieldCovariance energy spin observable (Y x) v ^ 2) ρ := by
exact ((hp v).add ((hH v).const_mul (1 / 2 : ℝ))).congr
(Filter.Eventually.of_forall fun x => hcancel x v)
have hstep (v : V) :
(∫ x, g x v * gaussianObservationVelocity (spin x.1) x.2 t v ∂ρ) =
∫ x, (1 + fieldMean energy spin observable (Y x) ^ 2) *
normalizedFieldCovariance energy spin observable (Y x) v ^ 2 ∂ρ := by
rw [hvEq, integral_add (hs v) ((hn v).const_mul c), integral_const_mul]
have hnoise : (∫ x, g x v * x.2 v ∂ρ) = Real.sqrt t * ∫ x, H x v ∂ρ :=
integral_finiteStandardEntropyGradient_mul_noise μ energy spin observable hspin hneg hpos hcoords t v
rw [hnoise]
have hcoef : c * Real.sqrt t = (1 / 2 : ℝ) := by
dsimp [c]
field_simp [ne_of_gt (Real.sqrt_pos.mpr ht)]
rw [← mul_assoc, hcoef, hposterior v]
rw [← integral_const_mul, ← integral_add (hp v) ((hH v).const_mul (1 / 2 : ℝ))]
exact integral_congr_ae (Filter.Eventually.of_forall fun x => hcancel x v)
change (∫ x, ∑ v, g x v * gaussianObservationVelocity (spin x.1) x.2 t v ∂ρ) = _
simp_rw [Finset.mul_sum]
rw [integral_finsetSum _ (fun v _ => hv v), integral_finsetSum _ (fun v _ => hc v)]
exact Finset.sum_congr rfl fun v _ => hstep v
omit [Nonempty S] [DecidableEq V] in
theorem integrable_finiteStandardSignal (μ : Measure S) [IsFiniteMeasure μ]
(spin : S → V → ℝ) (hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) :
Integrable (fun x : S × (V → ℝ) => spin x.1)
(μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) := by
apply Integrable.of_eval
intro v
apply Integrable.of_bound
(((measurable_of_finite (fun s => spin s v)).comp measurable_fst).aestronglyMeasurable) 1
exact Filter.Eventually.of_forall fun x => by
simpa only [Real.norm_eq_abs, Function.comp_def] using
(abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) x.1).le
omit [Nonempty S] [DecidableEq V] in
theorem integrable_finiteStandardObservation (μ : Measure S) [IsFiniteMeasure μ]
(spin : S → V → ℝ) (hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) :
Integrable (finiteStandardObservation spin t)
(μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) := by
have hn : Integrable (Prod.snd : S × (V → ℝ) → (V → ℝ))
(μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) :=
Integrable.of_eval (fun v => integrable_finiteStandardNoise μ v)
exact ((integrable_finiteStandardSignal μ spin hcoords).smul t).add
(hn.smul (Real.sqrt t))
omit [DecidableEq V] in
theorem integrable_finiteStandardEntropy (μ : Measure S) [IsFiniteMeasure μ]
(energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ)
(hspin : ∀ s, observable s = -1 ∨ observable s = 1)
(hneg : ∃ s, observable s = -1) (hpos : ∃ s, observable s = 1)
(hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) (t : ℝ) :
Integrable (fun x => fieldEntropy energy spin observable (finiteStandardObservation spin t x))
(μ.prod (Measure.pi (fun _ : V => gaussianReal 0 1))) :=
integrable_fieldEntropy_comp _ energy spin observable hspin hneg hpos
(fun s v => (abs_spin_eq_one (fun s => spin s v) (fun s => hcoords s v) s).le)
(integrable_finiteStandardObservation μ spin hcoords t)
end Planted
omit [DecidableEq V] in
theorem continuous_normalizedRowEnergy (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ)
(hspin : ∀ s, observable s = -1 ∨ observable s = 1)
(hneg : ∃ s, observable s = -1) (hpos : ∃ s, observable s = 1) :
Continuous (fun x => ∑ v, normalizedFieldCovariance energy spin observable x v ^ 2) := by
apply continuous_finsetSum
intro v _
exact (continuous_normalizedFieldCovariance_entropy energy spin observable v hspin hneg hpos).pow 2
omit [DecidableEq V] in
theorem abs_normalizedRowEnergy_le_card (energy : S → ℝ) (spin : S → V → ℝ)
(observable : S → ℝ) (x : V → ℝ)
(hspin : ∀ s, observable s = -1 ∨ observable s = 1)
(hneg : ∃ s, observable s = -1) (hpos : ∃ s, observable s = 1)
(hsignal : ∀ s v, |spin s v| ≤ 1) :
|∑ v, normalizedFieldCovariance energy spin observable x v ^ 2| ≤ Fintype.card V := by
rw [abs_of_nonneg (Finset.sum_nonneg (fun v _ => sq_nonneg _))]
calc
(∑ v, normalizedFieldCovariance energy spin observable x v ^ 2) ≤ ∑ _v : V, (1 : ℝ) := by
apply Finset.sum_le_sum
intro v _
exact (sq_le_one_iff_abs_le_one _).mpr
(abs_normalizedFieldCovariance_le_one_entropy energy spin observable x v
hspin hneg hpos (fun s => hsignal s v))
_ = _ := by simp
end Problem367.FinitePosterior
end
end OAI
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