| import OAI.Probability.FreeIsing.Posterior.DerivativeBounds |
| import OAI.Probability.FreeIsing.Posterior.SqrtObservationDerivative |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped BigOperators Topology |
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| namespace Problem367.FinitePosterior |
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| variable {V : Type*} [Fintype V] |
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| def gaussianObservationCurve (signal noise : V → ℝ) (t : ℝ) : V → ℝ := |
| t • signal + Real.sqrt t • noise |
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| def gaussianObservationVelocity (signal noise : V → ℝ) (t : ℝ) : V → ℝ := |
| signal + (1 / (2 * Real.sqrt t)) • noise |
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|
| theorem hasDerivAt_gaussianObservationCurve (signal noise : V → ℝ) |
| {t : ℝ} (ht : 0 < t) : |
| HasDerivAt (gaussianObservationCurve signal noise) |
| (gaussianObservationVelocity signal noise t) t := by |
| have hd := ((hasDerivAt_id t).smul_const signal).add |
| ((Real.hasDerivAt_sqrt ht.ne').smul_const noise) |
| change HasDerivAt (gaussianObservationCurve signal noise) |
| (1 • signal + (1 / (2 * Real.sqrt t)) • noise) t at hd |
| simpa only [gaussianObservationVelocity, one_smul] using hd |
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|
| theorem hasDerivAt_along_gaussianObservationCurve |
| (f : (V → ℝ) → ℝ) (g : (V → ℝ) → V → ℝ) |
| (hdf : ∀ x, HasFDerivAt f (∑ v, g x v • |
| (ContinuousLinearMap.proj v : (V → ℝ) →L[ℝ] ℝ)) x) |
| (signal noise : V → ℝ) {t : ℝ} (ht : 0 < t) : |
| HasDerivAt (fun r => f (gaussianObservationCurve signal noise r)) |
| (∑ v, g (gaussianObservationCurve signal noise t) v * |
| gaussianObservationVelocity signal noise t v) t := by |
| have hd := (hdf (gaussianObservationCurve signal noise t)).comp_hasDerivAt t |
| (hasDerivAt_gaussianObservationCurve signal noise ht) |
| simpa only [Function.comp_def, sum_apply, smul_apply, |
| ContinuousLinearMap.proj_apply, smul_eq_mul] using hd |
|
|
| omit [Fintype V] in |
| theorem gaussianObservationVelocity_bound (signal noise : V → ℝ) |
| {t r : ℝ} (ht : 0 < t) (hr : t / 2 < r) |
| (hsignal : ∀ v, |signal v| ≤ 1) (v : V) : |
| |gaussianObservationVelocity signal noise r v| ≤ |
| 1 + (1 / (2 * Real.sqrt (t / 2))) * |noise v| := by |
| have hhalf : 0 < t / 2 := by positivity |
| have hrpos : 0 < r := hhalf.trans hr |
| have hsqrt : 0 < Real.sqrt (t / 2) := Real.sqrt_pos.mpr hhalf |
| have hcoef : 1 / (2 * Real.sqrt r) ≤ 1 / (2 * Real.sqrt (t / 2)) := by |
| apply one_div_le_one_div_of_le |
| · positivity |
| · exact mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt hr.le) (by norm_num) |
| change |signal v + (1 / (2 * Real.sqrt r)) * noise v| ≤ _ |
| calc |
| _ ≤ |signal v| + |(1 / (2 * Real.sqrt r)) * noise v| := abs_add_le _ _ |
| _ = |signal v| + (1 / (2 * Real.sqrt r)) * |noise v| := by |
| rw [abs_mul, abs_of_nonneg (show 0 ≤ 1 / (2 * Real.sqrt r) by positivity)] |
| _ ≤ _ := add_le_add (hsignal v) |
| (mul_le_mul_of_nonneg_right hcoef (abs_nonneg _)) |
|
|
| section Expectation |
|
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| variable {Ω : Type*} [MeasurableSpace Ω] |
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| theorem hasDerivAt_integral_gaussianObservationCurve |
| (μ : Measure Ω) [IsFiniteMeasure μ] |
| (f : (V → ℝ) → ℝ) (g : (V → ℝ) → V → ℝ) |
| (hdf : ∀ x, HasFDerivAt f (∑ v, g x v • |
| (ContinuousLinearMap.proj v : (V → ℝ) →L[ℝ] ℝ)) x) |
| (hg : ∀ v, Continuous (fun x => g x v)) |
| (M C : ℝ) (hC : 0 ≤ C) |
| (hfbound : ∀ x, |f x| ≤ M) (hgbound : ∀ x v, |g x v| ≤ C) |
| (signal noise : Ω → V → ℝ) |
| (hsignal : Measurable signal) (hnoise : Measurable noise) |
| (hspin : ∀ ω v, |signal ω v| ≤ 1) |
| (hnoiseInt : ∀ v, Integrable (fun ω => noise ω v) μ) |
| {t : ℝ} (ht : 0 < t) : |
| HasDerivAt (fun r => ∫ ω, f (gaussianObservationCurve (signal ω) (noise ω) r) ∂μ) |
| (∫ ω, ∑ v, g (gaussianObservationCurve (signal ω) (noise ω) t) v * |
| gaussianObservationVelocity (signal ω) (noise ω) t v ∂μ) t := by |
| let D : (V → ℝ) → (V → ℝ) →L[ℝ] ℝ := fun x => ∑ v, g x v • |
| (ContinuousLinearMap.proj v : (V → ℝ) →L[ℝ] ℝ) |
| have hDcont : Continuous D := by |
| apply continuous_finsetSum |
| intro v _ |
| exact (hg v).smul continuous_const |
| have hDbound (x : V → ℝ) : ‖D x‖ ≤ (Fintype.card V : ℝ) * C := by |
| apply ContinuousLinearMap.opNorm_le_bound _ (mul_nonneg (by positivity) hC) |
| intro y |
| simp only [D, sum_apply, smul_apply, ContinuousLinearMap.proj_apply, smul_eq_mul] |
| calc |
| ‖∑ v, g x v * y v‖ ≤ ∑ v, ‖g x v * y v‖ := norm_sum_le _ _ |
| _ ≤ ∑ v : V, C * ‖y‖ := by |
| apply Finset.sum_le_sum |
| intro v _ |
| rw [norm_mul, Real.norm_eq_abs (g x v)] |
| exact mul_le_mul (hgbound x v) (norm_le_pi_norm y v) (norm_nonneg _) hC |
| _ = _ := by simp [Finset.sum_const, nsmul_eq_mul, mul_assoc] |
| have hsInt : Integrable signal μ := Integrable.of_eval fun v => |
| Integrable.of_bound ((measurable_pi_apply v).comp hsignal).aestronglyMeasurable 1 |
| (Filter.Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hspin ω v) |
| have hnInt : Integrable noise μ := |
| ⟨hnoise.aestronglyMeasurable, (Integrable.of_eval hnoiseInt).hasFiniteIntegral⟩ |
| have hd := (hasDerivAt_integral_sqrt_observation_vector hsInt hnInt hdf hDcont |
| hfbound (mul_nonneg (by positivity) hC) hDbound ht).2 |
| simpa only [gaussianObservationCurve, gaussianObservationVelocity, D, sum_apply, |
| smul_apply, ContinuousLinearMap.proj_apply, smul_eq_mul] using hd |
|
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| end Expectation |
|
|
| section Posterior |
|
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| variable {S : Type*} [Fintype S] [Nonempty S] |
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| theorem continuous_fieldCovariance (energy : S → ℝ) (spin : S → V → ℝ) |
| (observable : S → ℝ) (v : V) : |
| Continuous (fun x => fieldCovariance energy spin observable x v) := by |
| exact (continuous_fieldMean energy spin (fun s => observable s * spin s v)).sub |
| ((continuous_fieldMean energy spin observable).mul |
| (continuous_fieldMean energy spin (fun s => spin s v))) |
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| theorem hasFDerivAt_fieldMean_square (energy : S → ℝ) (spin : S → V → ℝ) |
| (observable : S → ℝ) (x : V → ℝ) : |
| HasFDerivAt (fun y => fieldMean energy spin observable y ^ 2) |
| (∑ v, (2 * fieldMean energy spin observable x * fieldCovariance energy spin observable x v) • |
| (ContinuousLinearMap.proj v : (V → ℝ) →L[ℝ] ℝ)) x := by |
| have hs : HasDerivAt (fun z : ℝ => z ^ 2) |
| (2 * fieldMean energy spin observable x) (fieldMean energy spin observable x) := by |
| have hd := (hasDerivAt_id (fieldMean energy spin observable x)).pow 2 |
| change HasDerivAt (fun z : ℝ => z ^ 2) _ (fieldMean energy spin observable x) at hd |
| simpa only [Nat.cast_ofNat, Nat.reduceSub, pow_one, mul_one, id_eq] using hd |
| have hd := hs.comp_hasFDerivAt x (hasFDerivAt_fieldMean energy spin observable x) |
| simpa only [Function.comp_def, fieldMeanDerivative, Finset.smul_sum, smul_smul] using hd |
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| variable {Ω : Type*} [MeasurableSpace Ω] |
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| theorem hasDerivAt_integral_fieldMean_observation |
| (μ : Measure Ω) [IsFiniteMeasure μ] |
| (energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ) |
| (A : ℝ) (hbound : ∀ s, |observable s| ≤ A) |
| (hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) |
| (signal noise : Ω → V → ℝ) |
| (hsignal : Measurable signal) (hnoise : Measurable noise) |
| (hspin : ∀ ω v, |signal ω v| ≤ 1) |
| (hnoiseInt : ∀ v, Integrable (fun ω => noise ω v) μ) |
| {t : ℝ} (ht : 0 < t) : |
| HasDerivAt (fun r => ∫ ω, fieldMean energy spin observable |
| (gaussianObservationCurve (signal ω) (noise ω) r) ∂μ) |
| (∫ ω, ∑ v, fieldCovariance energy spin observable |
| (gaussianObservationCurve (signal ω) (noise ω) t) v * |
| gaussianObservationVelocity (signal ω) (noise ω) t v ∂μ) t := by |
| have hA : 0 ≤ A := (abs_nonneg (observable (Classical.arbitrary S))).trans |
| (hbound (Classical.arbitrary S)) |
| exact hasDerivAt_integral_gaussianObservationCurve μ |
| (fieldMean energy spin observable) (fieldCovariance energy spin observable) |
| (hasFDerivAt_fieldMean energy spin observable) |
| (continuous_fieldCovariance energy spin observable) A (2 * A) (by positivity) |
| (fun x => abs_fieldMean_le energy spin observable x A hbound) |
| (fun x v => abs_fieldCovariance_le energy spin observable x v A (fun s => hcoords s v) hbound) |
| signal noise hsignal hnoise hspin hnoiseInt ht |
|
|
| theorem hasDerivAt_integral_fieldMean_square_observation |
| (μ : Measure Ω) [IsFiniteMeasure μ] |
| (energy : S → ℝ) (spin : S → V → ℝ) (observable : S → ℝ) |
| (A : ℝ) (hbound : ∀ s, |observable s| ≤ A) |
| (hcoords : ∀ s v, spin s v = -1 ∨ spin s v = 1) |
| (signal noise : Ω → V → ℝ) |
| (hsignal : Measurable signal) (hnoise : Measurable noise) |
| (hspin : ∀ ω v, |signal ω v| ≤ 1) |
| (hnoiseInt : ∀ v, Integrable (fun ω => noise ω v) μ) |
| {t : ℝ} (ht : 0 < t) : |
| HasDerivAt (fun r => ∫ ω, fieldMean energy spin observable |
| (gaussianObservationCurve (signal ω) (noise ω) r) ^ 2 ∂μ) |
| (∫ ω, ∑ v, (2 * fieldMean energy spin observable |
| (gaussianObservationCurve (signal ω) (noise ω) t) * |
| fieldCovariance energy spin observable (gaussianObservationCurve (signal ω) (noise ω) t) v) * |
| gaussianObservationVelocity (signal ω) (noise ω) t v ∂μ) t := by |
| have hA : 0 ≤ A := (abs_nonneg (observable (Classical.arbitrary S))).trans |
| (hbound (Classical.arbitrary S)) |
| apply hasDerivAt_integral_gaussianObservationCurve μ |
| (fun x => fieldMean energy spin observable x ^ 2) |
| (fun x v => 2 * fieldMean energy spin observable x * fieldCovariance energy spin observable x v) |
| (hasFDerivAt_fieldMean_square energy spin observable) |
| (fun v => ((continuous_fieldMean energy spin observable).const_mul 2).mul |
| (continuous_fieldCovariance energy spin observable v)) |
| (A ^ 2) (4 * A ^ 2) (by positivity) |
| (fun x => abs_fieldMean_square_le energy spin observable x A hbound) ?_ |
| signal noise hsignal hnoise hspin hnoiseInt ht |
| intro x v |
| rw [abs_mul, abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)] |
| have hm := abs_fieldMean_le energy spin observable x A hbound |
| have hc := abs_fieldCovariance_le energy spin observable x v A (fun s => hcoords s v) hbound |
| calc |
| 2 * |fieldMean energy spin observable x| * |fieldCovariance energy spin observable x v| ≤ |
| 2 * A * (2 * A) := by gcongr |
| _ = 4 * A ^ 2 := by ring |
|
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| end Posterior |
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| end Problem367.FinitePosterior |
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| end |
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| end OAI |
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