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b317718 | 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 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 | import OAI.Probability.FreeIsing.Posterior.DerivativeBounds
import OAI.Probability.FreeIsing.Posterior.SqrtObservationDerivative
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped BigOperators Topology
namespace Problem367.FinitePosterior
variable {V : Type*} [Fintype V]
def gaussianObservationCurve (signal noise : V → ℝ) (t : ℝ) : V → ℝ :=
t • signal + Real.sqrt t • noise
def gaussianObservationVelocity (signal noise : V → ℝ) (t : ℝ) : V → ℝ :=
signal + (1 / (2 * Real.sqrt t)) • noise
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
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
variable {Ω : Type*} [MeasurableSpace Ω]
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
end Expectation
section Posterior
variable {S : Type*} [Fintype S] [Nonempty S]
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)))
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
variable {Ω : Type*} [MeasurableSpace Ω]
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
end Posterior
end Problem367.FinitePosterior
end
end OAI
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