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namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped Topology BigOperators NNReal
namespace Problem367
theorem hasDerivAt_integral_sqrt_observation_of_integrable
{Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P]
{S G : Ω → ℝ} (hS : Integrable S P) (hG : Integrable G P)
{f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x) (hf'm : Measurable f')
{M : ℝ} (hM : 0 ≤ M)
(hf'bound : ∀ x, |f' x| ≤ M) {t : ℝ} (ht : 0 < t)
(hFint : Integrable (fun ω => f (t * S ω + Real.sqrt t * G ω)) P) :
Integrable (fun ω => f' (t * S ω + Real.sqrt t * G ω) *
(S ω + G ω / (2 * Real.sqrt t))) P ∧
HasDerivAt (fun r : ℝ => ∫ ω, f (r * S ω + Real.sqrt r * G ω) ∂P)
(∫ ω, f' (t * S ω + Real.sqrt t * G ω) *
(S ω + G ω / (2 * Real.sqrt t)) ∂P) t := by
have hfcont : Continuous f := continuous_iff_continuousAt.mpr
(fun x => (hf x).continuousAt)
let F : ℝ → Ω → ℝ := fun r ω => f (r * S ω + Real.sqrt r * G ω)
let F' : ℝ → Ω → ℝ := fun r ω => f' (r * S ω + Real.sqrt r * G ω) *
(S ω + G ω / (2 * Real.sqrt r))
let a : ℝ := 1 / (2 * Real.sqrt (t / 2))
have ha : 0 ≤ a := by dsimp [a]; positivity
have hFm (r : ℝ) : AEStronglyMeasurable (F r) P :=
hfcont.comp_aestronglyMeasurable
((hS.aestronglyMeasurable.const_mul r).add
(hG.aestronglyMeasurable.const_mul (Real.sqrt r)))
have hF'm (r : ℝ) : AEStronglyMeasurable (F' r) P := by
apply AEStronglyMeasurable.mul
· exact (hf'm.comp_aemeasurable
((hS.aemeasurable.const_mul r).add
(hG.aemeasurable.const_mul (Real.sqrt r)))).aestronglyMeasurable
· simpa only [div_eq_mul_inv, Pi.add_def] using hS.aestronglyMeasurable.add
(hG.aestronglyMeasurable.mul_const (2 * Real.sqrt r)⁻¹)
have hdom : Integrable (fun ω => M * (|S ω| + a * |G ω|)) P :=
(hS.abs.add (hG.abs.const_mul a)).const_mul M
have hbound : ∀ᵐ ω ∂P, ∀ r ∈ Ioi (t / 2),
‖F' r ω‖ ≤ M * (|S ω| + a * |G ω|) := by
filter_upwards [] with ω
intro r hr
have hrpos : 0 < r := lt_trans (by linarith) hr
have hden : 0 < 2 * Real.sqrt (t / 2) := by positivity
have hrecip : 1 / (2 * Real.sqrt r) ≤ a := by
apply one_div_le_one_div_of_le hden
exact mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt hr.le) (by norm_num)
have hinner : |S ω + G ω / (2 * Real.sqrt r)| ≤ |S ω| + a * |G ω| := by
calc
_ ≤ |S ω| + |G ω / (2 * Real.sqrt r)| := abs_add_le _ _
_ = |S ω| + (1 / (2 * Real.sqrt r)) * |G ω| := by
rw [abs_div, abs_of_nonneg (by positivity : 0 ≤ 2 * Real.sqrt r)]
ring
_ ≤ |S ω| + a * |G ω| := by gcongr
dsimp [F']
rw [abs_mul]
exact mul_le_mul (hf'bound _) hinner (abs_nonneg _) hM
have hdiff : ∀ᵐ ω ∂P, ∀ r ∈ Ioi (t / 2), HasDerivAt (fun q => F q ω) (F' r ω) r := by
filter_upwards [] with ω
intro r hr
have hrpos : 0 < r := lt_trans (by linarith) hr
have harg := ((hasDerivAt_id r).mul_const (S ω)).add
((Real.hasDerivAt_sqrt hrpos.ne').mul_const (G ω))
have harg' : HasDerivAt (fun q : ℝ => q * S ω + Real.sqrt q * G ω)
(S ω + G ω / (2 * Real.sqrt r)) r := by
simpa only [Pi.add_def, id_eq, one_mul, mul_one, one_div, div_eq_mul_inv, mul_comm] using harg
exact (hf (r * S ω + Real.sqrt r * G ω)).comp r harg'
exact hasDerivAt_integral_of_dominated_loc_of_deriv_le
(Ioi_mem_nhds (by linarith : t / 2 < t))
(Eventually.of_forall hFm) hFint (hF'm t) hbound hdom hdiff
theorem hasDerivAt_integral_sqrt_observation_vector_of_integrable
{Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
{P : Measure Ω} [IsFiniteMeasure P]
{S G : Ω → E} (hS : Integrable S P) (hG : Integrable G P)
{f : E → ℝ} {f' : E → E →L[ℝ] ℝ}
(hf : ∀ x, HasFDerivAt f (f' x) x) (hf'cont : Continuous f')
{M : ℝ} (hM : 0 ≤ M)
(hf'bound : ∀ x, ‖f' x‖ ≤ M) {t : ℝ} (ht : 0 < t)
(hFint : Integrable (fun ω => f (t • S ω + Real.sqrt t • G ω)) P) :
Integrable (fun ω => f' (t • S ω + Real.sqrt t • G ω)
(S ω + (1 / (2 * Real.sqrt t)) • G ω)) P ∧
HasDerivAt (fun r : ℝ => ∫ ω, f (r • S ω + Real.sqrt r • G ω) ∂P)
(∫ ω, f' (t • S ω + Real.sqrt t • G ω)
(S ω + (1 / (2 * Real.sqrt t)) • G ω) ∂P) t := by
have hfcont : Continuous f := continuous_iff_continuousAt.mpr
(fun x => (hf x).continuousAt)
let F : ℝ → Ω → ℝ := fun r ω => f (r • S ω + Real.sqrt r • G ω)
let F' : ℝ → Ω → ℝ := fun r ω => f' (r • S ω + Real.sqrt r • G ω)
(S ω + (1 / (2 * Real.sqrt r)) • G ω)
let a : ℝ := 1 / (2 * Real.sqrt (t / 2))
have ha : 0 ≤ a := by dsimp [a]; positivity
have hargm (r : ℝ) : AEStronglyMeasurable
(fun ω => r • S ω + Real.sqrt r • G ω) P :=
(hS.aestronglyMeasurable.const_smul r).add
(hG.aestronglyMeasurable.const_smul (Real.sqrt r))
have hFm (r : ℝ) : AEStronglyMeasurable (F r) P :=
hfcont.comp_aestronglyMeasurable (hargm r)
have hF'm (r : ℝ) : AEStronglyMeasurable (F' r) P :=
(isBoundedBilinearMap_apply (𝕜 := ℝ) (E := E) (F := ℝ)).continuous
|>.comp_aestronglyMeasurable
((hf'cont.comp_aestronglyMeasurable (hargm r)).prodMk
(hS.aestronglyMeasurable.add
(hG.aestronglyMeasurable.const_smul (1 / (2 * Real.sqrt r)))))
have hdom : Integrable (fun ω => M * (‖S ω‖ + a * ‖G ω‖)) P :=
(hS.norm.add (hG.norm.const_mul a)).const_mul M
have hbound : ∀ᵐ ω ∂P, ∀ r ∈ Ioi (t / 2),
‖F' r ω‖ ≤ M * (‖S ω‖ + a * ‖G ω‖) := by
filter_upwards [] with ω
intro r hr
have hrpos : 0 < r := lt_trans (by linarith) hr
have hden : 0 < 2 * Real.sqrt (t / 2) := by positivity
have hrecip : 1 / (2 * Real.sqrt r) ≤ a := by
apply one_div_le_one_div_of_le hden
exact mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt hr.le) (by norm_num)
have hinner : ‖S ω + (1 / (2 * Real.sqrt r)) • G ω‖ ≤
‖S ω‖ + a * ‖G ω‖ := by
calc
_ ≤ ‖S ω‖ + ‖(1 / (2 * Real.sqrt r)) • G ω‖ := norm_add_le _ _
_ = ‖S ω‖ + (1 / (2 * Real.sqrt r)) * ‖G ω‖ := by
rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (by positivity)]
_ ≤ ‖S ω‖ + a * ‖G ω‖ := by gcongr
exact ((f' (r • S ω + Real.sqrt r • G ω)).le_opNorm _).trans
(mul_le_mul (hf'bound _) hinner (norm_nonneg _) hM)
have hdiff : ∀ᵐ ω ∂P, ∀ r ∈ Ioi (t / 2),
HasDerivAt (fun q => F q ω) (F' r ω) r := by
filter_upwards [] with ω
intro r hr
have hrpos : 0 < r := lt_trans (by linarith) hr
have harg := ((hasDerivAt_id r).smul_const (S ω)).add
((Real.hasDerivAt_sqrt hrpos.ne').smul_const (G ω))
have harg' : HasDerivAt (fun q : ℝ => q • S ω + Real.sqrt q • G ω)
(S ω + (1 / (2 * Real.sqrt r)) • G ω) r := by
simpa only [Pi.add_def, id_eq, one_smul, one_div] using harg
exact (hf (r • S ω + Real.sqrt r • G ω)).comp_hasDerivAt r harg'
exact hasDerivAt_integral_of_dominated_loc_of_deriv_le
(Ioi_mem_nhds (by linarith : t / 2 < t))
(Eventually.of_forall hFm) hFint (hF'm t) hbound hdom hdiff
theorem continuous_integral_sqrt_observation_lipschitz
{Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
{P : Measure Ω} [IsFiniteMeasure P] {S G : Ω → E}
(hS : Integrable S P) (hG : Integrable G P)
{f : E → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) :
Continuous (fun t : ℝ => ∫ ω, f (t • S ω + Real.sqrt t • G ω) ∂P) := by
apply continuous_iff_continuousAt.mpr
intro t
let a : ℝ := |t| + 1
have ha : 0 ≤ a := by dsimp [a]; positivity
have hloc : ∀ᶠ r in 𝓝 t, |r| < a :=
continuous_abs.continuousAt.eventually
(gt_mem_nhds (by dsimp [a]; linarith : |t| < a))
apply continuousAt_of_dominated
(bound := fun ω => ‖f 0‖ + (K : ℝ) *
(a * ‖S ω‖ + Real.sqrt a * ‖G ω‖))
· exact Eventually.of_forall fun r => hf.continuous.comp_aestronglyMeasurable
((hS.aestronglyMeasurable.const_smul r).add
(hG.aestronglyMeasurable.const_smul (Real.sqrt r)))
· filter_upwards [hloc] with r hr
exact Eventually.of_forall fun ω => by
have harg : ‖r • S ω + Real.sqrt r • G ω‖ ≤
a * ‖S ω‖ + Real.sqrt a * ‖G ω‖ := by
calc
_ ≤ ‖r • S ω‖ + ‖Real.sqrt r • G ω‖ := norm_add_le _ _
_ = |r| * ‖S ω‖ + Real.sqrt r * ‖G ω‖ := by
rw [norm_smul, norm_smul, Real.norm_eq_abs, Real.norm_eq_abs,
abs_of_nonneg (Real.sqrt_nonneg r)]
_ ≤ _ := add_le_add
(mul_le_mul_of_nonneg_right hr.le (norm_nonneg _))
(mul_le_mul_of_nonneg_right
(Real.sqrt_le_sqrt ((le_abs_self r).trans hr.le)) (norm_nonneg _))
calc
_ ≤ ‖f 0‖ + ‖f (r • S ω + Real.sqrt r • G ω) - f 0‖ :=
norm_le_norm_add_norm_sub' _ _
_ ≤ ‖f 0‖ + (K : ℝ) * ‖r • S ω + Real.sqrt r • G ω‖ := by
gcongr
simpa only [sub_zero] using hf.norm_sub_le
(r • S ω + Real.sqrt r • G ω) 0
_ ≤ _ := add_le_add le_rfl (mul_le_mul_of_nonneg_left harg K.coe_nonneg)
· exact (integrable_const ‖f 0‖).add
(((hS.norm.const_mul a).add (hG.norm.const_mul (Real.sqrt a))).const_mul (K : ℝ))
· exact Eventually.of_forall fun ω => (hf.continuous.comp
((continuous_id.smul continuous_const).add
(Real.continuous_sqrt.smul continuous_const))).continuousAt
theorem tendsto_integral_sqrt_observation_zero_lipschitz
{Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E]
{P : Measure Ω} [IsFiniteMeasure P] {S G : Ω → E}
(hS : Integrable S P) (hG : Integrable G P)
{f : E → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) :
Tendsto (fun t : ℝ => ∫ ω, f (t • S ω + Real.sqrt t • G ω) ∂P)
(𝓝 0) (𝓝 (P.real Set.univ * f 0)) := by
simpa only [zero_smul, Real.sqrt_zero, zero_add, integral_const, smul_eq_mul] using
(continuous_integral_sqrt_observation_lipschitz hS hG hf).tendsto 0
namespace FinitePosterior
variable {V Ω : Type*} [Fintype V] [MeasurableSpace Ω]
theorem hasDerivAt_integral_gaussianObservationCurve_of_integrable
(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))
(C : ℝ) (hC : 0 ≤ C) (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)
(hFint : Integrable (fun ω =>
f (gaussianObservationCurve (signal ω) (noise ω) t)) μ) :
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_of_integrable hsInt hnInt hdf hDcont
(mul_nonneg (by positivity) hC) hDbound ht hFint).2
simpa only [gaussianObservationCurve, gaussianObservationVelocity, D, sum_apply,
smul_apply, ContinuousLinearMap.proj_apply, smul_eq_mul] using hd
end FinitePosterior
end Problem367
end
end OAI
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