File size: 3,128 Bytes
f61e547 | 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 | import OAI.Probability.FreeIsing.Gaussian.IntegrationByParts
import OAI.Probability.FreeIsing.Posterior.SqrtObservationDerivative
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped NNReal Topology
namespace Problem367
def gaussianObservationMean (f : ℝ → ℝ) (signal t : ℝ) : ℝ :=
∫ z, f (t * signal + Real.sqrt t * z) ∂gaussianReal 0 1
theorem hasDerivAt_gaussianObservationMean_raw
{f f' : ℝ → ℝ} {C D : ℝ} (signal : ℝ) {t : ℝ} (ht : 0 < t)
(hf : ∀ x, HasDerivAt f (f' x) x)
(hbound : ∀ x, |f x| ≤ C) (hbound' : ∀ x, |f' x| ≤ D) :
HasDerivAt (gaussianObservationMean f signal)
(∫ z, f' (t * signal + Real.sqrt t * z) *
(signal + z / (2 * Real.sqrt t)) ∂gaussianReal 0 1) t := by
have hfm' : Measurable f' := by
have heq : f' = deriv f := funext (fun x => (hf x).deriv.symm)
rw [heq]
exact measurable_deriv f
exact (hasDerivAt_integral_sqrt_observation (integrable_const signal)
((memLp_id_gaussianReal 1).integrable (by norm_num)) hf hfm' hbound
((abs_nonneg (f' 0)).trans (hbound' 0)) hbound' ht).2
theorem hasDerivAt_gaussianObservationMean
{f f' f'' : ℝ → ℝ} {C D E : ℝ} (signal : ℝ) {t : ℝ} (ht : 0 < t)
(hf : ∀ x, HasDerivAt f (f' x) x) (hf' : ∀ x, HasDerivAt f' (f'' x) x)
(hbound : ∀ x, |f x| ≤ C) (hbound' : ∀ x, |f' x| ≤ D)
(hbound'' : ∀ x, |f'' x| ≤ E) :
HasDerivAt (gaussianObservationMean f signal)
(signal * (∫ z, f' (t * signal + Real.sqrt t * z) ∂gaussianReal 0 1) +
(1 / 2 : ℝ) * ∫ z, f'' (t * signal + Real.sqrt t * z) ∂gaussianReal 0 1) t := by
have hraw := hasDerivAt_gaussianObservationMean_raw signal ht hf hbound hbound'
have hf'c : Continuous f' := continuous_iff_continuousAt.mpr (fun x => (hf' x).continuousAt)
have hfm : Measurable (fun z => f' (t * signal + Real.sqrt t * z)) :=
hf'c.measurable.comp (by fun_prop)
have hb : ∀ᵐ z ∂gaussianReal 0 1,
‖f' (t * signal + Real.sqrt t * z)‖ ≤ D :=
Eventually.of_forall (fun z => by simpa only [Real.norm_eq_abs] using hbound' _)
have hi : Integrable (fun z => f' (t * signal + Real.sqrt t * z)) (gaussianReal 0 1) :=
Integrable.of_bound hfm.aestronglyMeasurable D hb
have hzi : Integrable (fun z => z * f' (t * signal + Real.sqrt t * z))
(gaussianReal 0 1) :=
((memLp_id_gaussianReal 1).integrable (by norm_num)).mul_bdd hfm.aestronglyMeasurable hb
have hibp := gaussian_integral_mul_affine_of_bounded_deriv
(t * signal) (Real.sqrt t) hf' hbound' hbound''
have hsplit :
(fun z => f' (t * signal + Real.sqrt t * z) * (signal + z / (2 * Real.sqrt t))) =
(fun z => signal * f' (t * signal + Real.sqrt t * z) +
(1 / (2 * Real.sqrt t)) * (z * f' (t * signal + Real.sqrt t * z))) := by
funext z
ring
rw [hsplit, integral_add (hi.const_mul signal) (hzi.const_mul _),
integral_const_mul, integral_const_mul, hibp] at hraw
convert hraw using 1
have hroot : Real.sqrt t ≠ 0 := (Real.sqrt_pos.mpr ht).ne'
field_simp
end Problem367
end
end OAI
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